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Zhan, Mingquan. "Eulerian subgraphs and Hamiltonicity of claw-free graphs." Morgantown, W. Va. : [West Virginia University Libraries], 2003. http://etd.wvu.edu/templates/showETD.cfm?recnum=3024.
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Thesis (Ph. D.)--West Virginia University, 2003.<br>Title from document title page. Document formatted into pages; contains vi, 52 p. : ill. Includes abstract. Includes bibliographical references (p. 50-52).
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Ghenciu, Petre Ion. "Hamiltonian cycles in subset and subspace graphs." Thesis, University of North Texas, 2004. https://digital.library.unt.edu/ark:/67531/metadc4662/.
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In this dissertation we study the Hamiltonicity and the uniform-Hamiltonicity of subset graphs, subspace graphs, and their associated bipartite graphs. In 1995 paper "The Subset-Subspace Analogy," Kung states the subspace version of a conjecture. The study of this problem led to a more general class of graphs. Inspired by Clark and Ismail's work in the 1996 paper "Binomial and Q-Binomial Coefficient Inequalities Related to the Hamiltonicity of the Kneser Graphs and their Q-Analogues," we defined subset graphs, subspace graphs, and their associated bipartite graphs. The main emphasis of this dissertation is to describe those graphs and study their Hamiltonicity. The results on subset graphs are presented in Chapter 3, on subset bipartite graphs in Chapter 4, and on subspace graphs and subspace bipartite graphs in Chapter 5. We conclude the dissertation by suggesting some generalizations of our results concerning the panciclicity of the graphs.
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Smithers, Dayna Brown. "Graph Theory for the Secondary School Classroom." Digital Commons @ East Tennessee State University, 2005. https://dc.etsu.edu/etd/1015.
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After recognizing the beauty and the utility of Graph Theory in solving a variety of problems, the author decided that it would be a good idea to make the subject available for students earlier in their educational experience. In this thesis, the author developed four units in Graph Theory, namely Vertex Coloring, Minimum Spanning Tree, Domination, and Hamiltonian Paths and Cycles, which are appropriate for high school level.
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Dey, Sanjoy. "Structural properties of visibility and weak visibility graphs." Virtual Press, 1997. http://liblink.bsu.edu/uhtbin/catkey/1048394.
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Given a finite set S of n nonintersecting line segments with no three end points collinear, the segment end point visibility graph is defined as the graph whose vertices are the end points of the line segments in S and two vertices are adjacent if the straight line segment joining two end points does not intersect any element of S, or if they are end points of the same segment. Segment end point visibility graphs have a wide variety of applications in VLSI circuit design, study of art gallery problems, and other areas of computational geometry. This thesis contains a survey of the important results that are currently known regarding the characterization of these graphs. Also a weak visibility dual of a segment end point visibility graph is defined and some structural properties of such graphs are presented. Some open problems and questions related to the characterization of weak visibility graphs are also discussed.<br>Department of Mathematical Sciences
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Sun, Qiang. "A contribution to the theory of (signed) graph homomorphism bound and Hamiltonicity." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLS109/document.
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Dans cette thèse, nous etudions deux principaux problèmes de la théorie des graphes: problème d’homomorphisme des graphes planaires (signés) et problème de cycle hamiltonien.Comme une extension du théorème des quatre couleurs, il est conjecturé([80], [41]) que chaque graphe signé cohérent planaire de déséquilibré-maille d+1(d>1) admet un homomorphisme à cube projective signé SPC(d) de dimension d. La question suivant étalés naturelle:Est-ce que SPC(d) une borne optimale de déséquilibré-maille d+1 pour tous les graphes signés cohérente planaire de déséquilibré-maille d+1?Au Chapitre 2, nous prouvons que: si (B,Ω) est un graphe signé cohérente dedéséquilibré-maille d qui borne la classe des graphes signés cohérents planaires de déséquilibré-maille d+1, puis |B| ≥2^{d−1}. Notre résultat montre que si la conjecture ci-dessus est vérifiée, alors le SPC(d) est une borne optimale à la fois en terme du nombre des sommets et du nombre de arêtes.Lorsque d=2k, le problème est équivalent aux problème des graphes:est-ce que PC(2k) une borne optimale de impair-maille 2k+1 pour P_{2k+1} (tous les graphes planaires de impair-maille au moins 2k+1)? Notez que les graphes K_4-mineur libres sont les graphes planaires, est PC(2k) aussi une borne optimale de impair-maille 2k+1 pour tous les graphes K_4-mineur libres de impair-maille 2k+1? La réponse est négative, dans[6], est donné une famille de graphes d’ordre O(k^2) que borne les graphes K_4-mineur libres de impair-maille 2k+1. Est-ce que la borne optimale? Au Chapitre 3, nous prouvons que: si B est un graphe de impair-maille 2k+1 qui borne tous les graphes K_4-mineur libres de impair-maille 2k+1, alors |B|≥(k+1)(k+2)/2. La conjonction de nos résultat et le résultat dans [6] montre que l’ordre O(k^2) est optimal. En outre, si PC(2k) borne P_{2k+1}, PC(2k) borne également P_{2r+1}(r>k).Cependant, dans ce cas, nous croyons qu’un sous-graphe propre de P(2k) serait suffisant à borner P_{2r+1}, alors quel est le sous-graphe optimal de PC2k) qui borne P_{2r+1}? Le premier cas non résolu est k=3 et r= 5. Dans ce cas, Naserasr [81] a conjecturé que le graphe Coxeter borne P_{11}. Au Chapitre 4, nous vérifions cette conjecture pour P_{17}.Au Chapitres 5, 6, nous étudions les problèmes du cycle hamiltonien. Dirac amontré en 1952 que chaque graphe d’ordre n est hamiltonien si tout sommet a un degré au moins n/2. Depuis, de nombreux résultats généralisant le théorème de Dirac sur les degré ont été obtenus. Une approche consiste à construire un cycle hamiltonien à partir d'un ensemble de sommets en contrôlant leur position sur le cycle. Dans cette thèse, nous considérons deux conjectures connexes. La première est la conjecture d'Enomoto: si G est un graphe d’ordre n≥3 et δ(G)≥n/2+1, pour toute paire de sommets x,y dans G, il y a un cycle hamiltonien C de G tel que dist_C(x,y)=n/2.Notez que l’ ́etat de degre de la conjecture de Enomoto est forte. Motivé par cette conjecture, il a prouvé, dans [32], qu’une paire de sommets peut être posé des distances pas plus de n/6 sur un cycle hamiltonien. Dans [33], les cas δ(G)≥(n+k)/2 sont considérés, il a prouvé qu’une paire de sommets à une distance entre 2 à k peut être posé sur un cycle hamiltonien. En outre, Faudree et Li ont proposé une conjecture plus générale: si G est un graphe d’ordre n≥3 et δ(G)≥n/2+1, pour toute paire de sommets x,y dans G et tout entier 2≤k≤n/2, il existe un cycle hamiltonien C de G tel que dist_C(x,y)=k. Utilisant de Regularity Lemma et Blow-up Lemma, au chapitre 5, nous donnons une preuve de la conjeture d'Enomoto conjecture pour les graphes suffisamment grand, et dans le chapitre 6, nous donnons une preuve de la conjecture de Faudree et Li pour les graphe suffisamment grand<br>In this thesis, we study two main problems in graph theory: homomorphism problem of planar (signed) graphs and Hamiltonian cycle problem.As an extension of the Four-Color Theorem, it is conjectured ([80],[41]) that every planar consistent signed graph of unbalanced-girth d+1(d>1) admits a homomorphism to signed projective cube SPC(d) of dimension d. It is naturally asked that:Is SPC(d) an optimal bound of unbalanced-girth d+1 for all planar consistent signed graphs of unbalanced-girth d+1?In Chapter 2, we prove that: if (B,Ω) is a consistent signed graph of unbalanced-girth d which bounds the class of consistent signed planar graphs of unbalanced-girth d, then |B|≥2^{d-1}. Furthermore,if no subgraph of (B,Ω) bounds the same class, δ(B)≥d, and therefore,|E(B)|≥d·2^{d-2}.Our result shows that if the conjecture above holds, then the SPC(d) is an optimal bound both in terms of number of vertices and number of edges.When d=2k, the problem is equivalent to the homomorphisms of graphs: isPC(2k) an optimal bound of odd-girth 2k+1 for P_{2k+1}(the class of all planar graphs of odd-girth at least 2k+1)? Note that K_4-minor free graphs are planar graphs, is PC(2k) also an optimal bound of odd-girth 2k+1 for all K_4-minor free graphs of odd-girth 2k+1 ? The answer is negative, in [6], a family of graphs of order O(k^2) bounding the K_4-minor free graphs of odd-girth 2k+1 were given. Is this an optimal bound? In Chapter 3, we prove that: if B is a graph of odd-girth 2k+1 which bounds all the K_4-minor free graphs of odd-girth 2k+1,then |B|≥(k+1)(k+2)/2. Our result together with the result in [6] shows that order O(k^2) is optimal.Furthermore, if PC(2k) bounds P_{2k+1},then PC(2k) also bounds P_{2r+1}(r>k). However, in this case we believe that a proper subgraph of PC(2k) would suffice to bound P_{2r+1}, then what’s the optimal subgraph of PC(2k) that bounds P_{2r+1}? The first case of this problem which is not studied is k=3 and r=5. For this case, Naserasr [81] conjectured that the Coxeter graph bounds P_{11} . Supporting this conjecture, in Chapter 4, we prove that the Coxeter graph bounds P_{17}.In Chapter 5,6, we study the Hamiltonian cycle problems. Dirac showed in 1952that every graph of order n is Hamiltonian if any vertex is of degree at least n/2. This result started a new approach to develop sufficient conditions on degrees for a graph to be Hamiltonian. Many results have been obtained in generalization of Dirac’s theorem. In the results to strengthen Dirac’s theorem, there is an interesting research area: to control the placement of a set of vertices on a Hamiltonian cycle such that thesevertices have some certain distances among them on the Hamiltonian cycle.In this thesis, we consider two related conjectures, one is given by Enomoto: if G is a graph of order n≥3, and δ(G)≥n/2+1, then for any pair of vertices x, y in G, there is a Hamiltonian cycle C of G such that dist_C(x, y)=n/2. Motivated by this conjecture, it is proved,in [32],that a pair of vertices are located at distances no more than n/6 on a Hamiltonian cycle. In [33], the cases δ(G) ≥(n+k)/2 are considered, it is proved that a pair of vertices can be located at any given distance from 2 to k on a Hamiltonian cycle. Moreover, Faudree and Li proposed a more general conjecture: if G is a graph of order n≥3, and δ(G)≥n/2+1, then for any pair of vertices x, y in G andany integer 2≤k≤n/2, there is a Hamiltonian cycle C of G such that dist_C(x, y) = k. Using Regularity Lemma and Blow-up Lemma, in Chapter 5, we give a proof ofEnomoto’s conjecture for graphs of sufficiently large order, and in Chapter 6, we give a proof of Faudree and Li’s conjecture for graphs of sufficiently large order
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Steelman, Andrea Elizabeth. "Degree sum ensuring hamiltonicity." [Pensacola, Fla.] : University of West Florida, 2007. http://purl.fcla.edu/fcla/etd/WFE0000012.
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Bajo, Calderon Erica. "An Exploration on the Hamiltonicity of Cayley Digraphs." Youngstown State University / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=ysu161982054497591.
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Streib, Noah Sametz. "Planar and hamiltonian cover graphs." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/43744.
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This dissertation has two principal components: the dimension of posets with planar cover graphs, and the cartesian product of posets whose cover graphs have hamiltonian cycles that parse into symmetric chains. Posets of height two can have arbitrarily large dimension. In 1981, Kelly provided an infinite sequence of planar posets that shows that the dimension of planar posets can also be arbitrarily large. However, the height of the posets in this sequence increases with the dimension. In 2009, Felsner, Li, and Trotter conjectured that for each integer h at least 2, there exists a least positive integer c(h) so that if P is a poset with a planar cover graph (the class of posets with planar cover graphs includes the class of planar posets) and the height of P is h, then the dimension of P is at most c(h). In the first principal component of this dissertation we prove this conjecture. We also give the best known lower bound for c(h), noting that this lower bound is far from the upper bound. In the second principal component, we consider posets with the Hamiltonian Cycle--Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. This definition is motivated by a proof of Sperner's theorem that uses symmetric chains, and was intended as a possible method of attack on the Middle Two Levels Conjecture. We show that the subset lattices have the HC-SCP property by showing that the class of posets with the strong HC-SCP property, a slight strengthening of the HC-SCP property, is closed under cartesian product with a two-element chain. Furthermore, we show that the cartesian product of any two posets from this strong class has the (weak) HC-SCP property.
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Fernandes, Antonio M. "A study of nonlinear physical systems in generalized phase space." Virtual Press, 1996. http://liblink.bsu.edu/uhtbin/catkey/1020161.
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Classical mechanics provides a phase space representation of mechanical systems in terms of position and momentum state variables. The Hamiltonian system, a set of partial differential equations, defines a vector field in phase space and uniquely determines the evolutionary process of the system given its initial state.A closed form solution describing system trajectories in phase space is only possible if the system of differential equations defining the Hamiltonian is linear. For nonlinear cases approximate and qualitative methods are required.Generalized phase space methods do not confine state variables to position and momentum, allowing other observables to describe the system. Such a generalization adjusts the description of the system to the required information and provides a method for studying physical systems that are not strictly mechanical.This thesis presents and uses the methods of generalized phase space to compare linear to nonlinear systems.Ball State UniversityMuncie, IN 47306<br>Department of Physics and Astronomy
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Ozkan, Sibel. "Hamilton decompositions of graphs with primitive complements." Auburn, Ala., 2007. http://repo.lib.auburn.edu/2007%20Spring%20Dissertations/OZKAN_SIBEL_27.pdf.
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Bruno, Nicholas J. "A Sufficient Condition for Hamiltonian Connectedness in Standard 2-Colored Multigraphs." Miami University / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=miami1438385443.
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Yang, Weihua. "Supereulerian graphs, Hamiltonicity of graphes and several extremal problems in graphs." Phd thesis, Université Paris Sud - Paris XI, 2013. http://tel.archives-ouvertes.fr/tel-00877793.
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In this thesis, we focus on the following topics: supereulerian graphs, hamiltonian line graphs, fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees, and several extremal problems on the (minimum and/or maximum) size of graphs under a given graph property. The thesis includes six chapters. The first one is to introduce definitions and summary the main results of the thesis, and in the last chapter we introduce the furture research of the thesis. The main studies in Chapters 2 - 5 are as follows. In Chapter 2, we explore conditions for a graph to be supereulerian.In Section 1 of Chapter 2, we characterize the graphs with minimum degree at least 2 and matching number at most 3. By using the characterization, we strengthen the result in [93] and we also address a conjecture in the paper.In Section 2 of Chapter 2, we prove that if $d(x)+d(y)\geq n-1-p(n)$ for any edge $xy\in E(G)$, then $G$ is collapsible except for several special graphs, where $p(n)=0$ for $n$ even and $p(n)=1$ for $n$ odd. As a corollary, a characterization for graphs satisfying $d(x)+d(y)\geq n-1-p(n)$ for any edge $xy\in E(G)$ to be supereulerian is obtained. This result extends the result in [21].In Section 3 of Chapter 2, we focus on a conjecture posed by Chen and Lai [Conjecture~8.6 of [33]] that every 3-edge connected and essentially 6-edge connected graph is collapsible. We find a kind of sufficient conditions for a 3-edge connected graph to be collapsible.In Chapter 3, we mainly consider the hamiltonicity of 3-connected line graphs.In the first section of Chapter 3, we give several conditions for a line graph to be hamiltonian, especially we show that every 3-connected, essentially 11-connected line graph is hamilton- connected which strengthens the result in [91].In the second section of Chapter 3, we show that every 3-connected, essentially 10-connected line graph is hamiltonian-connected.In the third section of Chapter 3, we show that 3-connected, essentially 4-connected line graph of a graph with at most 9 vertices of degree 3 is hamiltonian. Moreover, if $G$ has 10 vertices of degree 3 and its line graph is not hamiltonian, then $G$ can be contractible to the Petersen graph.In Chapter 4, we consider edge fault-tolerant hamiltonicity of Cayley graphs generated by transposition trees. We first show that for any $F\subseteq E(Cay(B:S_{n}))$, if $|F|\leq n-3$ and $n\geq4$, then there exists a hamiltonian path in $Cay(B:S_{n})-F$ between every pair of vertices which are in different partite sets. Furthermore, we strengthen the above result in the second section by showing that $Cay(S_n,B)-F$ is bipancyclic if $Cay(S_n,B)$ is not a star graph, $n\geq4$ and $|F|\leq n-3$.In Chapter 5, we consider several extremal problems on the size of graphs.In Section 1 of Chapter 5, we bounds the size of the subgraph induced by $m$ vertices of hypercubes. We show that a subgraph induced by $m$ (denote $m$ by $\sum\limits_{i=0}^ {s}2^{t_i}$, $t_0=[\log_2m]$ and $t_i= [\log_2({m-\sum\limits_{r=0}^{i-1}2 ^{t_r}})]$ for $i\geq1$) vertices of an $n$-cube (hypercube) has at most $\sum\limits_{i=0}^{s}t_i2^{t_i-1} +\sum\limits_{i=0}^{s} i\cdot2^{t_i}$ edges. As its applications, we determine the $m$-extra edge-connectivity of hypercubes for $m\leq2^{[\frac{n}2]}$ and $g$-extra edge-connectivity of the folded hypercube for $g\leq n$.In Section 2 of Chapter 5, we partially study the minimum size of graphs with a given minimum degree and a given edge degree. As an application, we characterize some kinds of minimumrestricted edge connected graphs.In Section 3 of Chapter 5, we consider the minimum size of graphs satisfying Ore-condition.
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Costa, Polyanna Possani da [UNESP]. "Teoria dos grafos e suas aplicações." Universidade Estadual Paulista (UNESP), 2011. http://hdl.handle.net/11449/94358.
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costa_pp_me_rcla.pdf: 598986 bytes, checksum: 67c1c7e0c368ded41dbfb9631bdf1362 (MD5)<br>Neste trabalho estudamos a Teoria de Grafos e a aplicamos na solução de alguns problemas clássicos, como por exemplo O Problema das Pontes de Königsberg, O Problema do Caixeiro Viajante, Classificação dos Poliedros Regulares e Coloração de Mapas. As ferramentas básicas foram Topologia Geral e Álgebra<br>In this work we study Graph Theory and we apply it in the solution of some classical problems, for example Königsberg Bridges Problem, Travelling Salesman Problem, Classification of Regular Polyhedra and Map Coloring. The prerequisites are General Topology and Algebra
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Costa, Polyanna Possani da. "Teoria dos grafos e suas aplicações /." Rio Claro : [s.n.], 2011. http://hdl.handle.net/11449/94358.
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Orientador: Thiago de Melo<br>Banca: Elíris Cristina Rizziolli<br>Banca: Luiz Roberto Hartmann Junior<br>Resumo: Neste trabalho estudamos a Teoria de Grafos e a aplicamos na solução de alguns problemas clássicos, como por exemplo O Problema das Pontes de Königsberg, O Problema do Caixeiro Viajante, Classificação dos Poliedros Regulares e Coloração de Mapas. As ferramentas básicas foram Topologia Geral e Álgebra<br>Abstract: In this work we study Graph Theory and we apply it in the solution of some classical problems, for example Königsberg Bridges Problem, Travelling Salesman Problem, Classification of Regular Polyhedra and Map Coloring. The prerequisites are General Topology and Algebra<br>Mestre
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Johnson, Chase R. "Molecular Graph Theory." Digital WPI, 2010. https://digitalcommons.wpi.edu/etd-theses/1179.
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Graph Theory is a branch of mathematics that has a wealth of applications to other science and engineering disciplines, specifically Chemistry. The primary application of graphs to Chemistry is related to understanding of structure and symmetry at the molecular level. By projecting a molecule to the plane and examining it as a graph, a lot can be learned about the underlying molecular structure of a given compound. Using concepts of Graph Theory this masters project examines the underlying structures of two specific families of compounds, fullerenes and zeolites, from a chemical and mathematical perspective.
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Feghali, Carl. "Topics in graph colouring and extremal graph theory." Thesis, Durham University, 2016. http://etheses.dur.ac.uk/11790/.
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In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let $G$ be a connected graph with $n$ vertices and maximum degree $\Delta(G)$. Let $R_k(G)$ denote the graph with vertex set all proper $k$-colourings of $G$ and two $k$-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that $R_{\Delta(G)+1}(G)$ has a unique non-trivial component with diameter $O(n^2)$. This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours $a$, $b$ of a component of the subgraph induced by vertices with colour $a$ or $b$. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all $\Delta(G)$-colourings of a graph $G$ are Kempe equivalent unless $G$ is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees.
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Nikwigize, Adolphe. "Graph theory : Route problems." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-17397.
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Berg, Deborah. "Connections Between Voting Theory and Graph Theory." Scholarship @ Claremont, 2005. https://scholarship.claremont.edu/hmc_theses/178.
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Mathematical concepts have aided the progression of many different fields of study. Math is not only helpful in science and engineering, but also in the humanities and social sciences. Therefore, it seemed quite natural to apply my preliminary work with set intersections to voting theory, and that application has helped to focus my thesis. Rather than studying set intersections in general, I am attempting to study set intersections and what they mean in a voting situation. This can lead to better ways to model preferences and to predict which campaign platforms will be most popular. Because I feel that allowing people to only vote for one candidate results in a loss of too much information, I consider approval voting, where people can vote for as many platforms as they like.
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Keevash, Peter. "Topics in extremal graph theory." Thesis, University of Cambridge, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.619938.
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Law, Ka-ho, and 羅家豪. "Some results in graph theory." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2010. http://hub.hku.hk/bib/B44899816.
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Morrison, Julie Lindsay. "Computational graph theory in bioinformatics." Thesis, University of Strathclyde, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.435114.
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Richer, Duncan Christopher. "Graph theory and combinatorial games." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.621916.
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Nieh, Ari. "Fractional Analogues in Graph Theory." Scholarship @ Claremont, 2001. https://scholarship.claremont.edu/hmc_theses/131.
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Tait showed in 1878 that the Four Color Theorem is equivalent to being able to three-color the edges of any planar, three-regular, two-edge connected graph. Not surprisingly, this equivalent problem proved to be equally difficult. We consider the problem of fractional colorings, which resemble ordinary colorings but allow for some degree of cheating. Happily, it is known that every planar three-regular, two-edge connected graph is fractionally three-edge colorable. Is there an analogue to Tait’s Theorem which would allow us to derive the Fractional Four Color Theorem from this edge-coloring result?
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Eggemann, Nicole. "Some applications of graph theory." Thesis, Brunel University, 2009. http://bura.brunel.ac.uk/handle/2438/3953.
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Hatt, Justin Dale. "Online assessment of graph theory." Thesis, Brunel University, 2016. http://bura.brunel.ac.uk/handle/2438/13389.
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The objective of this thesis is to establish whether or not online, objective questions in elementary graph theory can be written in a way that exploits the medium of computer-aided assessment. This required the identification and resolution of question design and programming issues. The resulting questions were trialled to give an extensive set of answer files which were analysed to identify whether computer delivery affected the questions in any adverse ways and, if so, to identify practical ways round these issues. A library of questions spanning commonly-taught topics in elementary graph theory has been designed, programmed and added to the graph theory topic within an online assessment and learning tool used at Brunel University called Mathletics. Distracters coded into the questions are based on errors students are likely to make, partially evidenced by final examination scripts. Questions were provided to students in Discrete Mathematics modules with an extensive collection of results compiled for analysis. Questions designed for use in practice environments were trialled on students from 2007 – 2008 and then from 2008 to 2014 inclusive under separate testing conditions. Particular focus is made on the relationship of facility and discrimination between comparable questions during this period. Data is grouped between topic and also year group for the 2008 – 2014 tests, namely 2008 to 2011 and 2011 to 2014, so that it may then be determined what factors, if any, had an effect on the overall results for these questions. Based on the analyses performed, it may be concluded that although CAA questions provide students with a means for improving their learning in this field of mathematics, what makes a question more challenging is not solely based on the number of ways a student can work out his/her solution but also on several other factors that depend on the topic itself.
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Letzter, Shoham. "Extremal graph theory with emphasis on Ramsey theory." Thesis, University of Cambridge, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.709415.
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Knight, Christopher J. "Hydrogen bond topology order/disorder transitions in ice and the behavior of defects in a disordered ice lattice /." Columbus, Ohio : Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1236788109.
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Bessy, Stéphane. "Some problems in graph theory and graphs algorithmic theory." Habilitation à diriger des recherches, Université Montpellier II - Sciences et Techniques du Languedoc, 2012. http://tel.archives-ouvertes.fr/tel-00806716.
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This document is a long abstract of my research work, concerning graph theory and algorithms on graphs. It summarizes some results, gives ideas of the proof for some of them and presents the context of the different topics together with some interesting open questions connected to them The first part precises the notations used in the rest of the paper; the second part deals with some problems on cycles in digraphs; the third part is an overview of two graph coloring problems and one problem on structures in colored graphs; finally the fourth part focus on some results in algorithmic graph theory, mainly in parametrized complexity.
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Anderson, Jon K. "Genetic algorithms applied to graph theory." Virtual Press, 1999. http://liblink.bsu.edu/uhtbin/catkey/1136714.
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This thesis proposes two new variations on the genetic algorithm. The first attempts to improve clustering problems by optimizing the structure of a genetic string dynamically during the run of the algorithm. This is done by using a permutation on the allele which is inherited by the next generation. The second is a multiple pool technique which ensures continuing convergence by maintaining unique lineages and merging pools of similar age. These variations will be tested against two well-known graph theory problems, the Traveling Salesman Problem and the Maximum Clique Problem. The results will be analyzed with respect to string rates, child improvement, pool rating resolution, and average string age.<br>Department of Computer Science
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Peng, Richard. "Algorithm Design Using Spectral Graph Theory." Research Showcase @ CMU, 2013. http://repository.cmu.edu/dissertations/277.
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Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. Laplace’s equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial optimization, computer vision, computer graphics, and machine learning.
In this thesis, we develop highly efficient and parallelizable algorithms for solving linear systems involving graph Laplacian matrices. These solvers can also be extended to symmetric diagonally dominant matrices and M-matrices, both of which are closely related to graph Laplacians. Our algorithms build upon two decades of progress on combinatorial preconditioning, which connects numerical and combinatorial algorithms through spectral graph theory. They in turn rely on tools from numerical analysis, metric embeddings, and random matrix theory.
We give two solver algorithms that take diametrically opposite approaches. The first is motivated by combinatorial algorithms, and aims to gradually break the problem into several smaller ones. It represents major simplifications over previous solver constructions, and has theoretical running time comparable to sorting. The second is motivated by numerical analysis, and aims to rapidly improve the algebraic connectivity of the graph. It is the first highly efficient solver for Laplacian linear systems that parallelizes almost completely.
Our results improve the performances of applications of fast linear system solvers ranging from scientific computing to algorithmic graph theory. We also show that these solvers can be used to address broad classes of image processing tasks, and give some preliminary experimental results.
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Islam, Mustafa R. "A hypertext graph theory reference system." Virtual Press, 1993. http://liblink.bsu.edu/uhtbin/catkey/879844.
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G-Net system is being developed by the members of the G-Net research group under the supervision of Dr. K. Jay Bagga. The principle objective of the G-Net system is to provide an integrated tool for dealing with various aspects of graph theory. G-Net system is divided into two parts. GETS (Graph theory Experiments Tool Set) will provide a set of tools to experiment with graph theory, and HYGRES (HYpertext Graph theory Reference Service), the second subcomponent of the G-Net system to aid graph theory study and research. In this research a hypertext application is built to present the graph theory concepts, graph models and the algorithms. In other words, HYGRES (Guide Version) provides the hypertext facilities for organizing a graph theory database in a very natural and interactive way. An hypertext application development tool, called Guide, is used to implement this version of HYGRES. This project integrates the existing version of GETS so that it can also provide important services to HYGRES. The motivation behind this project is to study the initial criterion for developing a hypertext system, which can be used for future development of a stand alone version of the G-Net system.<br>Department of Computer Science
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Edwards, C. S. "Some extremal problems in graph theory." Thesis, University of Reading, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.373467.
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Florkowski, Stanley F. "Spectral graph theory of the Hypercube." Thesis, Monterey, Calif. : Naval Postgraduate School, 2008. http://edocs.nps.edu/npspubs/scholarly/theses/2008/Dec/08Dec%5FFlorkowski.pdf.
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Thesis (M.S. in Applied Mathematics)--Naval Postgraduate School, December 2008.<br>Thesis Advisor(s): Rasmussen, Craig W. "December 2008." Description based on title screen as viewed on January 29, 2009. Includes bibliographical references (p. 51-52). Also available in print.
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Robinson, Laura Ann. "Graph Theory for the Middle School." Digital Commons @ East Tennessee State University, 2006. https://dc.etsu.edu/etd/2226.
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After being introduced to graph theory and realizing how it can be utilized to solve real-world problems, the author decided to create modules of study on graph theory appropriate for middle school students. In this thesis, four modules were developed in the area of graph theory: an Introduction to Terms and Definitions, Graph Families, Graph Operations, and Graph Coloring. It is written as a guide for middle school teachers to prepare teaching units on graph theory.
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Garbe, Frederik. "Extremal graph theory via structural analysis." Thesis, University of Birmingham, 2018. http://etheses.bham.ac.uk//id/eprint/8869/.
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We discuss two extremal problems in extremal graph theory. First we establish a precise characterisation of 4-uniform hypergraphs with minimum codegree close to n/2 which contain a Hamilton 2-cycle. As a corollary we determine the exact Dirac threshold for Hamilton 2-cycles in 4-uniform hypergraphs, and we provide a polynomial-time algorithm which answers the corresponding decision problem for 4-graphs with minimum degree close to n/2. In contrast we also show that the corresponding decision problem for tight Hamilton cycles in dense k-graphs is NP-complete. Furthermore we study the following bootstrap percolation process: given a connected graph G, we infect an initial set A of vertices, and in each step a vertex v becomes infected if at least a p-proportion of its neighbours are infected. A set A which infects the whole graph is called a contagious set. Our main result states that for every pin (0,1] and for every connected graph G on n vertices the minimal size of a contagious set is less than 2pn or 1. This result is best-possible, but we provide a stronger bound in the case of graphs of girth at least five. Both proofs exploit the structure of a minimal counterexample.
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Grinshpun, Andrey Vadim. "Some problems in Graph Ramsey Theory." Thesis, Massachusetts Institute of Technology, 2015. http://hdl.handle.net/1721.1/97767.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.<br>This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.<br>Cataloged from student-submitted PDF version of thesis.<br>Includes bibliographical references (pages 149-156).<br>A graph G is r-Ramsey minimal with respect to a graph H if every r-coloring of the edges of G yields a monochromatic copy of H, but the same is not true for any proper subgraph of G. The study of the properties of graphs that are Ramsey minimal with respect to some H and similar problems is known as graph Ramsey theory; we study several problems in this area. Burr, Erdös, and Lovász introduced s(H), the minimum over all G that are 2- Ramsey minimal for H of [delta](G), the minimum degree of G. We find the values of s(H) for several classes of graphs H, most notably for all 3-connected bipartite graphs which proves many cases of a conjecture due to Szabó, Zumstein, and Zürcher. One natural question when studying graph Ramsey theory is what happens when, rather than considering all 2-colorings of a graph G, we restrict to a subset of the possible 2-colorings. Erdös and Hajnal conjectured that, for any fixed color pattern C, there is some [epsilon] > 0 so that every 2-coloring of the edges of a Kn, the complete graph on n vertices, which doesn't contain a copy of C contains a monochromatic clique on n[epsilon] vertices. Hajnal generalized this conjecture to more than 2 colors and asked in particular about the case when the number of colors is 3 and C is a rainbow triangle (a K3 where each edge is a different color); we prove Hajnal's conjecture for rainbow triangles. One may also wonder what would happen if we wish to cover all of the vertices with monochromatic copies of graphs. Let F = {F₁, F₂, . . .} be a sequence of graphs such that Fn is a graph on n vertices with maximum degree at most [delta]. If each Fn is bipartite, then the vertices of any 2-edge-colored complete graph can be partitioned into at most 2C[delta] vertex disjoint monochromatic copies of graphs from F, where C is an absolute constant. This result is best possible, up to the constant C.<br>by Andrey Vadim Grinshpun.<br>Ph. D.
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Parks, David J. "Graph theory in America, 1876-1950." Thesis, Open University, 2012. http://oro.open.ac.uk/54663/.
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This narrative is a history of the contributions made to graph theory in the United States of America by American mathematicians and others who supported the growth of scholarship in that country, between the years 1876 and 1950. The beginning of this period coincided with the opening of the first research university in the United States of America, The Johns Hopkins University (although undergraduates were also taught), providing the facilities and impetus for the development of new ideas. The hiring, from England, of one of the foremost mathematicians of the time provided the necessary motivation for research and development for a new generation of American scholars. In addition, it was at this time that home-grown research mathematicians were first coming to prominence. At the beginning of the twentieth century European interest in graph theory, and to some extent the four-colour problem, began to wane. Over three decades, American mathematicians took up this field of study - notably, Oswald Veblen, George Birkhoff, Philip Franklin, and Hassler Whitney. It is necessary to stress that these four mathematicians and all the other scholars mentioned in this history were not just graph theorists but worked in many other disciplines. Indeed, they not only made significant contributions to diverse fields but, in some cases, they created those fields themselves and set the standards for others to follow. Moreover, whilst they made considerable contributions to graph theory in general, two of them developed important ideas in connection with the four-colour problem. Grounded in a paper by Alfred Bray Kempe that was notorious for its fallacious 'proof' of the four-colour theorem, these ideas were the concepts of an unavoidable set and a reducible configuration. To place the story of these scholars within the history of mathematics, America, and graph theory, brief accounts are presented of the early years of graph theory, the early years of mathematics and graph theory in the USA, and the effects of the founding of the first institute for postgraduate study in America. Additionally, information has been included on other influences by such global events as the two world wars, the depression, the influx of European scholars into the United States of America, mainly during the 1930s, and the parallel development of graph theory in Europe. Until the end of the nineteenth century, graph theory had been almost entirely the prerogative of European mathematicians. Perhaps the first work in graph theory carried out in America was by Charles Sanders Peirce, arguably America's greatest logician and philosopher at the time. In the 1860s, he studied the four-colour conjecture and claimed to have written at least two papers on the subject during that decade, but unfortunately neither of these has survived. William Edward Story entered the field in 1879, with unfortunate consequences, but it was not until 1897 that an American mathematician presented a lecture on the subject, albeit only to have the paper disappear. Paul Wernicke presented a lecture on the four-colour problem to the American Mathematician Society, but again the paper has not survived. However, his 1904 paper has survived and added to the story of graph theory, and particularly the four-colour conjecture. The year 1912 saw the real beginning of American graph theory with Veblen and Birkhoff publishing major contributions to the subject. It was around this time that European mathematicians appeared to lose interest in graph theory. In the period 1912 to 1950 much of the progress made in the subject was from America and by 1950 not only had the United States of America become the foremost country for mathematics, it was the leading centre for graph theory.
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Loveland, Susan M. "The Reconstruction Conjecture in Graph Theory." DigitalCommons@USU, 1985. https://digitalcommons.usu.edu/etd/7022.
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In this paper we show that specific classes of graphs are reconstructible; we explore the relationship between the. reconstruction and edge-reconstruction conjectures; we prove that several classes of graphs are actually Harary to the reconstructible; and we give counterexamples reconstruction and edge-reconstruction conjectures for infinite graphs.
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Schuerger, Houston S. "Contributions to Geometry and Graph Theory." Thesis, University of North Texas, 2020. https://digital.library.unt.edu/ark:/67531/metadc1707341/.
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In geometry we will consider n-dimensional generalizations of the Power of a Point Theorem and of Pascal's Hexagon Theorem. In generalizing the Power of a Point Theorem, we will consider collections of cones determined by the intersections of an (n-1)-sphere and a pair of hyperplanes. We will then use these constructions to produce an n-dimensional generalization of Pascal's Hexagon Theorem, a classical plane geometry result which states that "Given a hexagon inscribed in a conic section, the three pairs of continuations of opposite sides meet on a straight line." Our generalization of this theorem will consider a pair of n-simplices intersecting an (n-1)-sphere, and will conclude with the intersections of corresponding faces lying in a hyperplane. In graph theory we will explore the interaction between zero forcing and cut-sets. The color change rule which lies at the center of zero forcing says "Suppose that each of the vertices of a graph are colored either blue or white. If u is a blue vertex and v is its only white neighbor, then u can force v to change to blue." The concept of zero forcing was introduced by the AIM Minimum Rank - Special Graphs Work Group in 2007 as a way of determining bounds on the minimum rank of graphs. Later, Darren Row established results concerning the zero forcing numbers of graphs with a cut-vertex. We will extend his work by considering graphs with arbitrarily large cut-sets, and the collections of components they yield, to determine results for the zero forcing numbers of these graphs.
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Weaver, Robert Wooddell. "Some problems in structural graph theory /." The Ohio State University, 1986. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487268021746449.
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Han, Lin. "Graph generative models from information theory." Thesis, University of York, 2012. http://etheses.whiterose.ac.uk/3726/.
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Generative models are commonly used in statistical pattern recognition to describe the probability distributions of patterns in a vector space. In recent years, sustained by the wide range of mathematical tools available in vector space, many algorithms for constructing generative models have been developed. Compared with the advanced development of the generative model for vectors, the development of a generative model for graphs has had less progress. In this thesis, we aim to solve the problem of constructing the generative model for graphs using information theory. Given a set of sample graphs, the generative model for the graphs we aim to construct should be able to not only capture the structural variation of the sample graphs, but to also allow new graphs which share similar properties with the original graphs to be generated. In this thesis, we pose the problem of constructing a generative model for graphs as that of constructing a supergraph structure for the graphs. In Chapter 3, we describe a method of constructing a supergraph-based generative model given a set of sample graphs. By adopting the a posteriori probability developed in a graph matching problem, we obtain a probabilistic framework which measures the likelihood of the sample graphs, given the structure of the supergraph and the correspondence information between the nodes of the sample graphs and those of the supergraph. The supergraph we aim to obtain is one which maximizes the likelihood of the sample graphs. The supergraph is represented here by its adjacency matrix, and we develop a variant of the EM algorithm to locate the adjacency matrix that maximizes the likelihood of the sample graphs. Experimental evaluations demonstrate that the constructed supergraph performs well on classifying graphs. In Chapter 4, we aim to develop graph characterizations that can be used to measure the complexity of graphs. The first graph characterization developed is the von Neumann entropy of a graph associated with its normalized Laplacian matrix. This graph characterization is defined by the eigenvalues of the normalized Laplacian matrix, therefore it is a member of the graph invariant characterization. By applying some transformations, we also develop a simplified form of the von Neumann entropy, which can be expressed in terms of the node degree statistics of the graphs. Experimental results reveal that effectiveness of the two graph characterizations. Our third contribution is presented in Chapter 5, where we use the graph characterization developed in Chapter 4 to measure the supergraph complexity and we develop a novel framework for learning a supergraph using the minimum description length criterion. We combine the Jensen-Shanon kernel with our supergraph construction and this provides us with a way of measuring graph similarity. Moreover, we also develop a method of sampling new graphs from the supergraph. The supergraph we present in this chapter is a generative model which can fulfil the tasks of graph classification, graph clustering, and of generating new graphs. We experiment with both the COIL and “Toy” datasets to illustrate the utility of our generative model. Finally, in Chapter 6, we propose a method of selecting prototype graphs of the most appropriate size from candidate prototypes. The method works by partitioning the sample graphs into two parts and approximating their hypothesis space using the partition functions. From the partition functions, the mutual information between the two sets is defined. The prototype which gives the highest mutual information is selected.
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Pappone, Francesco. "Graph neural networks: theory and applications." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23893/.
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Le reti neurali artificiali hanno visto, negli ultimi anni, una crescita vertiginosa nelle loro applicazioni e nelle architetture dei modelli impiegati. In questa tesi introduciamo le reti neurali su domini euclidei, in particolare mostrando l’importanza dell’equivarianza di traslazione nelle reti convoluzionali, e introduciamo, per analogia, un’estensione della convoluzione a dati strutturati come grafi. Inoltre presentiamo le architetture dei principali Graph Neural Network ed esponiamo, per ognuna delle tre architetture proposte (Spectral graph Convolutional Network, Graph Convolutional Network, Graph Attention neTwork) un’applicazione che ne mostri sia il funzionamento che l’importanza. Discutiamo, ulteriormente, l’implementazione di un algoritmo di classificazione basato su due varianti dell’architettura Graph Convolutional Network, addestrato e testato sul dataset PROTEINS, capace di classificare le proteine del dataset in due categorie: enzimi e non enzimi.
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Yi, Peipei. "Graph query autocompletion." HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/557.
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The prevalence of graph-structured data in modern real-world applications has led to a rejuvenation of research on graph data management and analytics. Several database query languages have been proposed for textually querying graph databases. Unfortunately, formulating a graph query using any of these query languages often demands considerable cognitive effort and requires "programming" skill at least similar to programming in SQL. Yet, in a wide spectrum of graph applications consumers need to query graph data but are not proficient query writers. Hence, it is important to devise intuitive techniques that can alleviate the burden of query formulation and thus increase the usability of graph databases. In this dissertation, we take the first step to study the graph query autocompletion problem. We provide techniques that take a user's graph query as input and generate top-k query suggestions as output, to help to alleviate the verbose and error-prone graph query formulation process in a visual environment. Firstly, we study visual query autocompletion for graph databases. Techniques for query autocompletion have been proposed for web search and XML search. However, a corresponding capability for graph query engine is in its infancy. We propose a novel framework for graph query autocompletion (called AutoG). The novelties of AutoG are as follows: First, we formalize query composition that specifies how query suggestions are formed. Second, we propose to increment a query with the logical units called c-prime features, that are (i) frequent subgraphs and (ii) constructed from smaller c-prime features in no more than c ways. Third, we propose algorithms to rank candidate suggestions. Fourth, we propose a novel index called feature DAG (FDAG) to further optimize the ranking. Secondly, we propose user focus-based graph query autocompletion. AutoG provides suggestions that are formed by adding subgraph increments to arbitrary places of an existing user query. However, humans can only interact with a small number of recent software artifacts in hand. Hence, many such suggestions could be irrelevant. We present the GFocus framework that exploits a novel notion of user focus of graph query formulation. Intuitively, the focus is the subgraph that a user is working on. We formulate locality principles to automatically identify and maintain the focus. We propose novel monotone submodular ranking functions for generating popular and comprehensive query suggestions only at the focus. We propose efficient algorithms and an index for ranking the suggestions. Thirdly, we propose graph query autocompletion for large graphs. Graph features that have been exploited in AutoG are either absent or rare in large graphs. To address this, we present Flexible graph query autocompletion for LArge Graphs, called FLAG. We propose wildcard label for query graph and query suggestions. In particular, FLAG allows augmenting users' queries using subgraph increments with wildcard labels, which summarize query suggestions that have similar increment structures but different labels. We propose an efficient ranking algorithm and a novel index, called Suggestion Summarization DAG (SSDAG), to optimize the online suggestion ranking. Detailed problem analysis and extensive experimental studies consistently demonstrate the effectiveness and robustness of our proposed techniques in a broad range of settings.
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Hegde, Rajneesh. "New Tools and Results in Graph Structure Theory." Diss., Georgia Institute of Technology, 2006. http://hdl.handle.net/1853/10481.
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We first prove a ``non-embeddable extensions' theorem for polyhedral graph embeddings. Let G be a ``weakly 4-connected' planar graph. We describe a set of constructions that produce a finite list of non-planar graphs, each having a minor isomorphic to G, such that every non-planar weakly 4-connected graph H that has a minor isomorphic to G has a minor isomorphic to one of the graphs in the list. The theorem is more general and applies in particular to polyhedral embeddings in any surface.
We discuss an approach to proving Jorgensen's conjecture, which states that if G is a 6-connected graph with no K_6 minor, then it is apex, that is, it has a vertex v such that deleting v yields a planar graph. We relax the condition of 6-connectivity, and prove Jorgensen's conjecture for a certain sub-class of these graphs.
We prove that every graph embedded in the Klein bottle with representativity at least 4 has a K_6 minor. Also, we prove that every ``locally 5-connected' triangulation of the torus, with one exception, has a K_6 minor. (Local 5-connectivity is a natural notion of local connectivity for a surface embedding.) The above theorem uses a locally 5-connected version of the well-known splitter theorem for triangulations of any surface.
We conclude with a theoretically optimal algorithm for the following graph connectivity problem. A shredder in an undirected graph is a set of vertices whose removal results in at least three components. A 3-shredder is a shredder of size three. We present an algorithm that, given a 3-connected graph, finds its 3-shredders in time proportional to the number of vertices and edges, when implemented on a RAM (random access machine).
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Waterhouse, Mary Alexandra Paula Royston Hastilow. "Coloured graph decompositions /." [St. Lucia, Qld.], 2005. http://www.library.uq.edu.au/pdfserve.php?image=thesisabs/absthe18769.pdf.
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Burns, Jonathan. "Recursive Methods in Number Theory, Combinatorial Graph Theory, and Probability." Scholar Commons, 2014. https://scholarcommons.usf.edu/etd/5193.
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Recursion is a fundamental tool of mathematics used to define, construct, and analyze mathematical objects. This work employs induction, sieving, inversion, and other recursive methods to solve a variety of problems in the areas of algebraic number theory, topological and combinatorial graph theory, and analytic probability and statistics. A common theme of recursively defined functions, weighted sums, and cross-referencing sequences arises in all three contexts, and supplemented by sieving methods, generating functions, asymptotics, and heuristic algorithms.
In the area of number theory, this work generalizes the sieve of Eratosthenes to a sequence of polynomial values called polynomial-value sieving. In the case of quadratics, the method of polynomial-value sieving may be characterized briefly as a product presentation of two binary quadratic forms. Polynomials for which the polynomial-value sieving yields all possible integer factorizations of the polynomial values are called recursively-factorable. The Euler and Legendre prime producing polynomials of the form n2+n+p and 2n2+p, respectively, and Landau's n2+1 are shown to be recursively-factorable. Integer factorizations realized by the polynomial-value sieving method, applied to quadratic functions, are in direct correspondence with the lattice point solutions (X,Y) of the conic sections aX2+bXY +cY2+X-nY=0. The factorization structure of the underlying quadratic polynomial is shown to have geometric properties in the space of the associated lattice point solutions of these conic sections.
In the area of combinatorial graph theory, this work considers two topological structures that are used to model the process of homologous genetic recombination: assembly graphs and chord diagrams. The result of a homologous recombination can be recorded as a sequence of signed permutations called a micronuclear arrangement. In the assembly graph model, each micronuclear arrangement corresponds to a directed Hamiltonian polygonal path within a directed assembly graph. Starting from a given assembly graph, we construct all the associated micronuclear arrangements. Another way of modeling genetic rearrangement is to represent precursor and product genes as a sequence of blocks which form arcs of a circle. Associating matching blocks in the precursor and product gene with chords produces a chord diagram. The braid index of a chord diagram can be used to measure the scope of interaction between the crossings of the chords. We augment the brute force algorithm for computing the braid index to utilize a divide and conquer strategy. Both assembly graphs and chord diagrams are closely associated with double occurrence words, so we classify and enumerate the double occurrence words based on several notions of irreducibility. In the area of analytic probability, moments abstractly describe the shape of a probability distribution. Over the years, numerous varieties of moments such as central moments, factorial moments, and cumulants have been developed to assist in statistical analysis. We use inversion formulas to compute high order moments of various types for common probability distributions, and show how the successive ratios of moments can be used for distribution and parameter fitting. We consider examples for both simulated binomial data and the probability distribution affiliated with the braid index counting sequence. Finally we consider a sequence of multiparameter binomial sums which shares similar properties with the moment sequences generated by the binomial and beta-binomial distributions. This sequence of sums behaves asymptotically like the high order moments of the beta distribution, and has completely monotonic properties.
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Fiala, Nick C. "Some topics in combinatorial design theory and algebraic graph theory /." The Ohio State University, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=osu1486402957198077.
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Narayanan, Bhargav. "Problems in Ramsey theory, probabilistic combinatorics and extremal graph theory." Thesis, University of Cambridge, 2015. https://www.repository.cam.ac.uk/handle/1810/252850.
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Srikanthan, T. "Bond graph analysis." Thesis, Coventry University, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.373896.
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Chen, Xujin, and 陳旭瑾. "Graph partitions and integer flows." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2004. http://hub.hku.hk/bib/B30286256.
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