Regular Bipartite Graphs Research Articles

Periodicity of bipartite walk on biregular graphs with conditional spectra

In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover’s walks. A discrete quantum walk is given by the powers of a unitary matrix U indexed by arcs or edges of the underlying graph. The walk is periodic if U k = I for some positive integer k. Kubota has given a characterization of periodicity of Grover’s walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota’s results—if a biregular graph G has eigenvalues whose squares are algebraic integers with degree at most two, we characterize periodicity of the bipartite walk over G in terms of its spectrum. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover’s walk on regular graphs.

On an infinite family of integral Cayley graphs of Pauli groups

The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph Cay(Pn,SPn) of the generalized Pauli group Pn on n-qubits; in fact, Pn may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set SPn allows us to recognize the structure of central product of graphs in Cay(Pn,SPn).Using this approach, we are able to recursively construct the adjacency matrix of Cay(Pn,SPn) for each n≥1, and to explicitly describe its spectrum and the associated eigenvectors. It turns out that Cay(Pn,SPn) is a (3n+2)-regular bipartite graph on 4n+1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral.

The local vertex anti-magic coloring for certain graph operations

This work proves the local vertex anti-magic coloring of even regular circulant bipartite graphs C(m;L). Let G be either Kr,r or Kr,r−F, F is a 1-factor. Also, we discover the local vertex anti-magic coloring for union of bipartite graphs; join graphs G∨H, where H∈{Or,Kr,Cr,Kr,s}; and the upper bound of corona product G⊙Or. It was a problem Lau and Shiu (2023) [1] that: For any G1 and G2, determine χℓva(G1×G2). We give partial answer to this problem by proving the followings:1.χℓva(C2m×C2n);2.χℓva(C2m+1×C2n+2); and3.χℓva(P3×H), where H∈{Kr,Km,m}.

Anti-Ramsey number for perfect matchings in 3-regular bipartite graphs

The anti-Ramsey number AR(G,H) is the maximum number of colors in an edge-coloring of a graph in the family G without any rainbow copy of H. The anti-Ramsey number for matchings has been studied extensively in several graph families, while the problem for perfect matchings was only solved in complete graphs. The anti-Ramsey number of matchings in the family of regular bipartite graphs of order large enough was determined by Li and Xu. Jin improved their bounds on the order of 3-regular bipartite graphs. In this paper, we consider the problem for perfect matchings in 3-regular bipartite graphs. Let G be the family of 3-regular bipartite graphs with m vertices in each partite set. First, we characterize the structure of extremal graphs for the Turán number of perfect matchings in graphs with maximum degree three. Using this characterization, we show that AR(G,mK2)=3m−3. Moreover, we show that AR(F,mK2)=3m−5, where F is the subfamily of 3-edge-connected graphs in G. In order to prove this result, we characterize the structure of bipartite graphs, which contains no perfect matchings, with maximum degree three and size Turán number minus one.

Subgraph of unitary Cayley graph of matrix algebras induced by idempotent matrices

In this work, we are interested in the subgraph of unitary Cayley graph of a matrix algebra over a finite field induced by the set of idempotent matrices. We study its structure and completely obtain all connected components with the degree on each vertex. Most of them turn out to be a regular bipartite graph such that each partite set has the same number of vertices. Our main tools are combinatorial results on matrix algebras. Moreover, we compute the clique number, chromatic number and independence number of the graph.

On security of GIS systems with N-tier architecture and family of graph based ciphers

Discovery of q-regular tree description in terms of an infinite system of quadratic equations over finite field Fq had an impact on Computer Science, theory of graph based cryptographic algorithms in particular. It stimulates the development of new graph based stream ciphers. It turns out that such encryption instruments can be efficiently used in GIS protection systems which use N-Tier Architecture. We observe known encryption algorithms based on the approximations of regular tree, their modifications defined over arithmetical rings and implementations of these ciphers. Additionally new more secure graph based ciphers suitable for GIS protection will be presented.Algorithms are constructed using vertices of bipartite regular graphs D(n,K) defined by a finite commutative ring K with a unit and a non-trivial multiplicative group K*. The partition of such graphs are n-dimensional affine spaces over the ring K. A walk of even length determines the transformation of the transition from the initial to the last vertex from one of the partitions of the graph. Therefore, the affine space Kn can be considered as a space of plaintexts, and walking on the graph is a password that defines the encryption transformation.With certain restrictions on the password the effect when different passwords with K*)2s, s <[(n+5)/2]/2 correspond to different ciphertexts of the selected plaintext with Kn can be achieved.In 2005, such an algorithm in the case of the finite field F127 was implemented for the GIS protection. Since then, the properties of encryption algorithms using D(n, K) graphs (execution speed, mixing properties, degree and density of the polynomial encryption transform) have been thoroughly investigated.The complexity of linearization attacks was evaluated and modifications of these algorithms with the resistance to linearization attacks were found. It turned out that together with D(n, K) graphs, other algebraic graphs with similar properties, such as A(n, K) graphs, can be effectively used.The article considers several solutions to the problem of protecting the geological information system from possible cyberattacks using stream ciphers based on graphs. They have significant advantages compared to the implemented earlier systems.

On the Locating Rainbow Connection Number of Trees and Regular Bipartite Graphs

Locating the rainbow connection number of graphs is a new mathematical concept that combines the concepts of the rainbow vertex coloring and the partition dimension. In this research, we determine the lower and upper bounds of the locating rainbow connection number of a graph and provide the characterization of graphs with the locating rainbow connection number equal to its upper and lower bounds to restrict the upper and lower bounds of the locating rainbow connection number of a graph. We also found the locating rainbow connection number of trees and regular bipartite graphs. The method used in this study is a deductive method that begins with a literature study related to relevant previous research concepts and results, making hypotheses, conducting proofs, and drawing conclusions. This research concludes that only path graphs with orders 2, 3, 4, and complete graphs have a locating rainbow connection number equal to 2 and the order of graph G, respectively. We also showed that the locating rainbow connection number of bipartite regular graphs is in the range of r-⌊n/4⌋+2 to n/2+1, and the locating rainbow connection number of a tree is determined based on the maximum number of pendants or the maximum number of internal vertices. Doi: 10.28991/ESJ-2023-07-04-016 Full Text: PDF

Distributed Average Consensus Algorithms in d-Regular Bipartite Graphs: Comparative Study

Consensus-based data aggregation in d-regular bipartite graphs poses a challenging task for the scientific community since some of these algorithms diverge in this critical graph topology. Nevertheless, one can see a lack of scientific studies dealing with this topic in the literature. Motivated by our recent research concerned with this issue, we provide a comparative study of frequently applied consensus algorithms for distributed averaging in d-regular bipartite graphs in this paper. More specifically, we examine the performance of these algorithms with bounded execution in this topology in order to identify which algorithm can achieve the consensus despite no reconfiguration and find the best-performing algorithm in these graphs. In the experimental part, we apply the number of iterations required for consensus to evaluate the performance of the algorithms in randomly generated regular bipartite graphs with various connectivities and for three configurations of the applied stopping criterion, allowing us to identify the optimal distributed consensus algorithm for this graph topology. Moreover, the obtained experimental results presented in this paper are compared to other scientific manuscripts where the analyzed algorithms are examined in non-regular non-bipartite topologies.

Edge proximity conditions for extendability in regular bipartite graphs

AbstractLet and be positive integers with , let be an ‐regular cyclically ‐edge‐connected bipartite graph and let be a matching of size in . Plummer showed that whenever , there is a perfect matching of containing . When , Aldred and Jackson, extended this result to the case when by showing there is a perfect matching in containing whenever the edges in are pairwise at least distance apart where In this paper, we relax the condition that and the distance restriction introduced by Aldred and Jackson to show that, for and an ‐regular cyclically ‐edge‐connected bipartite graph, for each matching in with and such that each pair of edges in is distance at least 3 apart, there is a perfect matching in containing .

Simultaneous coloring of vertices and incidences of outerplanar graphs

A v i -simultaneous proper k -coloring of a graph G is a coloring of all vertices and incidences of the graph in which any two adjacent or incident elements in the set V ( G )∪ I ( G ) receive distinct colors, where I ( G ) is the set of incidences of G . The v i -simultaneous chromatic number, denoted by χ v i ( G ) , is the smallest integer k such that G has a v i -simultaneous proper k -coloring. In [M. Mozafari-Nia, M. N. Iradmusa, A note on coloring of 3/3 -power of subquartic graphs, Vol. 79, No.3, 2021] v i -simultaneous proper coloring of graphs with maximum degree 4 is investigated and they conjectured that for any graph G with maximum degree Δ ≥ 2 , v i -simultaneous proper coloring of G is at most 2 Δ + 1 . In [M. Mozafari-Nia, M. N. Iradmusa, Simultaneous coloring of vertices and incidences of graphs, arXiv:2205.07189, 2022] the correctness of the conjecture for some classes of graphs such as k -degenerated graphs, cycles, forests, complete graphs, regular bipartite graphs is investigated. In this paper, we prove that the v i -simultaneous chromatic number of any outerplanar graph G is either Δ + 2 or Δ + 3 , where Δ is the maximum degree of G .

Inapproximability of Counting Hypergraph Colourings

Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovász Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this article is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting q -colourings on K -uniform hypergraphs with bounded degree Δ. An efficient algorithm exists if \({{\Delta \lesssim \frac{q^{K/3-1}}{4^KK^2}}}\) [Jain et al. 25 ; He et al. 23 ]. Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of K . To this end, we first establish that for general q computational hardness for finding a colouring on simple/linear hypergraphs occurs at Δ ≳ Kq K , almost matching the algorithm from the Lovász Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture is asymptotically tight (up to constant factors). We show in particular that for all even q ≥ 4 it is NP -hard to approximate the number of colourings when Δ ≳ q K/2 . Our approach is based on considering an auxiliary weighted binary CSP model on graphs, which is obtained by “halving” the K -ary hypergraph constraints. This allows us to utilise reduction techniques available for the graph case, which hinge upon understanding the behaviour of random regular bipartite graphs that serve as gadgets in the reduction. The major challenge in our setting is to analyse the induced matrix norm of the interaction matrix of the new CSP which captures the most likely solutions of the system. In contrast to previous analyses in the literature, the auxiliary CSP demonstrates both symmetry and asymmetry, making the analysis of the optimisation problem severely more complicated and demanding the combination of delicate perturbation arguments and careful asymptotic estimates.

Generalized Paley graphs equienergetic with their complements

We consider generalized Paley graphs , generalized Paley sum graphs , and their corresponding complements and , for k = 3, 4. Denote by either or . We compute the spectra of and and from them we obtain the spectra of and also. Then we show that, in the non-semiprimitive case, the spectrum of and with p prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs and for any , respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of such that and are equienergetic for k = 3, 4. In a previous work we have classified all bipartite regular graphs and all strongly regular graphs which are complementary equienergetic, i.e. and are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs which are neither bipartite nor strongly regular.

Permanents of almost regular complete bipartite graphs

Let G be a graph, and let A ( G ) be the adjacency matrix of G. The computation of permanent of A ( G ) is #p-complete. Computing permanent of A ( G ) is of great interest in quantum chemistry, statistical physics, among other disciplines. In this paper, we characterize the ordering of permanents of adjacency matrices of all graphs obtained from regular complete bipartite graph K p , p by deleting six edges. As an application, we show that all graphs with a perfect matching obtained from K p , p with six edges deleted are determined by their permanental polynomials.

Sharp Poincaré and log-Sobolev inequalities for the switch chain on regular bipartite graphs

Consider the switch chain on the set of d-regular bipartite graphs on n vertices with $$3\le d\le n^{c}$$ , for a small universal constant $$c>0$$ . We prove that the chain satisfies a Poincaré inequality with a constant of order O(nd); moreover, when d is fixed, we establish a log-Sobolev inequality for the chain with a constant of order $$O_d(n\log n)$$ . We show that both results are optimal. The Poincaré inequality implies that in the regime $$3\le d\le n^c$$ the mixing time of the switch chain is at most $$O\big ((nd)^2 \log (nd)\big )$$ , improving on the previously known bound $$O\big ((nd)^{13} \log (nd)\big )$$ due to Kannan et al. (Rand Struct Algorithm 14(4):293–308, 1999) and $$O\big (n^7d^{18} \log (nd)\big )$$ obtained by Dyer et al. (Sampling hypergraphs with given degrees (preprint). arXiv:2006.12021 ). The log-Sobolev inequality that we establish for constant d implies a bound $$O(n\log ^2 n)$$ on the mixing time of the chain which, up to the $$\log n$$ factor, captures a conjectured optimal bound. Our proof strategy relies on building, for any fixed function on the set of d-regular bipartite simple graphs, an appropriate extension to a function on the set of multigraphs given by the configuration model. We then establish a comparison procedure with the well studied random transposition model in order to obtain the corresponding functional inequalities. While our method falls into a rich class of comparison techniques for Markov chains on different state spaces, the crucial feature of the method—dealing with chains with a large distortion between their stationary measures—is a novel addition to the theory.

Positive matching decompositions of graphs

A matching M in a graph Γ is positive if Γ has a vertex-labeling such that M coincides with the set of edges with positive weights. A positive matching decomposition (pmd) of Γ is an edge-partition M 1 , … , M p of Γ such that M i is a positive matching in Γ − ( M 1 ∪ ⋯ ∪ M i − 1 ) , for i = 1 , … , p . The pmds of graphs are used to study algebraic properties of the Lovász–Saks–Schrijver ideals arising from orthogonal representations of graphs. We give a characterization of pmds of graphs in terms of alternating closed walks and apply it to study pmds of various classes of graphs including complete multipartite graphs, (regular) bipartite graphs, cacti, generalized Petersen graphs, etc. We further show that computation of pmds of a graph can be reduced to that of its maximum pendant-free subgraph.

Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

We determine connected bipartite regular graphs with four distinct adjacency eigenvalues that induce periodic Grover walks, and show that it is only C6. We also show that there are only three kinds of the second largest eigenvalues of bipartite regular periodic graphs with five distinct eigenvalues. Using walk-regularity, we enumerate feasible spectra for such graphs.

Anti-Ramsey numbers for cycles in the generalized Petersen graphs

For H⊆G, the anti-Ramsey number ar(G,H) is the maximum number of colors in an edge-coloring of G such that each subgraph isomorphic to H has at least two edges in the same color. The study of anti-Ramsey number ar(Kn,H) was introduced by Erdős et al. in 1973, and plentiful results were researched for some special graph H. Later, the problem was extended to ar(G,H) when replacing Kn by other graph G such as hypergraph, complete split graph, regular bipartite graph, triangulation and so on. In this paper, we consider a generalized Petersen graph Pn,k as the host graph and determine the exact anti-Ramsey numbers for cycles C5 and C6 in Pn,k, respectively.

Generalized polygons and star graphs of cyclic presentations of groups

Groups defined by presentations for which the components of the corresponding star graph are the incidence graphs of generalized polygons are of interest as they are small cancellation groups that – via results of Edjvet and Vdovina – are fundamental groups of polyhedra with the generalized polygons as links and so act on Euclidean or hyperbolic buildings; in the hyperbolic case the groups are SQ-universal. A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We obtain a classification of the concise cyclic presentations where the components of the corresponding star graph are generalized polygons. The classification reveals that both connected and disconnected star graphs are possible and that only generalized triangles (i.e. incidence graphs of projective planes) and regular complete bipartite graphs arise as the components. We list the presentations that arise in the Euclidean case and show that at most two of the corresponding groups are not SQ-universal (one of which is not SQ-universal, the other is unresolved). We obtain results that show that many of the SQ-universal groups are large.

Constructing the Mate of Cospectral 5-regular Graphs with and without a Perfect Matching

The problem of finding a perfect matching in an arbitrary simple graph is well known and popular in graph theory. It is used in various fields, such as chemistry, combinatorics, game theory etc. The matching of M in a simple graph G is a set of pairwise nonadjacent edges, ie, those that do not have common vertices. Matching is called perfect if it covers all vertices of the graph, ie each of the vertices of the graph is incidental to exactly one of the edges. By Koenig's theorem, regular bipartite graphs of positive degree always have perfect matching. However, graphs that are not bipartite need further research.
 Another interesting problem of graph theory is the search for pairwise nonisomorphic cospectral graphs. In addition, it is interesting to find cospectral graphs that have additional properties. For example, finding cospectral graphs with and without a perfect matching.
 The fact that for each there is a pair of cospectral connected k-regular graphs with and without a perfect matching had been investigated by Blazsik, Cummings and Haemers. The pair of cospectral connected 5-regular graphs with and without a perfect matching is constructed by using Godsil-McKay switching in the paper.

Disproof of a Conjecture of Erdős and Simonovits on the Turán Number of Graphs with Minimum Degree 3

Abstract In 1981, Erdős and Simonovits conjectured that for any bipartite graph $H$, we have $\textrm {ex}(n,H)=O(n^{3/2})$ if and only if $H$ is $2$-degenerate. Later, Erdős offered $250 for a proof and $500 for a counterexample. In this paper, we disprove the conjecture by finding, for any $\varepsilon>0$, a $3$-regular bipartite graph $H$ with $\textrm {ex}(n,H)=O(n^{4/3+\varepsilon })$.