Showing papers on "Strongly regular graph published in 2004"
TL;DR: In this article, it was shown that for any positive integer n⩾3, there exist two equienergetic graphs of order 4n that are not cospectral.
919 citations
TL;DR: The asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than $g$ is found.
Abstract: Consider random regular graphs of order $n$ and degree $d=d(n)\ge 3$ Let $g=g(n)\ge 3$ satisfy $(d-1)^{2g-1}=o(n)$ Then the number of cycles of lengths up to $g$ have a distribution similar to that of independent Poisson variables In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than $g$ A corresponding result is given for random regular bipartite graphs
142 citations
TL;DR: Using an orderly algorithm, the Steiner triple systems of order 19 are classied and the possibility of using the (strongly regular) block graphs of these designs in the isomorphism tests is utilized, leading to a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters.
Abstract: Using an orderly algorithm, the Steiner triple systems of order 19 are classied; there are 11;084;874;829 pairwise nonisomorphic such designs. For each design, the order of its automorphism group and the number of Pasch congurations it contains are recorded; 2;591 of the designs are anti-Pasch. There are three main parts of the classication: constructing an initial set of blocks, the seeds; completing the seeds to triple systems with an algorithm for exact cover; and carrying out isomorph rejection of the nal triple systems. Isomorph rejection is based on the graph canonical labeling software nauty supplemented with a vertex invariant based on Pasch congurations. The possibility of using the (strongly regular) block graphs of these designs in the isomorphism tests is utilized. The aforementioned value is in fact a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters (57; 24; 11; 9).
75 citations
TL;DR: In this paper, the authors investigated (G -)semisymmetric cubic graphs of order 6 p 2 for an odd prime p. They gave a group-theoretical construction of such graphs, and gave a classification of semisymmed cubic graphs.
Abstract: A graph Γ is said to be G -semisymmetric if it is regular and there exists a subgroup G of A := Aut( Γ ) acting transitively on its edge set but not on its vertex set. In the case of G = A , we call Γ a semisymmetric graph. The aim of this paper is to investigate ( G -)semisymmetric graphs of prime degree. We give a group-theoretical construction of such graphs, and give a classification of semisymmetric cubic graphs of order 6 p 2 for an odd prime p .
59 citations
TL;DR: In this paper, the eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the evaluation of eigen values of its generators.
Abstract: In this paper an efficient method is presented for calculating the eigenvalues of regular structural models. A structural model is called regular if they can be viewed as the direct or strong Cartesian product of some simple graphs known as their generators. The eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the evaluation of eigenvalues of its generators. The second eigenvalue of the Laplacian of a graph is also obtained using a much faster and much simple approach than the existing methods. Copyright © 2004 John Wiley & Sons, Ltd.
57 citations
TL;DR: The goal of this paper is to establish a connection between two classical models of random graphs: the random graph G ( n , p ) and the random regular graph G d ( n ).
57 citations
TL;DR: In this article, it was shown that the coherent algebra of a mixed directed strongly regular graph is a non-commutative algebra of rank at least 6, which is the same as that of a graph with 14 vertices.
53 citations
TL;DR: In this paper, the spectral radius of a simple connected graph with n vertices, m edges and degree sequence was shown to be upper bounded in terms of the degree sequence of the graph.
46 citations
TL;DR: In this paper, it was shown that λ 1(G) is the largest Laplacian eigenvalue of G if and only if G is a star graph, where d1, d2 are the highest and second highest degree, respectively.
45 citations
TL;DR: The goal of the paper is to initiate research towards a general, Blow-up Lemma type embedding statement for pseudo-random graphs with sublinear degrees, by showing that if the second eigenvalue λ of a d-regular graph G on 3n vertices is at most cd3/n2 log n, then G contains a triangle factor.
Abstract: The goal of the paper is to initiate research towards a general, Blow-up Lemma type embedding statement for pseudo-random graphs with sublinear degrees. In particular, we show that if the second eigenvalue λ of a d-regular graph G on 3n vertices is at most cd3/n2 log n, for some sufficiently small constant c > 0, then G contains a triangle factor. We also show that a fractional triangle factor already exists if λ < 0.1d2/n. The latter result is seen to be best possible up to a constant factor, for various values of the degree d = d(n).
40 citations
TL;DR: A variety of results are discussed, some quite recent, concerning the relationships between the embeddings of graphs in their complements and the structure of the embedding permutations.
TL;DR: In this paper, the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces were classified by direct analysis of the structure of arcregular subgroups of the automorphism groups of such graphs.
Abstract: In this paper, we classify the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces. Our classification is obtained by direct analysis of the structure of arc-regular subgroups (with cyclic vertex-stabilizers) of the automorphism groups of such graphs. This work is independent of the classification of primitive permutation groups of degree p or degree pq for p ≠ q and it is also independent of the classification of the arc-transitive graphs of order pq for p ≠ q.
TL;DR: In this article, it was shown that if G is positively curved then G is locally finite and every face of G is bounded by a cycle, and this conjecture holds also for cubic graphs.
Abstract: Let G be an infinite plane graph such that G is locally finite and every face of G is bounded by a cycle. Then G is said to be positively curved if, for every vertex x of G, , where the summation is taken over all facial cycles F of G containing x and |F| denotes the number of vertices in F. Note that if G is positively curved then the maximum degree of G is at most 5. As a discrete analog of a result in Riemannian geometry, Higuchi conjectured that if G is positively curved then G is finite. In this paper, we establish this conjecture for cubic graphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 241–274, 2004
TL;DR: In this article, the first known partial difference sets with negative Latin square type parameters are constructed in nonelementary abelian groups, the groups $\Z_4^{2k}times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all$k when ''ell'' is even.
Abstract: Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type parameters is constructed in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k when $\ell$ is even. Similarly, partial difference sets with Latin square type parameters are constructed in the same groups for all $k$ when $\ell$ is even and for all $k when $\ell$ is odd. These constructions provide the first example where the non-homomorphic bijection approach outlined by Hagita and Schmidt can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the partial difference sets are not isomorphic to the known graphs, and it is conjectured that the family of strongly regular graphs will be new.
Journal Article•
TL;DR: The energy of a graph is the sum of the absolute values of its eigenvalues as mentioned in this paper, i.e., it is a function of the number of vertices in the graph.
Abstract: The energy of a graph is the sum of the absolute values of its eigenvalues. Let G and L 2 (G) denote the complement and the second iterated line graph, respectively, of the graph G. If G1 and G2 are two regular graphs, both on n vertices, both of degree r ‚ 3, then L 2 (G1) and L 2 (G2) have equal energies, equal to (nr i 4)(2r i 3) i 2.
TL;DR: In this article, an efficient method for calculating the eigenvalues of space structures with regular topologies is presented, where the topology of a structure is formed as the Cartesian product of its generators.
TL;DR: For all odd integers n greater than or equal to 1, let G(n) denote the complete graph of order n, and for all even integers n equal to or more than 2, for all positive integers n, G, can be decomposed into h Hamilton cycles and t triangles if and only if nh + 3t is the number of edges in G n as mentioned in this paper.
Abstract: For all odd integers n greater than or equal to 1, let G(n) denote the complete graph of order n, and for all even integers n greater than or equal to 2 let G,, denote the complete graph of order n with the edges of a 1-factor removed. It is shown that for all non-negative integers h and t and all positive integers n, G, can be decomposed into h Hamilton cycles and t triangles if and only if nh + 3t is the number of edges in G(n). (C) 2004 Wiley Periodicals, Inc.
Posted Content•
TL;DR: The first known partial difference sets with negative Latin square type parameters in nonelementary abelian groups were constructed in this paper, where they were shown to be isomorphic to the strongly regular graphs associated to the PDSs.
Abstract: Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, we construct the first known family of partial difference sets with negative Latin square type parameters in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even. Similarly, we construct partial difference sets with Latin square type parameters in the same groups for all $k$ when $\ell$ is even and for all $k<\ell$ when $\ell$ is odd. These constructions provide the first example that the non-homomorphic bijection approach outlined by Hagita and Schmidt \cite{hagitaschmidt} can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the PDSs are not isomorphic to the known graphs, and we conjecture that the family of strongly regular graphs will be new.
Journal Article•
TL;DR: In this article, two necessary and sufficient conditions for a vertex-transitive graph G of degree k to admit λ′(G)=k are presented, and for any given integer s with 0≤s≤k-3, there is a connected vertextransitive G of G and G =k+s if and only if either k is odd or s is even.
Abstract: It is known that for connected vertex-transitive graphs of degree k(≥2), the restricted edge-connectivity k≤λ′≤2k-2 and the bounds can be attained. Two necessary and sufficient conditions for a vertex-transitive graph G of degree k to admit λ′(G)=k are presented. Afterwards, for any connected graph G 0, λ′(K 2×G 0) is determined to be λ′(K 2× G 0)=\%min\%{2δ(G 0),2λ′(G 0),v(G 0)}, and then for any given integer s with 0≤s≤k-3, there is a connected vertex-transitive graph G of degree k and λ′(G)=k+s if and only if either k is odd or s is even.
TL;DR: In this paper, infinitely many one-regular graphs of valency 4 and 6 are constructed, which are Cayley graphs on dihedral groups.
TL;DR: Krieger and Malabar as mentioned in this paper showed that a regular algebra over a commutative ring admits a uniform diagonalization formula where the entries of P and Q are algebra expressions in the a i and the a ii, if and only if R is strongly regular.
Abstract: In connection with the fundamental Separativity Problem for regular rings, we show that a regular algebra R over a commutative ring admits a uniform diagonalisation formula where the entries of P and Q are algebra expressions in the a i and the a i ', if and only if R is strongly regular (abelian regular in the terminology of Goodearl, K.R. (1979). Von Neumann Regular Rings. London: Pitman. 2nd ed. Krieger, Malabar, CFI. 1991). Next, we study regular algebras R over a field F such that for any a ∈ R there exist b ∈ F[a] and b' ∈ R such that bb'b = b, b'bb' = b' and the subalgebra of R generated by a and b' is regular. Such algebras are called one-accessible. We show that a finite product of matrix rings over a field is one-accessible and that a regular algebra over an uncountable perfect field is one-accessible if and only if it is algebraic. Tangentially, we elucidate and characterize when a nilpotent element has all its powers regular (or unit-regular) in an arbitrary algebra R over a commutati...
TL;DR: In this paper, it was shown that there are finitely many triangle-free distance-regular graphs with degree 8, 9 or 10, which are not triangle-regular regular graphs.
Abstract: In this paper we prove that there are finitely many triangle-free distance-regular graphs with degree 8, 9 or 10.
TL;DR: In this article, the authors analyzed the spectra of strongly regular graphs in the environment of Euclidean Jordan algebras and established necessary conditions for the existence of a strongly regular graph.
Abstract: We analyze the spectra of strongly regular graphs in the environment of Euclidean Jordan algebras In particular we obtain the spectra of the strongly regular graphs constructed in the Euclidean Jordan algebra studied in Cardoso and Vieira (J Math Sci 120:881–894, 2004) recurring to homogeneous linear difference equations of second order with constant coefficients Next, we associate a three dimensional Euclidean Jordan algebra V to the adjacency matrix of a strongly regular graph τ with three distinct eigenvalues and we define the generalized Krein parameters of τ Finally, we establish necessary conditions for the existence of a strongly regular graph
TL;DR: An upper bound for ∏(n), the maximum number of edges in a strongly multiplicative graph of order n, is given, which is sharper than the upper bound obtained by Beineke and Hegde [1].
Abstract: In this note we give an upper bound for ∏(n), the maximum number of edges in a strongly multiplicative graph of order
n, which is sharper than the upper bound obtained by Beineke and Hegde [1]. Keywords and phrases:graph labeling, strongly multiplicative graphs.
TL;DR: This note examines the connection between vertices of high eccentricity and the existence of k-factors in regular graphs and results are obtained that lead to new results in the case that the radius of the graph is small.
Abstract: In this note we examine the connection between vertices of high eccentricity and the existence of $k$-factors in regular graphs. This leads to new results in the case that the radius of the graph is small ($\leq 3$), namely that a $d$-regular graph $G$ has all $k$-factors, for $k|V(G)|$ even and $k\le d$, if it has at most $2d+2$ vertices of eccentricity $>3$. In particular, each regular graph $G$ of diameter $\leq3$ has every $k$-factor, for $k|V(G)|$ even and $k\le d$.
TL;DR: The Chvatal-Erdos theorem for 2-connected triangle-free graphs has been extended in this paper, showing that every longest cycle in G is dominating, and G has a cycle of length at least min{n - α(G) + κ(G), n}.
TL;DR: It is proved that if G is a k-regular bipartite graph and all 2-factors of G are isomorphic then k ≤ 3.
TL;DR: In this article, the authors obtained a sequence (b p ) p = 1 ∞ of upper bounds on the largest eigenvalue λ 1 (G ) of the Laplacian matrix of G.
TL;DR: In this article, it was shown that every connected edge-regular graph with (equivalently, either satisfies, or has parameters or, or is strongly regular) is not strongly regular.
Abstract: An undirected graph is said to be edge-regular with parameters if it has vertices, each vertex has degree , and each edge belongs to triangles. We put . Brouwer, Cohen, and Neumaier proved that every connected edge-regular graph with (equivalently, with ) is strongly regular. In this paper we construct an example of an edge-regular, not strongly regular graph on 36 vertices with . This shows that the estimate above is sharp. We prove that every connected edge-regular graph with (equivalently, either satisfies , or has parameters or , or is strongly regular.