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Journal articles on the topic 'Eulerian graph theory. Hamiltonian graph theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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1

Tamizh Chelvam, T., and S. Anukumar Kathirvel. "Generalized unit and unitary Cayley graphs of finite rings." Journal of Algebra and Its Applications 18, no. 01 (2019): 1950006. http://dx.doi.org/10.1142/s0219498819500063.

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Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient co

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Medvedev, Paul, and Mihai Pop. "What do Eulerian and Hamiltonian cycles have to do with genome assembly?" PLOS Computational Biology 17, no. 5 (2021): e1008928. http://dx.doi.org/10.1371/journal.pcbi.1008928.

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Many students are taught about genome assembly using the dichotomy between the complexity of finding Eulerian and Hamiltonian cycles (easy versus hard, respectively). This dichotomy is sometimes used to motivate the use of de Bruijn graphs in practice. In this paper, we explain that while de Bruijn graphs have indeed been very useful, the reason has nothing to do with the complexity of the Hamiltonian and Eulerian cycle problems. We give 2 arguments. The first is that a genome reconstruction is never unique and hence an algorithm for finding Eulerian or Hamiltonian cycles is not part of any as

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3

Metsidik, Metrose. "Eulerian and Even-Face Graph Partial Duals." Symmetry 13, no. 8 (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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Broersma, H. J. "A note on K4-closures in hamiltonian graph theory." Discrete Mathematics 121, no. 1-3 (1993): 19–23. http://dx.doi.org/10.1016/0012-365x(93)90533-y.

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5

Broersma, H. J. "On some intriguing problems in hamiltonian graph theory—a survey." Discrete Mathematics 251, no. 1-3 (2002): 47–69. http://dx.doi.org/10.1016/s0012-365x(01)00325-9.

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6

Li, Hao. "Generalizations of Dirac’s theorem in Hamiltonian graph theory—A survey." Discrete Mathematics 313, no. 19 (2013): 2034–53. http://dx.doi.org/10.1016/j.disc.2012.11.025.

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7

Ceulemans, A., E. Lijnen, P. W. Fowler, R. B. Mallion, and T. Pisanski. "Graph theory and the Jahn–Teller theorem." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2140 (2011): 971–89. http://dx.doi.org/10.1098/rspa.2011.0508.

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The Jahn–Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons,

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Blanco, Rocío, and Melody García-Moya. "Graph Theory for Primary School Students with High Skills in Mathematics." Mathematics 9, no. 13 (2021): 1567. http://dx.doi.org/10.3390/math9131567.

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Graph theory is a powerful representation and problem-solving tool, but it is not included in present curriculum at school levels. In this study we perform a didactic proposal based in graph theory, to provide students useful and motivational tools for problem solving. The participants, who were highly skilled in mathematics, worked on map coloring, Eulerian cycles, star polygons and other related topics. The program included six sessions in a workshop format and four creative sessions where participants invented their own mathematical challenges. Throughout the experience they applied a wide

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9

Thomassen, Carsten. "On the Number of Hamiltonian Cycles in Bipartite Graphs." Combinatorics, Probability and Computing 5, no. 4 (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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10

Horák, Peter, Tomáš Kaiser, Moshe Rosenfeld, and Zdeněk Ryjáček. "The Prism Over the Middle-levels Graph is Hamiltonian." Order 22, no. 1 (2005): 73–81. http://dx.doi.org/10.1007/s11083-005-9008-7.

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11

Schlueter-Kuck, Kristy L., and John O. Dabiri. "Coherent structure colouring: identification of coherent structures from sparse data using graph theory." Journal of Fluid Mechanics 811 (December 13, 2016): 468–86. http://dx.doi.org/10.1017/jfm.2016.755.

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We present a frame-invariant method for detecting coherent structures from Lagrangian flow trajectories that can be sparse in number, as is the case in many fluid mechanics applications of practical interest. The method, based on principles used in graph colouring and spectral graph drawing algorithms, examines a measure of the kinematic dissimilarity of all pairs of fluid trajectories, measured either experimentally, e.g. using particle tracking velocimetry, or numerically, by advecting fluid particles in the Eulerian velocity field. Coherence is assigned to groups of particles whose kinemati

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12

Paulraja, P., and Kumar Sampath. "On hamiltonian decompositions of tensor products of graphs." Applicable Analysis and Discrete Mathematics 13, no. 1 (2019): 178–202. http://dx.doi.org/10.2298/aadm170803003p.

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Finding a hamiltonian decomposition of G is one of the challenging problems in graph theory. We do not know for what classes of graphs G and H, their tensor product G x H is hamiltonian decomposable. In this paper, we have proved that, if G is a hamiltonian decomposable circulant graph with certain properties and H is a hamiltonian decomposable multigraph, then G x H is hamiltonian decomposable. In particular, tensor products of certain sparse hamiltonian decomposable circulant graphs are hamiltonian decomposable.

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13

Montgomery, Richard. "Hamiltonicity in random directed graphs is born resilient." Combinatorics, Probability and Computing 29, no. 6 (2020): 900–942. http://dx.doi.org/10.1017/s0963548320000140.

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AbstractLet $\{D_M\}_{M\geq 0}$ be the n-vertex random directed graph process, where $D_0$ is the empty directed graph on n vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $$\varepsilon > 0$$ , we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $1/2-\varepsilon$ of both the in- and out-edges incident to each vertex are removed. We say such a dir

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Mickewich, Tom. "HAMILTONIAN GRAPHS – COMPARING FOUR SUFFICIENT CONDITIONS IN AN UNDERGRADUATE GRAPH THEORY COURSE." PRIMUS 4, no. 2 (1994): 173–81. http://dx.doi.org/10.1080/10511979408965747.

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15

AMORIM, LINO, YONG–GEUN OH, and JOANA OLIVEIRA DOS SANTOS. "Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry–Mather theory." Mathematical Proceedings of the Cambridge Philosophical Society 165, no. 3 (2017): 411–34. http://dx.doi.org/10.1017/s0305004117000561.

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AbstractWe construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants, developed by Oh, and results, by Abouzaid, about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalisation of graph selector. We then, following Bernard–Oliveira dos Santos, use these results to give a new characterisation of the Aubry and Mañé sets of a Tonelli Hamiltonian and to generalise a result of Arnaud on

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Caravelli, Francesco, Michael Saccone, and Cristiano Nisoli. "On the degeneracy of spin ice graphs, and its estimate via the Bethe permanent." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2252 (2021): 20210108. http://dx.doi.org/10.1098/rspa.2021.0108.

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The concept of spin ice can be extended to a general graph. We study the degeneracy of spin ice graph on arbitrary interaction structures via graph theory. We map spin ice graphs to the Ising model on a graph and clarify whether the inverse mapping is possible via a modified Krausz construction. From the gauge freedom of frustrated Ising systems, we derive exact, general results about frustration and degeneracy. We demonstrate for the first time that every spin ice graph, with the exception of the one-dimensional Ising model, is degenerate. We then study how degeneracy scales in size, using th

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17

Häggkvist, Roland, and Anders Johansson. "(1, 2)-Factorizations of General Eulerian Nearly Regular Graphs." Combinatorics, Probability and Computing 3, no. 1 (1994): 87–95. http://dx.doi.org/10.1017/s0963548300001000.

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Every general graph with degrees 2k and 2k − 2, k ≥ 3, with zero or at least two vertices of degree 2k − 2 in each component, has a k-edge-colouring such that each monochromatic subgraph has degree 1 or 2 at every vertex.In particular, if T is a triangle in a 6-regular general graph, there exists a 2-factorization of G such that each factor uses an edge in T if and only if T is non-separating.

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18

HIRANO, YOSHIYASU. "IMPROVED LOWER BOUND FOR THE NUMBER OF KNOTTED HAMILTONIAN CYCLES IN SPATIAL EMBEDDINGS OF COMPLETE GRAPHS." Journal of Knot Theory and Its Ramifications 19, no. 05 (2010): 705–8. http://dx.doi.org/10.1142/s0218216510007991.

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We prove that every spatial embedding of the complete graph K8 contains at least 3 knotted Hamiltonian cycles, and that every spatial embedding of Kn contains at least 3(n - 1)(n - 2) ⋯ 8 knotted Hamiltonian cycles, for n > 8.

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19

Fleischner, H. "(Some of) the many uses of Eulerian graphs in graph theory (plus some applications)." Discrete Mathematics 230, no. 1-3 (2001): 23–43. http://dx.doi.org/10.1016/s0012-365x(00)00067-4.

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20

LEE, CHOONGBUM, BENNY SUDAKOV, and DAN VILENCHIK. "Getting a Directed Hamilton Cycle Two Times Faster." Combinatorics, Probability and Computing 21, no. 5 (2012): 773–801. http://dx.doi.org/10.1017/s096354831200020x.

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Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected ca

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21

Rappaport, David. "The visibility graph of congruent discs is Hamiltonian." Computational Geometry 25, no. 3 (2003): 257–65. http://dx.doi.org/10.1016/s0925-7721(02)00113-x.

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22

Bauer, Douglas, and Edward Schmeichel. "Hamiltonian degree conditions which imply a graph is pancyclic." Journal of Combinatorial Theory, Series B 48, no. 1 (1990): 111–16. http://dx.doi.org/10.1016/0095-8956(90)90133-k.

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23

Naumowicz, Adam. "A Note on the Seven Bridges of Königsberg Problem." Formalized Mathematics 22, no. 2 (2014): 177–78. http://dx.doi.org/10.2478/forma-2014-0018.

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Summary In this paper we account for the formalization of the seven bridges of Königsberg puzzle. The problem originally posed and solved by Euler in 1735 is historically notable for having laid the foundations of graph theory, cf. [7]. Our formalization utilizes a simple set-theoretical graph representation with four distinct sets for the graph’s vertices and another seven sets that represent the edges (bridges). The work appends the article by Nakamura and Rudnicki [10] by introducing the classic example of a graph that does not contain an Eulerian path. This theorem is item #54 from the “Fo

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24

BEN-SHIMON, SONNY, MICHAEL KRIVELEVICH, and BENNY SUDAKOV. "Local Resilience and Hamiltonicity Maker–Breaker Games in Random Regular Graphs." Combinatorics, Probability and Computing 20, no. 2 (2010): 173–211. http://dx.doi.org/10.1017/s0963548310000453.

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For an increasing monotone graph propertythelocal resilienceof a graphGwith respect tois the minimalrfor which there exists a subgraphH⊆Gwith all degrees at mostr, such that the removal of the edges ofHfromGcreates a graph that does not possess. This notion, which was implicitly studied for somead hocproperties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the binomial random graph model(n, p) and some families of pseudo-random graphs with respect to several graph properties, such as containi

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25

Takaoka, Asahi. "Complexity of Hamiltonian Cycle Reconfiguration." Algorithms 11, no. 9 (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 pre

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26

Fahrenthold, E. P., and M. Venkataraman. "System Dynamics Modeling of Porous Media." Journal of Dynamic Systems, Measurement, and Control 119, no. 2 (1997): 251–59. http://dx.doi.org/10.1115/1.2801241.

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A wide range of engineering problems involve porous media modeling. General porous media models are highly nonlinear, geometrically complex, and must account for energy transfer between fluid and solid constituents normally modeled in distinct Lagrangian and Eulerian reference frames. Combining finite element discretization techniques with bond graph methods greatly simplifies the model formulation process, as compared to alternative schemes based on weighted residual solutions of the governing partial differential equations. The result generalizes existing numerical models of porous media and

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27

Yang, Bin, Ziqiong Lin, and William Zhu. "Covering-Based Rough Sets on Eulerian Matroids." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/254797.

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Rough set theory is an efficient and essential tool for dealing with vagueness and granularity in information systems. Covering-based rough set theory is proposed as a significant generalization of classical rough sets. Matroid theory is a vital structure with high applicability and borrows extensively from linear algebra and graph theory. In this paper, one type of covering-based approximations is studied from the viewpoint of Eulerian matroids. First, we explore the circuits of an Eulerian matroid from the perspective of coverings. Second, this type of covering-based approximations is repres

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Ma, Jingjing. "Application of DNA Nanoparticle Conjugation on the Hamiltonian Path Problem." Journal of Nanoelectronics and Optoelectronics 16, no. 3 (2021): 501–5. http://dx.doi.org/10.1166/jno.2021.2930.

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A DNA computing algorithm is proposed in this paper. The algorithm uses the assembly of DNA/Au nanoparticle conjugation to solve an NP-complete problem in the Graph theory, the Hamiltonian Path problem. According to the algorithm, I designed the special DNA/Au nanoparticle conjugations which assembled based on a specific graph, then, a series of experimental techniques are utilized to get the final result. This biochemical algorithm can reduce the complexity of the Hamiltonian Path problem greatly, which will provide a practical way to the best use of DNA self-assembly model.

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29

Baril, Jean-Lue. "Hamiltonian paths for involutions in the square of a Cayley graph." Journal of Discrete Mathematical Sciences and Cryptography 10, no. 4 (2007): 473–84. http://dx.doi.org/10.1080/09720529.2007.10698133.

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30

Abrosimov, M. B. "COMPARISON OF SUFFICIENT DEGREE BASED CONDITIONS FOR HAMILTONIAN GRAPH." Prikladnaya Diskretnaya Matematika, no. 45 (September 1, 2019): 55–63. http://dx.doi.org/10.17223/20710410/45/6.

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31

Lai, Hong-Jian, Yehong Shao, Hehui Wu, and Ju Zhou. "Every 3-connected, essentially 11-connected line graph is Hamiltonian." Journal of Combinatorial Theory, Series B 96, no. 4 (2006): 571–76. http://dx.doi.org/10.1016/j.jctb.2005.11.002.

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32

Gokan, Yusuke, Hayato Katsumata, Katsuya Nakajima, Ayaka Shimizu, and Yoshiro Yaguchi. "A note on the cross-index of a complete graph based on a linear tree." Journal of Knot Theory and Its Ramifications 27, no. 11 (2018): 1843010. http://dx.doi.org/10.1142/s0218216518430101.

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In this paper, it is shown that a complete graph with [Formula: see text] vertices has an optimal diagram, i.e. a diagram whose crossing number equals the value of Guy’s formula, with a free maximal linear tree and without free Hamiltonian cycles for any odd integer [Formula: see text].

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Zarei, Mohammad Hossein, та Yahya Khalili. "Systematic study of the completeness of two-dimensional classical ϕ4 theory". International Journal of Quantum Information 15, № 07 (2017): 1750051. http://dx.doi.org/10.1142/s0219749917500514.

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The completeness of some classical statistical mechanical (SM) models is a recent result that has been developed by quantum formalism for the partition functions. In this paper, we consider a 2D classical [Formula: see text] filed theory whose completeness has been proved in [V. Karimipour and M. H. Zarei, Phys. Rev. A 85 (2012) 032316]. We give a new and general systematic proof for the completeness of such a model where, by a few simple steps, we show how the partition function of an arbitrary classical field theory can be derived from a 2D classical [Formula: see text] model. To this end, w

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Dumitru, Violeta. "Economical Model Based on Graph Theory for Optimization Execution Order of Automotive Products on the Manufacturing Lines Served by Robots." Applied Mechanics and Materials 822 (January 2016): 443–51. http://dx.doi.org/10.4028/www.scientific.net/amm.822.443.

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This paper presents an economic model based on graph theory to optimize the execution order for automotive products on the manufacturing lines served by robots, so that the total manufacturing cost is minimized. The starting point is a representation of the range of automotive products with similar design attributes but with different construction requirements (material, tolerances, production volume). This representation is distinguished by a directed graph G = (X, Γ) consisting of the set X of nodes (products) and the set Γ of arcs (paths with minimal cost). In terms of mathematical optimiza

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35

HUANG, HAO, JIE MA, ASAF SHAPIRA, BENNY SUDAKOV, and RAPHAEL YUSTER. "Large Feedback Arc Sets, High Minimum Degree Subgraphs, and Long Cycles in Eulerian Digraphs." Combinatorics, Probability and Computing 22, no. 6 (2013): 859–73. http://dx.doi.org/10.1017/s0963548313000394.

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A minimum feedback arc set of a directed graph G is a smallest set of arcs whose removal makes G acyclic. Its cardinality is denoted by β(G). We show that a simple Eulerian digraph with n vertices and m arcs has β(G) ≥ m2/2n2+m/2n, and this bound is optimal for infinitely many m, n. Using this result we prove that a simple Eulerian digraph contains a cycle of length at most 6n2/m, and has an Eulerian subgraph with minimum degree at least m2/24n3. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in E

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36

Bansal, N., S. Bravyi, and B. M. Terhal. "Classical approximation schemes for the ground-state energy of quantum and classical Ising spin Hamiltonians on planar graphs." Quantum Information and Computation 9, no. 7&8 (2009): 701–20. http://dx.doi.org/10.26421/qic9.7-8-12.

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We describe a classical approximation algorithm for evaluating the ground state energy of the classical Ising Hamiltonian with linear terms on an arbitrary planar graph. The running time of the algorithm grows linearly with the number of spins and exponentially with $1/\epsilon$, where $\epsilon$ is the worst-case relative error. This result contrasts the well known fact that exact computation of the ground state energy for the two-dimensional Ising spin glass model is NP-hard. We also present a classical approximation algorithm for the quantum Local Hamiltonian Problem or Quantum Ising Spin G

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Zhang, Rong, and Shu-Guang Guo. "On the least Q -eigenvalue of a non-bipartite hamiltonian graph." Linear Algebra and its Applications 538 (February 2018): 89–102. http://dx.doi.org/10.1016/j.laa.2017.10.012.

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He, Qing-Bi, Hong-Gang Li, Mao-Ming Jin, Hui-Ming Duan, and Qing-Hua Zhang. "New necessary and sufficient condition and algorithm for directed hamiltonian graph based on boolean determinant theory." Journal of Discrete Mathematical Sciences and Cryptography 20, no. 3 (2017): 725–45. http://dx.doi.org/10.1080/09720529.2016.1226618.

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Andrianova, E. G., V. K. Raev, and D. I. Filgus. "Determination of the Shortest Hamiltonian Paths in an Arbitrary Graph of Distributed Databases." Russian Technological Journal 7, no. 4 (2019): 7–20. http://dx.doi.org/10.32362/2500-316x-2019-7-4-7-20.

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A method has been developed for finding the shortest Hamiltonian path in an arbitrary graph based on the rank approach, which provides high efficiency and a significant reduction in the error in solving the problem of organizing the process of managing multiple transactions and queries when they are implemented in network databases. In many cases, existing solutions do not provide the necessary results in terms of access time and accuracy of the found solution. Using the developed method allows minimizing the idle time of computing devices, reducing the volume and time of data transfer from on

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Häggkvist, Roland. "Hamilton Cycles in Oriented Graphs." Combinatorics, Probability and Computing 2, no. 1 (1993): 25–32. http://dx.doi.org/10.1017/s0963548300000468.

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41

Kirkland, Steve, Sarah Plosker, and Xiaohong Zhang. "Switching and partially switching the hypercube while maintaining perfect state transfer." Quantum Information and Computation 19, no. 7&8 (2019): 541–54. http://dx.doi.org/10.26421/qic19.7-8-1.

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A graph is said to exhibit perfect state transfer (PST) if one of its corresponding Hamiltonian matrices, which are based on the vertex-edge structure of the graph, gives rise to PST in a quantum information-theoretic context, namely with respect to inter-qubit interactions of a quantum system. We perform various perturbations to the hypercube graph---a graph that is known to exhibit PST---to create graphs that maintain many of the same properties of the hypercube, including PST as well as the distance for which PST occurs. We show that the sensitivity with respect to readout time errors remai

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BOSE, PROSENJIT, HAZEL EVERETT, and STEPHEN WISMATH. "PROPERTIES OF ARRANGEMENT GRAPHS." International Journal of Computational Geometry & Applications 13, no. 06 (2003): 447–62. http://dx.doi.org/10.1142/s0218195903001281.

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An arrangement graph G is the abstract graph obtained from an arrangement of lines L, in general position by associating vertices of G with the intersection points of L, and the edges of G with the line segments joining the intersection points of L. A simple polygon (respectively path) of n sides in general position, induces a set of n lines by extension of the line segments into lines. The main results of this paper are: • Given a graph G, it is NP-Hard to determine if G is the arrangement graph of some set of lines. • There are non-Hamiltonian arrangement graphs for arrangements of six lines

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"Fuzzy Eulerian and Fuzzy Hamiltonian Graphs with Their Applications." International Journal of Recent Technology and Engineering 8, no. 4 (2019): 7995–99. http://dx.doi.org/10.35940/ijrte.d4386.118419.

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In this article we discussed prominence of Fuzzy Eulerian and Fuzzy Hamiltonian graphs. Fuzzy logic is introduced to study the uncertainty of the event. In Fuzzy set theory we assign a membership value to each element of the set which ranges from 0 to 1. The earnest efforts of the researchers are perceivable in the relevant establishment of the subject integrating coherent practicality and reality. Fuzzy graphs found an escalating number of applications in day to day life system where the information intrinsic in the system varies with different levels of accuracy. In this article we initiated

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Asratian, Armen S., Jonas B. Granholm, and Nikolay K. Khachatryan. "A localization method in Hamiltonian graph theory." Journal of Combinatorial Theory, Series B, May 2020. http://dx.doi.org/10.1016/j.jctb.2020.04.005.

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de Mello Koch, Robert, Eunice Gandote, and Augustine Larweh Mahu. "Scrambling in Yang-Mills." Journal of High Energy Physics 2021, no. 1 (2021). http://dx.doi.org/10.1007/jhep01(2021)058.

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Abstract Acting on operators with a bare dimension ∆ ∼ N2 the dilatation operator of U(N) $$ \mathcal{N} $$ N = 4 super Yang-Mills theory defines a 2-local Hamiltonian acting on a graph. Degrees of freedom are associated with the vertices of the graph while edges correspond to terms in the Hamiltonian. The graph has p ∼ N vertices. Using this Hamiltonian, we study scrambling and equilibration in the large N Yang-Mills theory. We characterize the typical graph and thus the typical Hamiltonian. For the typical graph, the dynamics leads to scrambling in a time consistent with the fast scrambling

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46

Bao, Bo, Rong Chen, and Genghua Fan. "Circuit Covers of Signed Eulerian Graphs." Electronic Journal of Combinatorics 28, no. 1 (2021). http://dx.doi.org/10.37236/9084.

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A signed circuit cover of a signed graph is a natural analog of a circuit cover of a graph, and is equivalent to a covering of its corresponding signed-graphic matroid with circuits. It was conjectured that a signed graph whose signed-graphic matroid has no coloops has a 6-cover. In this paper, we prove that the conjecture holds for signed Eulerian graphs.

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Qiu, Lihong, Yizhe Ji, and Wei Wang. "On the Generalized Spectral Characterizations of Eulerian Graphs." Electronic Journal of Combinatorics 26, no. 1 (2019). http://dx.doi.org/10.37236/8257.

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A graph $G$ is said to be determined by its generalized spectra (DGS for short) if, for any graph $H$, graphs $H$ and $G$ are cospectral with cospectral complements imply that $H$ is isomorphic to $G$. In Wang (J. Combin. Theory, Ser. B, 122 (2017) 438-451), the author gave a simple method for a graph to be DGS. However, the method does not apply to Eulerian graphs. In this paper, we gave a simple method for a large family of Eulerian graphs to be DGS. Numerical experiments are also presented to illustrate the effectiveness of the proposed method.

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Farrell, Matthew, and Lionel Levine. "Multi-Eulerian Tours of Directed Graphs." Electronic Journal of Combinatorics 23, no. 2 (2016). http://dx.doi.org/10.37236/5588.

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Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least once, and uses edges e and f the same number of times whenever tail(e)=tail(f). This definition leads to a simple generalization of the BEST Theorem. We then show that the minimal length of a multi-Eulerian tour is bounded in terms of the Pham index, a measure of 'Eulerianness'.

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Creed, Páidí, and Mary Cryan. "The Number of Euler Tours of Random Directed Graphs." Electronic Journal of Combinatorics 20, no. 3 (2013). http://dx.doi.org/10.37236/2377.

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In this paper we obtain the expectation and variance of the number of Euler tours of a random Eulerian directed graph with fixed out-degree sequence. We use this to obtain the asymptotic distribution of the number of Euler tours of a random $d$-in/$d$-out graph and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows

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Pruesse, Gara, and Frank Ruskey. "The Prism of the Acyclic Orientation Graph is Hamiltonian." Electronic Journal of Combinatorics 2, no. 1 (1995). http://dx.doi.org/10.37236/1199.

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Every connected simple graph $G$ has an acyclic orientation. Define a graph ${AO}(G)$ whose vertices are the acyclic orientations of $G$ and whose edges join orientations that differ by reversing the direction of a single edge. It was known previously that ${AO}(G)$ is connected but not necessarily Hamiltonian. However, Squire proved that the square ${AO}(G)^2$ is Hamiltonian. We prove the slightly stronger result that the prism ${AO}(G) \times e$ is Hamiltonian. If $G$ is a mixed graph (some edges directed, but not necessarily all), then ${AO}(G)$ can be defined as before. The graph ${AO}(G)$

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