Showing papers on "Strongly regular graph published in 2008"

A Classical approach to the graph isomorphism problem using quantum walks

Abstract: Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.

On Regular Fuzzy Graphs

Abstract: In this paper, regular fuzzy graphs, total degree and totally regular fuzzy graphs are introduced. Regular fuzzy graphs and totally regular fuzzy graphs are compared through various examples. A necessary and sufficient condition under which they are equivalent is provided. A characterization of regular fuzzy graphs on a cycle is provided. Some properties of regular fuzzy graphs are studied and they are examined for totally regular fuzzy graphs.

A connection between ordinary and Laplacian spectra of bipartite graphs

Abstract: Let G be a bipartite graph with n vertices and m edges. Let S(G) be the subdivision of G, obtained by inserting a new vertex on each edge of G. The ordinary characteristic polynomial of S(G) and the Laplacian characteristic polynomial of G are related as φ(S(G),λ) = λm−nψ(G,λ2). If μi, i=1,2, … ,h, are the non-zero Laplacian eigenvalues of G, then the ordinary spectrum of S(G) consists of the numbers , and of n+m−2h zeros. As a corollary, we demonstrate that if ψ (G,λ) is written in the form , then for any tree T of order n and for any k, ck(Sn)≤ ck(T) ≤ ck(Pn), where Sn and Pn are, respectively, the star and the path of order n.

The spectral excess theorem for distance-regular graphs: a global (over)view

Abstract: Distance-regularity of a graph is in general not determined by the spectrum of the graph. The spectral excess theorem states that a connected regular graph is distance-regular if for every vertex, the number of vertices at extremal distance (the excess) equals some given expression in terms of the spectrum of the graph. This result was proved by Fiol and Garriga [From local adjacency polynomials to locally pseudo-distance-regular graphs, J. Combinatorial Th. B 71 (1997), 162-183] using a local approach. This approach has the advantage that more general results can be proven, but the disadvantage that it is quite technical. The aim of the current paper is to give a less technical proof by taking a global approach.

A note on energy of some graphs

Abstract: Eigenvalue of a graph is the eigenvalue of its adjacency matrix. The energy E(G) of a graph G is the sum of the absolute values of its eigenvalues. In this paper we obtain analytic expressions for the energy of two classes of regular graphs.

Ring geometries, two-weight codes, and strongly regular graphs

Abstract: It is known that a projective linear two-weight code C over a finite field $${\mathbb{F}}_q$$ corresponds both to a set of points in a projective space over $${\mathbb{F}}_q$$ that meets every hyperplane in either a or b points for some integers a < b, and to a strongly regular graph whose vertices may be identified with the codewords of C. Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homogeneous weights and sets of points in an associated projective ring geometry. We will introduce regular projective two-weight codes over finite Frobenius rings, we will show that such a code gives rise to a strongly regular graph, and we will give some constructions of two-weight codes using ring geometries. All these examples yield infinite families of strongly regular graphs with non-trivial parameters.

Strong difference families over arbitrary graphs

Abstract: The concept of a strong difference family formally introduced in Buratti [J Combin Designs 7 (1999), 406–425] with the aim of getting group divisible designs with an automorphism group acting regularly on the points, is here extended for getting, more generally, sharply-vertex-transitive Γ-decompositions of a complete multipartite graph for several kinds of graphs Γ. We show, for instance, that if Γ has e edges, then it is often possible to get a sharply-vertex-transitive Γ-decomposition of Km × e for any integer m whose prime factors are not smaller than the chromatic number of Γ. This is proved to be true whenever Γ admits an α-labeling and, also, when Γ is an odd cycle or the Petersen graph or the prism T5 or the wheel W6. We also show that sometimes strong difference families lead to regular Γ-decompositions of a complete graph. We construct, for instance, a regular cube-decomposition of K16m for any integer m whose prime factors are all congruent to 1 modulo 6. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 443–461, 2008

Semisymmetric cubic graphs as regular covers of K 3,3

Abstract: A regular graph X is called semisymmetric if it is edge-transitive but not vertex-transitive. For G ≤ AutX, we call a G-cover X semisymmetric if X is semisymmetric, and call a G-cover X one-regular if AutX acts regularly on its arc-set. In this paper, we give the sufficient and necessary conditions for the existence of one-regular or semisymmetric Z n -covers of K 3,3. Also, an infinite family of semisymmetric Z n × Z n -covers of K 3,3 are constructed.

Cores of Geometric Graphs

Abstract: Cameron and Kazanidis have recently shown that rank-3 graphs are either cores or have complete cores, and they asked whether this holds for all strongly regular graphs. We prove that this is true for the point graphs and line graphs of generalized quadrangles and that when the number of points is sufficiently large, it is also true for the block graphs of Steiner systems and orthogonal arrays.

There are exactly five biplanes with k = 11

Abstract: A biplane is a 2-(k(k − 1)/2 + 1,k,2) symmetric design. Only sixteen nontrivial biplanes are known: there are exactly nine biplanes with k < 11, at least five biplanes with k = 11, and at least two biplanes with k = 13. It is here shown by exhaustive computer search that the list of five known biplanes with k = 11 is complete. This result further implies that there exists no 3-(57, 12, 2) design, no 11211 symmetric configuration, and no (324, 57, 0, 12) strongly regular graph. The five biplanes have 16 residual designs, which by the Hall–Connor theorem constitute a complete classification of the 2-(45, 9, 2) designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 117–127, 2008

Maximum acyclic and fragmented sets in regular graphs

Abstract: We show that a typical d-regular graph G of order n does not contain an induced forest with around 2Ind/d vertices, when n >> d >> 1, this bound being best possible because of a result of Frieze and Luczak [6]. We then deduce an affirmative answer to an open question of Edwards and Farr (see [4]) about fragmentability, which concerns large subgraphs with components of bounded size. An alternative, direct answer to the question is also given. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 149156, 2008

Can a Graph Have Distinct Regular Partitions

Abstract: The regularity lemma of Szemeredi gives a concise approximate description of a graph via a so-called regular partition of its vertex set. In this paper we address the following problem: Can a graph have two “distinct” regular partitions? It turns out that (as observed by several researchers) for the standard notion of a regular partition, one can construct a graph that has very distinct regular partitions. On the other hand, we show that for the stronger notion of a regular partition that has been recently studied, all such regular partitions of the same graph must be very “similar.” En route, we also give a short argument for deriving a recent variant of the regularity lemma obtained independently by Rodl and Schacht and by Lovasz and Szegedy from a previously known variant of the regularity lemma due to Alon et al. in 2000. The proof also provides a deterministic polynomial time algorithm for finding such partitions.

Properties of graphs with large girth

Abstract: This thesis is devoted to the analysis of a class of iterative probabilistic algorithms in regular graphs, called locally greedy algorithms, which will provide bounds for graph functions in regular graphs with large girth. This class is useful because, by conveniently setting the parameters associated with it, we may derive algorithms for some well-known graph problems, such as algorithms to find a large independent set, a large induced forest, or even a small dominating set in an input graph G. The name "locally greedy" comes from the fact that, in an algorithm of this class, the probability associated with the random selection of a vertex v is determined by the current state of the vertices within some fixed distance of v. Given r ≥ 3 and an r-regular graph G, we determine the expected performance of a locally greedy algorithm in G, depending on the girth g of the input and on the degree r of its vertices. When the girth of the graph is sufficiently large, this analysis leads to new lower bounds on the independence number of G and on the maximum number of vertices in an induced forest in G, which, in both cases, improve the bounds previously known. It also implies bounds on the same functions in graphs with large girth and maximum degree r and in random regular graphs. As a matter of fact, the asymptotic lower bounds on the cardinality of a maximum induced forest in a random regular graph improve earlier bounds, while, for independent sets, our bounds coincide with asymptotic lower bounds first obtained by Wormald. Our result provides an alternative proof of these bounds which avoids sharp concentration arguments. The main contribution of this work lies in the method presented rather than in these particular new bounds. This method allows us, in some sense, to directly analyse prioritised algorithms in regular graphs, so that the class of locally greedy algorithms, or slight modications thereof, may be applied to a wider range of problems in regular graphs with large girth.