Showing papers on "Strongly regular graph published in 1996"
TL;DR: Sharp upper bounds are given on the upper signed domination number of a regular graph and on the signed dominationNumber of a connected cubic graph.
92 citations
TL;DR: The distribution of eigenvalues of regular graphs, regular hypergraphs, and biregular bipartite graphs of given girth is studied by considering the polynomials orthogonal with respect to the measures attached to theSpectra of such graphs and to the continuous spectra of their ‘universal covers’.
Abstract: In this paper we study the distribution of eigenvalues of regular graphs, regular hypergraphs, and biregular bipartite graphs of given girth by considering the polynomials orthogonal with respect to the measures attached to the spectra of such graphs and to the continuous spectra of their ‘universal covers’. Our estimates are tight for Biggs graphs and generalized polygons. We also give an application to the distribution of eigenvalues of Hecke operators acting on weight 2 cusp forms for certain congruence subgroups.
88 citations
TL;DR: It is found out that there are no genuine “global” pseudo-distance-regular graphs: when pseudo- Distance-Regularity is shared by all the vertices, the graph turns out to be distance-regular.
86 citations
01 Jul 1996
TL;DR: It is demonstrated that isomorphism of strongly regular graphs may be tested in time n in light of Neumaier’s claw bound, which implies that low degree stronglyRegular graphs have a small second-largest eigenvalue, unless they are Steiner or Latin square graphs.
Abstract: We demonstrate that isomorphism of strongly regular graphs may be tested in time n~m’’’’ogm). Our approach is to analyze the standard individualization and refinement algorithm in light of Neumaier’s claw bound, which implies that low degree strongly regular graphs have a small second-largest eigenvalue, unless they are Steiner or Latin square graphs.
83 citations
TL;DR: This paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings of closed surfaces exhibiting the highest possible symmetry, and shows that ifGsatisfies certain natural conditions, then all orientable regular embeddeds of its canonical double covering, isomorphic to the tensor productG?K2, can be described in terms of regular embeds ofG.
56 citations
TL;DR: A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte's bound (also called Hoffman's bound), which gives rise to colorings meeting Hoffman's lower bound for the chromatic number.
Abstract: A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte's bound (also called Hoffman's bound). Such spreads give rise to colorings meeting Hoffman's lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences lead to conditions for existence. Most examples come from spreads and fans in (partial) geometries. We give other examples, including a spread in the McLaughlin graph. For strongly regular graphs related to regular two-graphs, spreads give lower bounds for the number of non-isomorphic strongly regular graphs in the switching class of the regular two-graph.
48 citations
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TL;DR: In this article, the eigenvalues of the adjacency matrix and the Laplace matrix of a graph have been studied in the context of algebraic generalizations of strongly regular graphs.
Abstract: Two standard matrix representations of a graph are the adjacency matrix and the Laplace matrix. The eigenvalues of these matrices are interesting parameters of the graph. Graphs with few eigenvalues in general have nice combinatorial properties and a rich structure. A well investigated family of such graphs comprises the strongly regular graphs (the regular graphs with three eigenvalues), and we may see other graphs with few eigenvalues as algebraic generalizations of such graphs. We study the (nonregular) graphs with three adjacency eigenvalues, graphs with three Laplace eigenvalues, and regular graphs with four eigenvalues. The last ones are also studied in relation with three-class association schemes. We also derive bounds on the diameter and on the size of special subsets in terms of the eigenvalues of the graph. Included are lists of feasible parameter sets of graphs with three Laplace eigenvalues, regular graphs with four eigenvalues, and three-class association schemes.
40 citations
01 Dec 1996
TL;DR: It is shown that in this way exactly the class of C-edNCE graph languages (generated by C-EDNCEgraph grammars) is obtained, one of the largest known classes of context-free graph languages.
Abstract: A set of (labeled) graphs can be defined by a regular tree language and one regular string language for each possible edge label, as follows. For each treetfrom the regular tree language the graph gr(t) has the same nodes ast(with the same labels), and there is an edge with label?from nodexto nodeyif the string of labels of the nodes on the shortest path fromxtoyintbelongs to the regular string language for?. Slightly generalizing this definition scheme, we allow gr(t) to have only those nodes oftthat have certain labels, and we allow a relabeling of these nodes. It is shown that in this way exactly the class of C-edNCE graph languages (generated by C-edNCE graph grammars) is obtained, one of the largest known classes of context-free graph languages.
37 citations
TL;DR: It is shown that there exists a constant c such that if A‰f 1 ; 2;:::;ngwithjAj >f (n; 2) = 2 n, then the coprime graph induced by A not only contains a triangle, but also a cycle of length 2l + 1 for every positive integer lcn.
Abstract: In this paper we study cycles in the coprime graph of integers. We denote byf(n;k) the number of positive integers mnwith a prime factor among the flrst k primes. We show that there exists a constant c such that if A‰f 1 ; 2;:::;ngwithjAj >f (n; 2) (if 6jn then f(n; 2) = 2 n), then the coprime graph induced by A not only contains a triangle, but also a cycle of length 2l + 1 for every positive integer lcn.
23 citations
TL;DR: In this paper, it was shown that the exponent of a primitive directed graph with diameter d is at most d 2 + 1 for all positive integers k. This bound was later generalized to the class of irreducible matrices.
TL;DR: In this paper, a quasi-cyclic two-weight code over GF(3) was constructed using heuristic optimization with a local search, a technique which has been successfully employed to obtain many optimal codes.
TL;DR: In this article, the number of non-isomorphic 2-(45, 12, 3, 3) designs with automorphisms of order 5 or 11 was analyzed and a trivial automorphism group of order 360 was discovered.
Abstract: Under the assumption that the incidence matrix of a 2-(45, 12, 3) design has a certain block structure, we determine completely the number of nonisomorphic designs involved. We discover 1136 such designs with trivial automorphism group. In addition we analyze all 2-(45, 12, 3) designs having an automorphism of order 5 or 11. Altogether, the total number of nonisomorphic 2-(45, 12, 3) designs found in 3752. Many of these designs are self-dual and each of these self-dual designs possess a polarity. Some have polarities with no absolute points, giving rise to strongly regular (45, 12, 3, 3) graphs. In total we discovered 58 pairwise nonisomorphic strongly regular graphs, one of which has a trivial automorphism group. Further, we analyzed completely all the designs for subdesigns with parameters 2-(12, 4, 3), 2-(9, 3, 3), and 2-(5, 4, 3). In the first case, the number of 2-(12, 4, 3) subdesigns that a design possessed, if non-zero, turned out to be a multiple of 3, whereas 2-(9, 3, 3) subdesigns were so abundant it was more unusual to find a design without them. Finally, in the case of 2-(5, 4, 3) subdesigns there is a design, unique amongst the ones discovered, that has precisely 9 such subdesigns and these form a partition of the point set of the design. This design has a transitive group of automorphisms of order 360. © 1996 John Wiley & Sons, Inc.
01 Jan 1996
TL;DR: In this paper, a graph (simple, undirected, and loopless) of order v is called strongly regular with parameters v, k,λ,μ whenever it is not complete or edgeless.
Abstract: A graph (simple, undirected, and loopless) of order v is called strongly regular with parameters v, k,λ,μ whenever it is not complete or edgeless.
TL;DR: A new parameter π(G) of graph G and some recursive formulas are introduced as tools, and connected graphs with π ( G ) = 0 or 1 are characterized and the chromatic uniqueness of certain kinds of graphs is discussed.
TL;DR: In this paper, the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes is investigated, and a classification of association schemes whose symmetrizedness is obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes is given.
Abstract: If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes. In particular, we give a classification of association schemes whose symmetrizations are obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes. Also the cyclotomic schemes associated to Paley graphs and their symmetrizable fission schemes are discussed in terms of their character tables.
TL;DR: In this article, a notion of integrality, or resonance, for finite commutative hypergroups and their generalizations, signed hypergroups, was introduced, and Lagrange's theorem for sub-hypergroups was established using a condition on integrality of weights for the dual signed hypergroup.
TL;DR: All binary projective two-weight codes with parameters [27, 6,12] and [35,6,16] are enumerated up to equivalence, and the automorphism groups of the codes and related strongly regular graphs computed.
TL;DR: The “height” of a graphG is defined to be the number of steps to construct G by two simple graph operations such that every graph in G has height at most h.
TL;DR: This work investigates resolvable regular graph designs with block size 4.5 and determines the parameters for which such designs exist, for v ⩽ 16.
TL;DR: A graph G is called ( K 3 , K 3 )-co-critical if the edges of G can be coloured with two colours without getting a monochromatic triangle, but adding any new edge to the graph, this kind of 'good' colouring is impossible.
TL;DR: A quasi-cyclic two-weight code over GF(3) is presented and it is shown that these codes result in strongly regular graphs if the code is projective.
TL;DR: It is shown that the number of columns (1,k − 2, 1) in the intersection array of distance-regular graphs is at most 20.
Abstract: We show that the number of columns (1,k � 2, 1) in the intersection array of distance-regular graphs is at most 20
TL;DR: It is proved that every connected strongly regular graph with even order is 2-extendable when k ⩾ 3, except the Petersen graph, the (6, 4, 2, 4) graph and K 4.
TL;DR: It is shown that the problem to find the T-span for a complete graph is NP-complete.
TL;DR: If G is odd order and regular of degree d ⩾[ 37−1) 6]|V(G)| , then a necessary and sufficient condition for ξ T ( G ) = d + 1 is given.
TL;DR: For every positive integern, the construction of a strongly regular graph of order at most 2n+2 which contains everygraph of ordern as a subgraph is shown.
Abstract: For every positive integern we show the construction of a strongly regular graph of order at most 2
n+2 which contains every graph of ordern as a subgraph. The estimation concerning the construction is best possible.