Journal articles: 'Graphe biparti'

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KIYOMI, MASASHI, TOSHIKI SAITOH, and RYUHEI UEHARA. "BIPARTITE PERMUTATION GRAPHS ARE RECONSTRUCTIBLE." Discrete Mathematics, Algorithms and Applications 04, no. 03 (August 6, 2012): 1250039. http://dx.doi.org/10.1142/s1793830912500395.

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The graph reconstruction conjecture is a long-standing open problem in graph theory. The conjecture has been verified for all graphs with at most 11 vertices. Further, the conjecture has been verified for regular graphs, trees, disconnected graphs, unit interval graphs, separable graphs with no pendant vertex, outer-planar graphs, and unicyclic graphs. We extend the list of graph classes for which the conjecture holds. We give a proof that bipartite permutation graphs are reconstructible.

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Farooq, Rashid, Mehar Ali Malik, Qudsia Naureen, and Shariefuddin Pirzada. "On the nullity of a family of tripartite graphs." Acta Universitatis Sapientiae, Informatica 8, no. 1 (June 1, 2016): 96–107. http://dx.doi.org/10.1515/ausi-2016-0006.

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Abstract The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplicity of the eigenvalue zero in the spectrum of a graph is called nullity of the graph. Fan and Qian (2009) obtained the nullity set of n-vertex bipartite graphs and characterized the bipartite graphs with nullity n − 4 and the regular n-vertex bipartite graphs with nullity n − 6. In this paper, we study similar problem for a class of tripartite graphs. As observed the nullity problem in tripartite graphs does not follow as an extension to that of the nullity of bipartite graphs, this makes the study of nullity in tripartite graphs interesting. In this direction, we obtain the nullity set of a class of n-vertex tripartite graphs and characterize these tripartite graphs with nullity n − 4. We also characterize some tripartite graphs with nullity n − 6 in this class.

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Prasetyo, Joko. "FAKTORISASI PADA GRAF REGULER." EDUPEDIA 4, no. 1 (April 18, 2020): 75. http://dx.doi.org/10.24269/ed.v4i1.434.

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This research aims to: (1) know the criteria of a graph that has a -factor, (2) know the conditions of a regular graph that has a 1-factorization , (3) know the conditions of a regular graph that has a 2-factorization.This research is a qualitative descriptive study using the method of literature study or literature review where a study of books, scientific journals, and other literature languages is carried out relating to factorization on regular graphs. This research begins by discussing the definitions and examples of euler graphs and regular bipartite multigraphs. Next in reviewing the terms of a regular graph which has a 1-factorization and which has a 2-factorization, it starts by discussing the definition and theorem of matching on bipartite graphs, definitions and examples of factorization graphs, then discussing the proof of theorem of regular graphs that have a 1-factor and a regular graph which has a 2-factor.The results of this study indicate that: (1) Graph is said to be -factorable or can be factored into -factor , if can be decomposed or be eksplained into spanning subgraphs , where each has a -factor and is edge-disjoint from , that is 1) 2) … n) = . (2) The condition for a graph that has a 1-factorization is, if the graph is a -regular bipartite multigraph, with . (3) The condition for a graph that has a 2-factorization is, if the graph is a -regular graph, with . Key words: Bipartite graphs, Factorization, Decomposition, Regular graph.

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A. Antony mary, A., A. Amutha, and M. S. Franklin Thamil Selvi. "A Study on Slope Number of Certain Classes of Bipartite Graphs." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 440. http://dx.doi.org/10.14419/ijet.v7i4.10.21036.

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Graph drawing is the most important area of mathematics and computer science which combines methods from geometric graph theory and information visualization. Generally, graphs are represented to explore some intellectual ideas. Graph drawing is the familiar concept of graph theory. It has many quality measures and one among them is the slope number. Slope number problem is an optimization problem and is NP-hard to determine the slope number of any arbitrary graph. In the present paper, the investigation on slope number of bipartite graph is studied elaborately. Since the bipartite graphs creates one of the most intensively investigated classes of graphs, we consider few classes of graphs and discussed structural behavior of such graphs.

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PANDA, SWARUP. "Inverses of bicyclic graphs." Electronic Journal of Linear Algebra 32 (February 6, 2017): 217–31. http://dx.doi.org/10.13001/1081-3810.3322.

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A graph G is said to be nonsingular (resp., singular) if its adjacency matrix A(G) is nonsingular (resp., singular). The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of Â±1s. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite. In [C.D. Godsil. Inverses of trees. Combinatorica, 5(1):33â39, 1985.], Godsil proved that such graphs are invertible. He posed the question of characterizing the bipartite graphs with unique perfect matchings possessing inverses. In this article, Godsilâs question for the class of bicyclic graphs is answered.

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FÜRER, MARTIN, and SHIVA PRASAD KASIVISWANATHAN. "Approximately Counting Embeddings into Random Graphs." Combinatorics, Probability and Computing 23, no. 6 (July 9, 2014): 1028–56. http://dx.doi.org/10.1017/s0963548314000339.

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LetHbe a graph, and letCH(G) be the number of (subgraph isomorphic) copies ofHcontained in a graphG. We investigate the fundamental problem of estimatingCH(G). Previous results cover only a few specific instances of this general problem, for example the case whenHhas degree at most one (the monomer-dimer problem). In this paper we present the first general subcase of the subgraph isomorphism counting problem, which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labelling of the vertices such that every edge is between vertices with different labels, and for every vertex all neighbours with a higher label have identical labels. The labelling implicitly generates a sequence of bipartite graphs, which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphsHand all graphsG, the algorithm is an unbiased estimator. Furthermore, for all graphsHhaving a decomposition where each of the bipartite graphs generated is small and almost all graphsG, the algorithm is a fully polynomial randomized approximation scheme.We show that the graph classes ofHfor which we obtain a fully polynomial randomized approximation scheme for almost allGincludes graphs of degree at most two, bounded-degree forests, bounded-width grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs of large girth, whereas unbounded-width grid graphs are excluded. Moreover, our general technique can easily be applied to proving many more similar results.

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Metsidik, Metrose. "Eulerian and Even-Face Graph Partial Duals." Symmetry 13, no. 8 (August 11, 2021): 1475. http://dx.doi.org/10.3390/sym13081475.

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Eulerian and bipartite graph is a dual symmetric concept in Graph theory. It is well-known that a plane graph is Eulerian if and only if its geometric dual is bipartite. In this paper, we generalize the well-known result to embedded graphs and partial duals of cellularly embedded graphs, and characterize Eulerian and even-face graph partial duals of a cellularly embedded graph by means of half-edge orientations of its medial graph.

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FANKHAUSER, STEFAN, KASPAR RIESEN, HORST BUNKE, and PETER DICKINSON. "SUBOPTIMAL GRAPH ISOMORPHISM USING BIPARTITE MATCHING." International Journal of Pattern Recognition and Artificial Intelligence 26, no. 06 (September 2012): 1250013. http://dx.doi.org/10.1142/s0218001412500139.

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Graphs provide us with a flexible and powerful way to represent objects in various areas of computer science. One of the main drawbacks is, however, that many standard algorithms on graphs have a high computational complexity. The present paper considers the problem of graph isomorphism, i.e. checking two graphs for identity. A novel approach for the efficient computation of graph isomorphism is presented. The proposed algorithm is based on bipartite graph matching by means of an assignment algorithm. The algorithmic framework is suboptimal in the sense of possibly rejecting pairs of graphs without making a decision. As an advantage, however, it offers polynomial runtime. In experiments on diverse graph data sets we demonstrate substantial speedups of our proposed method over several standard procedures for graph isomorphism. Furthermore, although the computational framework for isomorphism is suboptimal, we show that the proposed algorithm rejects only very few pairs of graphs and otherwise returns correct results.

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D'Souza, S., K. P. Girija, and H. J. Gowtham. "Цветовая энергия некоторых кластерных графов." Владикавказский математический журнал, no. 2 (June 24, 2021): 51–64. http://dx.doi.org/10.46698/x5522-9720-4842-z.

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Let $G$ be a simple connected graph. The energy of a graph $G$ is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph $G$. It represents a proper generalization of a formula valid for the total $\pi$-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph $G$ is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph $G$ is called the chromatic number of $G$ and is denoted by $\chi(G)$. The color energy of a graph $G$ is defined as the sum of absolute values of the color eigenvalues of $G$. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.

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Basavanagoud, B., and Roopa S. Kusugal. "On the Line Degree Splitting Graph of a Graph." Bulletin of Mathematical Sciences and Applications 18 (May 2017): 1–10. http://dx.doi.org/10.18052/www.scipress.com/bmsa.18.1.

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In this paper, we introduce the concept of the line degree splitting graph of a graph. We obtain some properties of this graph. We find the girth of the line degree splitting graphs. Further, we establish the characterization of graphs whose line degree splitting graphs are eulerian, complete bipartite graphs and complete graphs.

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Tian, Yingzhi, Huaping Ma, and Liyun Wu. "The Connectivity of a Bipartite Graph and Its Bipartite Complementary Graph." Parallel Processing Letters 30, no. 03 (September 2020): 2040005. http://dx.doi.org/10.1142/s0129626420400058.

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In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph [Formula: see text] and the same invariant in the complement [Formula: see text] of [Formula: see text] is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph [Formula: see text] with bipartition ([Formula: see text]), its bipartite complementary graph [Formula: see text] is a bipartite graph with [Formula: see text] and [Formula: see text] and [Formula: see text]. In this paper, we obtain the Nordhaus-Gaddum type inequalities for connectivity of bipartite graphs and its bipartite complementary graphs. Furthermore, we prove that these inequalities are best possible.

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Furtula, Boris, Slavko Radenkovic, and Ivan Gutman. "Bicyclic molecular graphs with the greatest energy." Journal of the Serbian Chemical Society 73, no. 4 (2008): 431–33. http://dx.doi.org/10.2298/jsc0804431f.

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The molecular graph Qn is obtained by attaching hexagons to the end vertices of the path graph Pn-12. Earlier empirical studies indicated that Qn has greatest energy among all bicyclic n-vertex (molecular) graphs. Recently, Li and Zhang proved that Qn has greatest energy among all bipartite bicyclic graphs, with the exception of the graphs Ra,b, a + b = n, where Ra,b is the graph obtained by joining the cycles Ca and Cb by an edge. This result is now completed by showing that Qn has the greatest energy among all bipartite bicyclic n-vertex graphs.

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V J Kaneria, H P Chudasama, and P P Andharia. "Absolute Mean Graceful Labeling in Path Union of Various Graphs." Mathematical Journal of Interdisciplinary Sciences 7, no. 1 (September 6, 2018): 51–56. http://dx.doi.org/10.15415/mjis.2018.71008.

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Present paper aims to focus on absolute mean graceful labeling in path union of various graphs. We proved path union of graphs like tree, path Pn, cycle Cn, complete bipartite graph Km, n, grid graph PM × Pn, step grid graph Stn and double step grid graph DStn are absolute mean graceful graphs.

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Groshaus, M., A. L. P. Guedes, and J. P. Puppo. "Biclique graph of bipartite permutation graphs." Electronic Notes in Discrete Mathematics 62 (November 2017): 33–38. http://dx.doi.org/10.1016/j.endm.2017.10.007.

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PERARNAU, G., and B. REED. "Existence of Spanning ℱ-Free Subgraphs with Large Minimum Degree." Combinatorics, Probability and Computing 26, no. 3 (December 7, 2016): 448–67. http://dx.doi.org/10.1017/s0963548316000328.

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Let ℱ be a family of graphs and letdbe large enough. For everyd-regular graphG, we study the existence of a spanning ℱ-free subgraph ofGwith large minimum degree. This problem is well understood if ℱ does not contain bipartite graphs. Here we provide asymptotically tight results for many families of bipartite graphs such as cycles or complete bipartite graphs. To prove these results, we study a locally injective analogue of the question.

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Campeña, Francis Joseph H., and Severino V. Gervacio. "On the fold thickness of graphs." Arabian Journal of Mathematics 9, no. 2 (February 19, 2020): 345–55. http://dx.doi.org/10.1007/s40065-020-00276-z.

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Abstract The graph $$G'$$ G ′ obtained from a graph G by identifying two nonadjacent vertices in G having at least one common neighbor is called a 1-fold of G. A sequence $$G_0, G_1, G_2, \ldots , G_k$$ G 0 , G 1 , G 2 , … , G k of graphs such that $$G_0=G$$ G 0 = G and $$G_i$$ G i is a 1-fold of $$G_{i-1}$$ G i - 1 for each $$i=1, 2, \ldots , k$$ i = 1 , 2 , … , k is called a uniform k-folding of G if the graphs in the sequence are all singular or all nonsingular. The fold thickness of G is the largest k for which there is a uniform k-folding of G. We show here that the fold thickness of a singular bipartite graph of order n is $$n-3$$ n - 3 . Furthermore, the fold thickness of a nonsingular bipartite graph is 0, i.e., every 1-fold of a nonsingular bipartite graph is singular. We also determine the fold thickness of some well-known families of graphs such as cycles, fans and some wheels. Moreover, we investigate the fold thickness of graphs obtained by performing operations on these families of graphs. Specifically, we determine the fold thickness of graphs obtained from the cartesian product of two graphs and the fold thickness of a disconnected graph whose components are all isomorphic.

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Takaoka, Asahi. "Complexity of Hamiltonian Cycle Reconfiguration." Algorithms 11, no. 9 (September 17, 2018): 140. http://dx.doi.org/10.3390/a11090140.

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The Hamiltonian cycle reconfiguration problem asks, given two Hamiltonian cycles C 0 and C t of a graph G, whether there is a sequence of Hamiltonian cycles C 0 , C 1 , … , C t such that C i can be obtained from C i − 1 by a switch for each i with 1 ≤ i ≤ t , where a switch is the replacement of a pair of edges u v and w z on a Hamiltonian cycle with the edges u w and v z of G, given that u w and v z did not appear on the cycle. We show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete, settling an open question posed by Ito et al. (2011) and van den Heuvel (2013). More precisely, we show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete for chordal bipartite graphs, strongly chordal split graphs, and bipartite graphs with maximum degree 6. Bipartite permutation graphs form a proper subclass of chordal bipartite graphs, and unit interval graphs form a proper subclass of strongly chordal graphs. On the positive side, we show that, for any two Hamiltonian cycles of a bipartite permutation graph and a unit interval graph, there is a sequence of switches transforming one cycle to the other, and such a sequence can be obtained in linear time.

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Wang, Hong. "Packing two bipartite graphs into a complete bipartite graph." Journal of Graph Theory 26, no. 2 (October 1997): 95–104. http://dx.doi.org/10.1002/(sici)1097-0118(199710)26:2<95::aid-jgt4>3.0.co;2-a.

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Kwun, Young Chel, Hafiz Mutee ur Rehman, Muhammad Yousaf, Waqas Nazeer, and Shin Min Kang. "The Entropy of Weighted Graphs with Atomic Bond Connectivity Edge Weights." Discrete Dynamics in Nature and Society 2018 (December 16, 2018): 1–10. http://dx.doi.org/10.1155/2018/8407032.

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The aim of this report to solve the open problem suggested by Chen et al. We study the graph entropy with ABC edge weights and present bounds of it for connected graphs, regular graphs, complete bipartite graphs, chemical graphs, tree, unicyclic graphs, and star graphs. Moreover, we compute the graph entropy for some families of dendrimers.

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Dong, Gui Xiang, and Xiu Fang Liu. "Incidence Coloring Number of Some Join Graphs." Applied Mechanics and Materials 602-605 (August 2014): 3185–88. http://dx.doi.org/10.4028/www.scientific.net/amm.602-605.3185.

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The incidence coloring of a graph is a mapping from its incidence set to color set in which neighborly incidences are assigned different colors. In this paper, we determined the incidence coloring numbers of some join graphs with paths and paths, cycles, complete graphs, complete bipartite graphs, respectively, and the incidence coloring numbers of some join graphs with complete bipartite graphs and cycles, complete graphs, respectively.

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Chen, Yichao, and Xuluo Yin. "The Thickness of the Cartesian Product of Two Graphs." Canadian Mathematical Bulletin 59, no. 4 (December 1, 2016): 705–20. http://dx.doi.org/10.4153/cmb-2016-020-1.

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AbstractThe thickness of a graph G is the minimum number of planar subgraphs whose union is G. A t-minimal graph is a graph of thickness t that contains no proper subgraph of thickness t. In this paper, upper and lower bounds are obtained for the thickness, t(G ⎕ H), of the Cartesian product of two graphs G and H, in terms of the thickness t(G) and t(H). Furthermore, the thickness of the Cartesian product of two planar graphs and of a t-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph K4k,4k, the thickness of the Cartesian product of two complete bipartite graphs Kn,n and Kn,n is also given for n≠4k + 1.

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Nikmehr, M. J., and S. M. Hosseini. "When the annihilator-ideal graph is planar or complete bipartite graph." Asian-European Journal of Mathematics 12, no. 02 (April 2019): 1950024. http://dx.doi.org/10.1142/s1793557119500244.

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Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.

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Alameda, Joseph S., Emelie Curl, Armando Grez, Leslie Hogben, O’Neill Kingston, Alex Schulte, Derek Young, and Michael Young. "Families of graphs with maximum nullity equal to zero forcing number." Special Matrices 6, no. 1 (January 1, 2018): 56–67. http://dx.doi.org/10.1515/spma-2018-0006.

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Abstract The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i =6 j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartite 3- semiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to find bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen graphs.

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Wu, Ziteng, Chengyun Song, Yunqing Chen, and Lingxuan Li. "A review of recommendation system research based on bipartite graph." MATEC Web of Conferences 336 (2021): 05010. http://dx.doi.org/10.1051/matecconf/202133605010.

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The interaction history between users and items is usually stored and displayed in the form of bipartite graphs. Neural network recommendation based on the user-item bipartite graph has a significant effect on alleviating the long-standing data sparseness and cold start of the recommendation system. The whole paper is based on the bipartite graph. An review of the recommendation system of graphs summarizes the three characteristics of graph neural network processing bipartite graph data in the recommendation field: interchangeability, Multi-hop transportability, and strong interpretability. The biggest contribution of the full paper is that it summarizes the general framework of graph neural network processing bipartite graph recommendation from the models with the best recommendation effect in the past three years: embedding layer, propagation update layer, and prediction layer. Although there are subtle differences between different models, they are all this framework can be applied, and different models can be regarded as variants of this general model, that is, other models are fine-tuned on the basis of this framework. At the end of the paper, the latest research progress is introduced, and the main challenges and research priorities that will be faced in the future are pointed out.

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Simic, Slobodan, and Zoran Stanic. "On Q-integral (3,s)-semiregular bipartite graphs." Applicable Analysis and Discrete Mathematics 4, no. 1 (2010): 167–74. http://dx.doi.org/10.2298/aadm1000002s.

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A graph is called Q-integral if its signless Laplacian spectrum consists entirely of integers. We establish some general results regarding signless Laplacians of semiregular bipartite graphs. Especially, we consider those semiregular bipartite graphs with integral signless Laplacian spectrum. In some particular cases we determine the possible Q-spectra and consider the corresponding graphs.

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Shang, Yilun. "Isoperimetric Numbers of Randomly Perturbed Intersection Graphs." Symmetry 11, no. 4 (April 1, 2019): 452. http://dx.doi.org/10.3390/sym11040452.

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Social networks describe social interactions between people, which are often modeled by intersection graphs. In this paper, we propose an intersection graph model that is induced by adding a sparse random bipartite graph to a given bipartite graph. Under some mild conditions, we show that the vertex–isoperimetric number and the edge–isoperimetric number of the randomly perturbed intersection graph on n vertices are Ω ( 1 / ln n ) asymptomatically almost surely. Numerical simulations for small graphs extracted from two real-world social networks, namely, the board interlocking network and the scientific collaboration network, were performed. It was revealed that the effect of increasing isoperimetric numbers (i.e., expansion properties) on randomly perturbed intersection graphs is presumably independent of the order of the network.

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Voorhees, Burton, and Bergerud Ryder. "Simple graph models of information spread in finite populations." Royal Society Open Science 2, no. 5 (May 2015): 150028. http://dx.doi.org/10.1098/rsos.150028.

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We consider several classes of simple graphs as potential models for information diffusion in a structured population. These include biases cycles, dual circular flows, partial bipartite graphs and what we call ‘single-link’ graphs. In addition to fixation probabilities, we study structure parameters for these graphs, including eigenvalues of the Laplacian, conductances, communicability and expected hitting times. In several cases, values of these parameters are related, most strongly so for partial bipartite graphs. A measure of directional bias in cycles and circular flows arises from the non-zero eigenvalues of the antisymmetric part of the Laplacian and another measure is found for cycles as the value of the transition probability for which hitting times going in either direction of the cycle are equal. A generalization of circular flow graphs is used to illustrate the possibility of tuning edge weights to match pre-specified values for graph parameters; in particular, we show that generalizations of circular flows can be tuned to have fixation probabilities equal to the Moran probability for a complete graph by tuning vertex temperature profiles. Finally, single-link graphs are introduced as an example of a graph involving a bottleneck in the connection between two components and these are compared to the partial bipartite graphs.

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ELLIS-MONAGHAN, JOANNA A., and IRASEMA SARMIENTO. "Distance Hereditary Graphs and the Interlace Polynomial." Combinatorics, Probability and Computing 16, no. 6 (November 2007): 947–73. http://dx.doi.org/10.1017/s0963548307008723.

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The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [3] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [5], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, using the medial graph of a planar graph, we relate the one-variable vertex-nullity interlace polynomial to the classical Tutte polynomial when x=y, and conclude that, like the Tutte polynomial, it is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan in [13].We define the γ invariant as the coefficient of x1in the vertex-nullity interlace polynomial, analogously to the β invariant, which is the coefficientof x1in the Tutte polynomial. We then turn to distance hereditary graphs, characterized by Bandelt and Mulder in [9] as being constructed by a sequence ofadding pendant and twin vertices, and show that graphs in this class have γ invariant of 2n+1whenntrue twins are added intheir construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with γ invariant 2, just as the series-parallel graphs are exactly the class of graphs with β invariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of an Euler circuitin the oriented medial graph of a series-parallel graph. From this we conclude that the vertex-nullity interlace polynomial is polynomial time to compute for bipartite distancehereditary graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs.

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Thomassen, Carsten. "On the Number of Hamiltonian Cycles in Bipartite Graphs." Combinatorics, Probability and Computing 5, no. 4 (December 1996): 437–42. http://dx.doi.org/10.1017/s0963548300002182.

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We prove that a bipartite uniquely Hamiltonian graph has a vertex of degree 2 in each color class. As consequences, every bipartite Hamiltonian graph of minimum degree d has at least 21−dd! Hamiltonian cycles, and every bipartite Hamiltonian graph of minimum degree at least 4 and girth g has at least (3/2)g/8 Hamiltonian cycles. We indicate how the existence of more than one Hamiltonian cycle may lead to a general reduction method for Hamiltonian graphs.

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ERDŐS, PÉTER L., ISTVÁN MIKLÓS, and ZOLTÁN TOROCZKAI. "New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling." Combinatorics, Probability and Computing 27, no. 2 (November 2, 2017): 186–207. http://dx.doi.org/10.1017/s0963548317000499.

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In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.

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Gharibyan, Aram H., and Petros A. Petrosyan. "ON LOCALLY-BALANCED 2-PARTITIONS OF BIPARTITE GRAPHS." Proceedings of the YSU A: Physical and Mathematical Sciences 54, no. 3 (253) (December 15, 2020): 137–45. http://dx.doi.org/10.46991/pysu:a/2020.54.3.137.

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A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood}, if for every $v\in V(G)$, $\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1$. A bipartite graph is \emph{$(a,b)$-biregular} if all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. In this paper we prove that the problem of deciding, if a given graph has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete even for $(3,8)$-biregular bipartite graphs. We also prove that a $(2,2k+1)$-biregular bipartite graph has a locally-balanced $2$-partition with an open neighbourhood if and only if it has no cycle of length $2 \pmod{4}$. Next, we prove that if $G$ is a subcubic bipartite graph that has no cycle of length $2 \pmod{4}$, then $G$ has a locally-balanced $2$-partition with an open neighbourhood. Finally, we show that all doubly convex bipartite graphs have a locally-balanced $2$-partition with an open neighbourhood.

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Erdős, P., and L. Pyber. "Covering a graph by complete bipartite graphs." Discrete Mathematics 170, no. 1-3 (June 1997): 249–51. http://dx.doi.org/10.1016/s0012-365x(96)00124-0.

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33

Lee, Jon, and Jennifer Ryan. "Local bipartite turán graphs and graph partitioning." Networks 24, no. 3 (May 1994): 131–45. http://dx.doi.org/10.1002/net.3230240302.

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Das, Sandip, Prantar Ghosh, Shamik Ghosh, and Sagnik Sen. "Oriented bipartite graphs and the Goldbach graph." Discrete Mathematics 344, no. 9 (September 2021): 112497. http://dx.doi.org/10.1016/j.disc.2021.112497.

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Shaveisi, Farzad. "A neigborhood union condition for nonadjacent vertices in graphs." Discrete Mathematics, Algorithms and Applications 09, no. 05 (October 2017): 1750060. http://dx.doi.org/10.1142/s1793830917500604.

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A simple graph [Formula: see text] is called [Formula: see text]-bounded if for every two nonadjacent vertices [Formula: see text] of [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text], where [Formula: see text] denotes the set of neighbors of the vertex [Formula: see text] in [Formula: see text]. In this paper, some properties of [Formula: see text]-bounded graphs are studied. It is shown that any bipartite [Formula: see text]-bounded graph is a complete bipartite graph with at most two horns; in particular, any [Formula: see text]-bounded tree is either a star or a two-star graph. Also, we prove that any non-end vertex of every [Formula: see text]-bounded graph is contained in either a triangle or a rectangle. Among other results, it is shown that all regular [Formula: see text]-bounded graphs are strongly regular graphs. Finally, we determine that how many edges can an [Formula: see text]-bounded graph have?

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Alkam, Osama, and Emad Abu Osba. "Zero Divisor Graph for the Ring of Eisenstein Integers Modulo n." Algebra 2014 (December 15, 2014): 1–6. http://dx.doi.org/10.1155/2014/146873.

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Let En be the ring of Eisenstein integers modulo n. In this paper we study the zero divisor graph Γ(En). We find the diameters and girths for such zero divisor graphs and characterize n for which the graph Γ(En) is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.

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PRADHAN, D. "COMPLEXITY OF CERTAIN FUNCTIONAL VARIANTS OF TOTAL DOMINATION IN CHORDAL BIPARTITE GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 03 (August 6, 2012): 1250045. http://dx.doi.org/10.1142/s1793830912500450.

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In this paper, we consider minimum total domination problem along with two of its variations namely, minimum signed total domination problem and minimum minus total domination problem for chordal bipartite graphs. In the minimum total domination problem, the objective is to find a smallest size subset TD ⊆ V of a given graph G = (V, E) such that |TD∩NG(v)| ≥ 1 for every v ∈ V. In the minimum signed (minus) total domination problem for a graph G = (V, E), it is required to find a function f : V → {-1, 1} ({-1, 0, 1}) such that f(NG(v)) = ∑u∈NG(v)f(u) ≥ 1 for each v ∈ V, and the cost f(V) = ∑v∈V f(v) is minimized. We first show that for a given chordal bipartite graph G = (V, E) with a weak elimination ordering, a minimum total dominating set can be computed in O(n + m) time, where n = |V| and m = |E|. This improves the complexity of the minimum total domination problem for chordal bipartite graphs from O(n2) time to O(n + m) time. We then adopt a unified approach to solve the minimum signed (minus) total domination problem for chordal bipartite graphs in O(n + m) time. The method is also able to solve the minimum k-tuple total domination problem for chordal bipartite graphs in O(n + m) time. For a fixed integer k ≥ 1 and a graph G = (V, E), the minimum k-tuple total domination problem is to find a smallest subset TDk ⊆ V such that |TDk ∩ NG(v)| ≥ k for every v ∈ V.

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Strapasson, João Eloir, Sueli Irene Rodrigues Costa, and Marcelo Muniz. "A Note on Quadrangular Embedding of Abelian Cayley Graphs." TEMA (São Carlos) 17, no. 3 (December 20, 2016): 331. http://dx.doi.org/10.5540/tema.2016.017.03.0331.

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The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound.

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Bakonyi, Mihály, and Erik M. Varness. "Algorithmic aspects of bipartite graphs." International Journal of Mathematics and Mathematical Sciences 18, no. 2 (1995): 299–304. http://dx.doi.org/10.1155/s0161171295000378.

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We generalize previous work done by Donald J. Rose and Robert E. Tarjan [2], who developed efficient algorithms for use on directed graphs. This paper considers an edge elimination process on bipartite graphs, presenting several theorems which lead to an algorithm for computing the minimal fill-in of a given ordered graph.

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Talmaciu, Mihai, and Elena Nechita. "On Polar, Trivially Perfect Graphs." International Journal of Computers Communications & Control 5, no. 5 (December 1, 2010): 939. http://dx.doi.org/10.15837/ijccc.2010.5.2257.

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<p>During the last decades, different types of decompositions have been processed in the field of graph theory. In various problems, for example in the construction of recognition algorithms, frequently appears the so-called weakly decomposition of graphs.<br />Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. Recognizing a polar graph is known to be NP-complete. For this class of graphs, polynomial algorithms for the maximum stable set problem are unknown and algorithms for the dominating set problem are also NP-complete.<br />In this paper we characterize the polar graphs using the weakly decomposition, give a polynomial time algorithm for recognizing graphs that are both trivially perfect and polar, and directly calculate the domination number. For the stability number and clique number, we give polynomial time algorithms. </p>

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41

Sason, Igal. "A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs." Entropy 23, no. 3 (February 25, 2021): 270. http://dx.doi.org/10.3390/e23030270.

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This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn’s information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but may be irregular on the other, the extended entropy-based proof technique yields the same bound as was conjectured by Kahn (2001) and proved by Sah et al. (2019).

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Wang, Hongzhuan, and Piaoyang Yin. "Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index." Symmetry 11, no. 5 (May 2, 2019): 615. http://dx.doi.org/10.3390/sym11050615.

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Resistance distance is a concept developed from electronic networks. The calculation of resistance distance in various circuits has attracted the attention of many engineers. This report considers the resistance-based graph invariant, the Resistance–Harary index, which represents the sum of the reciprocal resistances of any vertex pair in the figure G, denoted by R H ( G ) . Vertex bipartiteness in a graph G is the minimum number of vertices removed that makes the graph G become a bipartite graph. In this study, we give the upper bound and lower bound of the R H index, and describe the corresponding extremal graphs in the bipartite graph of a given order. We also describe the graphs with maximum R H index in terms of graph parameters such as vertex bipartiteness, cut edges, and matching numbers.

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Smbatyan, Khachik S. "TWO RESULTS ON THE PALETTE INDEX OF GRAPHS." Proceedings of the YSU A: Physical and Mathematical Sciences 55, no. 1 (254) (May 21, 2021): 36–43. http://dx.doi.org/10.46991/pysu:a/2021.55.1.036.

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Given a proper edge coloring $\alpha$ of a graph $G$, we define the palette $S_G(v,\alpha)$ of a vertex $v\in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check{s}(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. A graph $G$ is called nearly bipartite if there exists $ v\in V(G)$ so that $G-v$ is a bipartite graph. In this paper, we give an upper bound on the palette index of a nearly bipartite graph $G$ by using the decomposition of $G$ into cycles. We also provide an upper bound on the palette index of Cartesian products of graphs. In particular, we show that for any graphs $G$ and $H$, $\check{s}(G\square H)\leq \check{s}(G)\check{s}(H)$.

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T, Bharathi, Jeya Rowena, and Ashwini Sibiya Rani P. "Fuzzy square difference labeling of some graphs." Journal of Computational Mathematica 5, no. 1 (June 18, 2021): 70–75. http://dx.doi.org/10.26524/cm93.

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We introduced a new concept called the Fuzzy square difference labeling. We proved that the path graph (Pn), the cycle graph (Cn), the star graph (Sn) and the complete bipartite graph (Km,n, n ≤ 3) are Fuzzy square difference graphs.

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Ma, Xintao, Liyan Dong, Yuequn Wang, Yongli Li, and Minghui Sun. "AIRC: Attentive Implicit Relation Recommendation Incorporating Content Information for Bipartite Graphs." Mathematics 8, no. 12 (November 30, 2020): 2132. http://dx.doi.org/10.3390/math8122132.

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With users being exposed to the growing volume of online information, the recommendation system aiming at mining the important or interesting information is becoming a modern research topic. One approach of recommendation is to integrate the graph neural network with deep learning algorithms. However, some of them are not tailored for bipartite graphs, which is a unique type of heterogeneous graph having two entity types. Others, though customized, neglect the importance of implicit relation and content information. In this paper, we propose the attentive implicit relation recommendation incorporating content information (AIRC) framework that is designed for bipartite graphs based on the GC–MC algorithm. First, through reconstructing the bipartite graphs, we obtain the implicit relation graphs. Then we analyze the content information of users and items with a CNN process, so that each user and item has its feature-tailored embeddings. Besides, we expand the GC–MC algorithms by adding a graph attention mechanism layer, which handles the implicit relation graph by highlighting important features and neighbors. Therefore, our framework takes into consideration both the implicit relation and content information. Finally, we test our framework on Movielens dataset and the results show that our framework performs better than other state-of-art recommendation algorithms.

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KIM, DONGSEOK, and JAEUN LEE. "THE QUANTUM 𝔰𝔩(3) INVARIANTS OF CUBIC BIPARTITE PLANAR GRAPHS." Journal of Knot Theory and Its Ramifications 17, no. 03 (March 2008): 361–75. http://dx.doi.org/10.1142/s0218216508006142.

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Temperley–Lieb algebras have been generalized to 𝔰𝔩(3,ℂ) web spaces. Since a cubic bipartite planar graph with suitable directions on edges is a web, the quantum 𝔰𝔩(3) invariants naturally extend to all cubic bipartite planar graphs. First we completely classify them as a connected sum of prime webs. We provide a method to find all prime webs. Using the quantum 𝔰𝔩(3) invariants, we provide a criterion which determines the symmetry of cubic bipartite planar graphs. We also provide an example that our criterions for graphs is different from that of links.

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Nguyen, Ngoc Tuy, Jörg Bornemann, and Van Bang Le. "Graph classes related to chordal graphs and chordal bipartite graphs." Electronic Notes in Discrete Mathematics 27 (October 2006): 73–74. http://dx.doi.org/10.1016/j.endm.2006.08.062.

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El-Shanawany, R. A., and M. Sh Higazy. "General Symmetric Starter of Orthogonal Double Covers of Complete Bipartite Graph." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–8. http://dx.doi.org/10.1155/2007/42892.

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An orthogonal double cover (ODC) of the complete graph is a collection of graphs such that every two of them share exactly one edge and every edge of the complete graph belongs to exactly two of the graphs. In this paper, we consider the case where the graph to be covered twice is the complete bipartite graphKmn,mn(for any values ofm,n) and all graphs in the collection are isomorphic to certain spanning subgraphs. Furthermore, the ODCs ofKn,nby certain disjoint stars are constructed.

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FRATI, FABRIZIO. "ON MINIMUM AREA PLANAR UPWARD DRAWINGS OF DIRECTED TREES AND OTHER FAMILIES OF DIRECTED ACYCLIC GRAPHS." International Journal of Computational Geometry & Applications 18, no. 03 (June 2008): 251–71. http://dx.doi.org/10.1142/s021819590800260x.

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It has been shown that there exist planar digraphs that require exponential area in every upward straight-line planar drawing. On the other hand, upward poly-line planar drawings of planar graphs can be realized in Θ(n2) area. In this paper we consider families of DAGs that naturally arise in practice, like DAGs whose underlying graph is a tree (directed trees), is a bipartite graph (directed bipartite graphs), or is an outerplanar graph (directed outerplanar graphs). Concerning directed trees, we show that optimal Θ(n log n) area upward straight-line/poly-line planar drawings can be constructed. However, we prove that if the order of the neighbors of each node is assigned, then exponential area is required for straight-line upward drawings and quadratic area is required for poly-line upward drawings, results surprisingly and sharply contrasting with the area bounds for planar upward drawings of undirected trees. After having established tight bounds on the area requirements of planar upward drawings of several families of directed trees, we show how the results obtained for trees can be exploited to determine asymptotic optimal values for the area occupation of planar upward drawings of directed bipartite graphs and directed outerplanar graphs.

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LIN, CHENG-KUAN, TUNG-YANG HO, JIMMY J. M. TAN, and LIH-HSING HSU. "FAULT-TOLERANT HAMILTONIAN LACEABILITY AND FAULT-TOLERANT CONDITIONAL HAMILTONIAN FOR BIPARTITE HYPERCUBE-LIKE NETWORKS." Journal of Interconnection Networks 10, no. 03 (September 2009): 243–51. http://dx.doi.org/10.1142/s0219265909002558.

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A bipartite graph G is hamiltonian laceable if there is a hamiltonian path between any two vertices of G from distinct vertex bipartite sets. A bipartite graph G is k-edge fault-tolerant hamiltonian laceable if G - F is hamiltonian laceable for every F ⊆ E(G) with |F| ≤ k. A graph G is k-edge fault-tolerant conditional hamiltonian if G - F is hamiltonian for every F ⊆ E(G) with |F| ≤ k and δ(G - F) ≥ 2. Let G0 = (V0, E0) and G1 = (V1, E1) be two disjoint graphs with |V0| = |V1|. Let Er = {(v,ɸ(v)) | v ϵ V0,ɸ(v) ϵ V1, and ɸ: V0 → V1 is a bijection}. Let G = G0 ⊕ G1 = (V0 ⋃ V1, E0 ⋃ E1 ⋃ Er). The set of n-dimensional hypercube-like graphHn is defined recursively as (a) H1 = K2, K2 is the complete graph with two vertices, and (b) if G0 and G1 are in Hn, then G = G0 ⊕ G1 is in Hn+1. Let Bn be the set of graphs G where G is bipartite and G ϵ Hn. In this paper, we show that every graph in Bn is (n - 2)-edge fault-tolerant hamiltonian laceable if n ≥ 2 and every graph in Bn is (2n - 5)-edge fault-tolerant conditional hamiltonian if n ≥ 3.

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Journal articles on the topic 'Graphe biparti'

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