Showing papers on "Strongly regular graph published in 1995"
TL;DR: In this article, the authors consider the problem of finding all pairs of nodes connected by a simple path such that the concatenation of the labels along the path satisfies a regular expression.
Abstract: We consider the following problem: given a labelled directed graph $G$ and a regular expression $R$, find all pairs of nodes connected by a simple path such that the concatenation of the labels along the path satisfies $R$ The problem is motivated by the observation that many recursive queries in relational databases can be expressed in this form, and by the implementation of a query language, ${\bf G}^+$, based on this observation We show that the problem is in general intractable, but present an algorithm than runs in polynomial time in the size of the graph when the regular expression and the graph are free of conflicts We also present a class of languages whose expressions can always be evaluated in time polynomial in the size of both the graph and the expression, and characterize syntactically the expressions for such languages
396 citations
TL;DR: In this article, the authors studied connected regular graphs with four distinct eigenvalues, and properties and feasibility conditions of the eigen values were given, as well as some uniqueness and nonexistence results.
105 citations
TL;DR: A graph G is vertex domination-critical if for any vertex v of G the domination number of G - v is less than the dominationNumber of G, which is called γ-critical.
Abstract: A graph G is vertex domination-critical if for any vertex v of G the domination number of G - v is less than the domination number of G. If such a graph G has domination number γ, it is called γ-critical. Brigham et al. studied γ-critical graphs and posed the following questions: (1) If G is a γ-critical graph, is |V| ≥ (δ + 1)(γ - 1) + 1?(2) If a γ-critical graph G has (Δ + 1)(γ - 1) + 1 vertices, is G regular? (3) Does i = γ for all γ-critical graphs? (4) Let d be the diameter of the γ-critical graph G. Does d ≤ 2(γ - 1) always hold? We show that the first and third questions have a negative answer and the others have a positive answer.
48 citations
TL;DR: In this article, it was shown that for every t and g there are t-tough graphs of girth strictly greater than g. This strengthened a recent result of Bauer, van den Heuvel and Schmeichel who proved the above for g e 3, and hence disproved in a strong sense a conjecture of Chvatal that there exists an absolute constant t 0 so that every t 0 -tough graph is pancyclic.
Abstract: A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. It is shown that for every t and g there are t-tough graphs of girth strictly greater than g. This strengthens a recent result of Bauer, van den Heuvel and Schmeichel who proved the above for g e 3, and hence disproves in a strong sense a conjecture of Chvatal that there exists an absolute constant t0 so that every t0-tough graph is pancyclic. The proof is by an explicit construction based on the tight relationship between the spectral properties of a regular graph and its expansion properties. A similar technique provides a simple construction of triangle-free graphs with independence number m on Ω(m4/3) vertices, improving previously known explicit constructions by Erdos and by Chung, Cleve and Dagum.
46 citations
TL;DR: In this article, the authors study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme.
Abstract: Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.
45 citations
TL;DR: In this article, the authors determined all graphs with the spectrum of a distance-regular graph with at most 30 vertices, except possibly for the Taylor graph on 28 vertices.
Abstract: We determine all graphs with the spectrum of a distance-regular graph with at most 30 vertices (except possibly for the Taylor graph on 28 vertices)
42 citations
TL;DR: The o-tolerance competition graph as discussed by the authors is a generalization of the p-competition graph defined by Kim, McKee, McMorris, and Roberts, and it is defined as a symmetric function defined from N × N into N, where N denotes the nonnegative integers.
40 citations
TL;DR: This work investigates the maximum value of the product of the independent domination numbers of a graph and its complement, as a function of n, and proves that if G is regular then i(G) · i( G ) 2 /12.68 .
30 citations
TL;DR: In this paper, the p-rank of strongly regular graphs with adjacency matrix A + cI for integral c was studied and it was shown that for any strongly regular graph with a given parameter set, there are no graphs for which a p -rank is minimal.
30 citations
01 Jan 1995
TL;DR: A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte's bound (alsocalled Hoffman's bound), which gives lower bounds for the number of non-isomorphic strongly regular graphs in the switching class of the regular two-graph.
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.
28 citations
TL;DR: This work studies the existence conditions of strongly closed subgraphs in highly regular graphs such as distance-regular graphs or distance-biregular graphs.
Abstract: A geodetically closed induced subgraph Δ of a graph Γ is defined to be strongly closed if Γ i ( α ) ∩ Γ 1 ( β ) stays in Δ for every i and α, β ϵ Δ with ∂ ( α , β ) = i . We study the existence conditions of strongly closed subgraphs in highly regular graphs such as distance-regular graphs or distance-biregular graphs.
02 Mar 1995
TL;DR: A random graph of type-H is obtained by selecting edges of H independently and with probability p and this can be represented as a communication network H in which the links fail independently andwith probability f=1−p.
Abstract: Let H be an undirected graph. A random graph of type-H is obtained by selecting edges of H independently and with probability p. We can thus represent a communication network H in which the links fail independently and with probability f=1−p. A fundamental type of H is the clique of n nodes (leading to the well-known random graph G n,p ). Another fundamental type of H is a random member of the set G n d of all regular graphs of degree d (leading to a new type of random graphs, of the class G n,p d ). Note that G n,p =G n,p n−1 . The G n,p d model was introduced in ([11]).
TL;DR: This work establishes the matching uniqueness of many specific regular graphs; each of these graphs is either a cage, or a graph whose components are isomorphic to Moore graphs.
TL;DR: It is shown that the independence ratio for connected 3-regular triangle-free graphs must be at least 11/30 - 2/15n, where n is the number of vertices in the graph.
TL;DR: In this paper, the authors showed that there is a one-to-one correspondence between sets D closed under multiplication by elements of the prime field of F (and yielding a strongly regular F), and subsets of projective spaces such that the cardinality of the intersection with a hyperplane takes only two values.
Abstract: Given a finite field F and a subset D of F* such that D = —D, we can define a graph F with vertex set F by letting x ~ y whenever y x 6 D. (Here ~ denotes adjacency.) The spectrum of F consists of the numbers £deD X (d), where x runs through the (additive) characters of F. In particular, the trivial character x0 yields the eigenvalue | D \, the valency of F. One might wonder in what cases the graph F is strongly regular, and there has been done a lot of work on this question, see, e.g., Delsarte [4], van Lint and Schrijver [6], Calderbank and Kantor [3], Brouwer [1], de Resmini [7] and de Resmini and Migliori [8]. (As Delsarte showed, there is a one-to-one correspondence between (i) sets D closed under multiplication by elements of the prime field of F (and yielding a strongly regular F), and (ii) projective two-weight codes, and (iii) subsets of projective spaces such that the cardinality of the intersection with a hyperplane takes only two values. Work on this problem occurs in each of these three terminologies.) In [5], we constructed four new examples, which will be described below. Our sets D will be unions of a number of cosets of a subgroup K of F*, i.e., D = Z K for some set Z c F*. The field F is described by its characteristic p and a primitive polynomial defining it over its prime field. For the resulting strongly regular graphs we give the standard parameters v, k, A, m, r, s, f, g (cf. Brouwer and van Lint [2]).
TL;DR: The purpose of this paper is to classify isomorphism of regular graph covering projections of a graph G that arise from finite abelian groups as voltage groups if the voltage group is a finite-dimensional vector space over a finite field and the automorphism group is trivial.
TL;DR: Several new ternary linear codes of dimension 6 are found, including one two-weight code giving rise to a new strongly regular graph.
Abstract: It is proved that ternary codes with parameters [40,5,25] and [148,5,98] do not exist. A new ternary code is constructed with parameters [47.5,30]. The results solve the problem of finding the smallest n for which a ternary [n, 5, d] code exists for d=25; 29, 30, and 98. Several new ternary linear codes of dimension 6 are found, including one two-weight code giving rise to a new strongly regular graph.
TL;DR: An extension of this theory of equitable partitions of graphs to matrix algebras acting on the space of functions on a finite set is developed to establish a quotient theorem which has standard results on quotients in association schemes and distance regular graphs.
TL;DR: It is presented necessary conditions for { HL k ( G ) to converge to a connected limit graph and sufficient conditions for the sequence { HLk ( G)} to diverge.
TL;DR: In this article, the authors investigated regular graphs with a star partition X 1 ∪ X 2 ∪ √ X m such that G − Xi is regular for some i.i.d.
01 Jan 1995
TL;DR: Spectral graph theory has a long history in representation theory and number theory and has been very useful for examining the spectra of strongly regular graphs with symmetries as mentioned in this paper, however, spectral graph theory is mostly algebraic.
Abstract: The study of eigenvalues of graphs has a long history. Since the early days, representation theory and number theory have been very useful for examining the spectra of strongly regular graphs with symmetries. In contrast, recent developments in spectral graph theory concern the effectiveness of eigenvalues in studying general (unstructured) graphs. The concepts and techniques, in large part, use essentially geometric methods.(Still, extremal and explicit constructions are mostly algebraic [20].) There has been a significant increase in the interaction between spectral graph theory and many areas of mathematics as well as other disciplines, such as physics, chemistry, communication theory, and computer sciences.
TL;DR: In this paper, it was shown that a distance-regular graph with vertices of valency 3 can be denoted by at most four parameters, i.e., intersection numbers ci, ai, bi (0 ⩽ i⩽ d) can be defined by four parameters.
Abstract: We show that a distance-regular graph Г with Γ(x) ≃ 3 ∗ K a1+1 (a 1 ⩾ 2) for every x ∈ Γ and d ⩾ r(Γ) + 3 is a distance-2 graph of a distance-biregular graph with vertices of valency 3. In particular, intersection numbers ci, ai, bi (0 ⩽ i ⩽ d) can be denoted by at most four parameters.
TL;DR: In this article, it was shown that every regular 3-valent polyhedral graph has a cycle through at least 4/5 of its vertices, where vertices are all 5gons and 6gons.
Abstract: It is shown that every regular 3-valent polyhedral graph whose faces are all 5-gons and 6-gons contains a cycle through at least 4/5 of its vertices.
TL;DR: In this paper, a canonical basis of R n associated with a graph G on n vertices has been defined in connection with eigenspaces and star partitions of G. The canonical star basis together with eigenvalues of G determines G to an isomorphism.
Abstract: A canonical basis of R n associated with a graph G on n vertices has been defined in [15] in connection with eigenspaces and star partitions of G. The canonical star basis together with eigenvalues of G determines G to an isomorphism. We study algorithms for finding the canonical basis and some of its variations. The emphasis is on the following three special cases; graphs with distinct eigenvalues, graphs with bounded eigenvalue multiplicities and strongly regular graphs. We show that the procedure is reduced in some parts to special cases of some well known combinatorial optimization problems, such as the maximal matching problem. the minimal cut problem, the maximal clique problem etc. This technique provides another proof of a result of L. Babai et al. [2] that isomorphism testing for graphs with bounded eigenvalue multiplicities can be performend in a polynomial time. We show that the canonical basis in strongly regular graphs is related to the graph decomposition into two strongly regular induced su...
TL;DR: In this paper, the authors generalize and prove Hofmeister's modification of a conjecture of Runge characterizing almost regular graphs, and discuss related matters, and show that the conjecture is correct.
TL;DR: It is shown that every 2-connected, k-regular claw-free graph on n vertices contains a cycle of length at least min 4k?2, n (k?8), and this result is best possible.
TL;DR: In this paper, it was shown that distance regular graphs with λ =μ=2,a2=4 are locally unions of triangles, hexagons or heptagons and that a distance regular graph with intersection array (13,10,7; 1,2,7) does not exist.
Abstract: Several properties of graphs with λ=μ=2,a2=4 are studied. It is proved that such graphs are locally unions of triangles, hexagons or heptagons. As a consequence, a distance regular graph with intersection array (13,10,7;1,2,7) does not exist.
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TL;DR: A graph G has constant u = u(G) if any two vertices that are not adjacent have u common neighbours as discussed by the authors, which is the same as the Bruck-Ryser condition that a graph has constant U and u if and only if it has two distinct restricted Laplace eigenvalues.
Abstract: A graph G has constant u = u(G) if any two vertices that are not adjacent have u common neighbours. G has constant u and u if G has constant u = u(G), and its complement G has constant u = u(G). If such a graph is regular, then it is strongly regular, otherwise precisely two vertex degrees occur. We shall prove that a graph has constant u and u if and only if it has two distinct restricted Laplace eigenvalues. Bruck-Ryser type conditions are found. Several constructions are given and characterized. A list of feasible parameter sets for graphs with at most 40 vertices is generated.
TL;DR: It is proved that there are no edge-chromatic 4-critical graphs of order 12, and Yap has posed the following question, which is motivated by the fact that smallest counter examples usually possess interesting properties.
TL;DR: It is proved that, in a distance-regular interval-regular graph having a triangle, I(u,v) is convex for every vertices u,v at distance 3.