Showing papers on "Strongly regular graph published in 2006"
TL;DR: In this paper, it was shown that if G and H are two non-abelian finite groups such that Γ G ≅ Γ H, then | G | = | H |, then H is nilpotent.
304 citations
31 Aug 2006
TL;DR: In this paper, it was shown that if Γ(G) is a complete graph, then G is a solvable group, and that if G is not complete, then it is not solvable.
Abstract: Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define F(G) to be the graph whose vertex set is cd(G) - {1}, and there is an edge between a and b if (a, b) > 1. We prove that if Γ(G) is a complete graph, then G is a solvable group.
135 citations
01 Jan 2006
TL;DR: The parameterized complexity of the k-Sizer-Regular Induced Subgraph with k as a parameter was shown to be W[1]-hard in this article, and a kernel of size O(k(r+k)^2 was shown in this paper.
Abstract: The r-Regular Induced Subgraph problem asks, given a graph G and a non-negative integer k, whether G contains an r-regular induced subgraph of size at least k, that is, an induced subgraph in which every vertex has degree exactly r. In this paper we examine its parameterization k-Sizer-Regular Induced Subgraph with k as parameter and prove that it is W[1]-hard. We also examine the parameterized complexity of the dual parameterized problem, namely, the k-Almostr-Regular Graph problem, which asks for a given graph G and a non-negative integer k whether G can be made r-regular by deleting at most k vertices. For this problem, we prove the existence of a problem kernel of size O(kr(r+k)^2).
92 citations
TL;DR: In this paper, the authors use the line digraph construction to associate an orthogonal matrix with each graph and derive two further matrices from these matrices, each of which is considered as a graph invariant.
Abstract: We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the first two cases, we compute the spectrum explicitly and show that it is determined by the spectrum of the adjacency matrix of the original graph. We then show by computation that the isomorphism classes of many known families of strongly regular graphs (up to 64 vertices) are characterized by the spectrum of this matrix. We conjecture that this is always the case for strongly regular graphs and we show that the conjecture is not valid for general graphs. We verify that the smallest regular graphs which are not distinguished with our method are on 14 vertices.
84 citations
21 May 2006
TL;DR: It is proved that if (2) could be generalized to all regular directed graphs (including ones that are not consistently labelled) then L=RL, and it is shown that such a problem can be solved in deterministic logarithmic space given a log-space pseudorandom walk generator forregular directed graphs.
Abstract: We revisit the general RL vs. L question, obtaining the following results. Generalizing Reingold's techniques to directed graphs, we present a deterministic, log-space algorithm that given a regular directed graph G (or, more generally, a digraph with Eulerian connected components) and two vertices s and t, finds a path between s and t if one exists.If we restrict ourselves to directed graphs that are regular and consistently labelled, then we are able to produce pseudorandom walks for such graphs in logarithmic space (this result already found an independent application).We prove that if (2) could be generalized to all regular directed graphs (including ones that are not consistently labelled) then L=RL. We do so by exhibiting a new complete promise problem for RL, and showing that such a problem can be solved in deterministic logarithmic space given a log-space pseudorandom walk generator for regular directed graphs.
82 citations
TL;DR: In this article, a graph is called s-regular if its automorphism group acts regularly on the set of its s-arcs, and all s -regular cubic graphs of order 10 p or 10 p 2 are classified for each s ≥ 1 and each prime p.
Abstract: A graph is called s -regular if its automorphism group acts regularly on the set of its s -arcs. In this paper, the s -regular cyclic or elementary abelian coverings of the Petersen graph for each s ≥1 are classified when the fibre-preserving automorphism groups act arc-transitively. As an application of these results, all s -regular cubic graphs of order 10 p or 10 p 2 are also classified for each s ≥1 and each prime p , of which the proof depends on the classification of finite simple groups.
62 citations
TL;DR: Hellwig and Volkmann as discussed by the authors gave a sufficient condition for λ 2-optimality in graphs of diameter g - 1, g being the girth of the graph, and showed that a graph with diameter at most g - 2 is λ2-optimal.
Abstract: For a connected graph the restricted edge-connectivity λ2(G) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in GS. A graph G is said to be λ2-optimal if λ2(G) = ξ(G), where ξ(G) is the minimum edge-degree in G defined as ξ(G) = min{d(u) + d(v) - 2:uv e E(G)}, d(u) denoting the degree of a vertex u. A. Hellwig and L. Volkmann [Sufficient conditions for λ2-optimality in graphs of diameter 2, Discrete Math 283 (2004), 113120] gave a sufficient condition for λ2-optimality in graphs of diameter 2. In this paper, we generalize this condition in graphs of diameter g - 1, g being the girth of the graph, and show that a graph G with diameter at most g - 2 is λ2-optimal. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 7386, 2006
55 citations
TL;DR: This paper analyzes a simple algorithm introduced by Steger and Wormald and proves that it produces an asymptotically uniform random regular graph in a polynomial time, confirming a conjecture of Wormald.
Abstract: Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [10] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d = O(n1/3−e), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd2). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author.
The algorithm works for relatively large d in practical (quadratic) time and can be used to derive many properties of uniform random regular graphs.
51 citations
TL;DR: In this paper, a new family of strongly regular graphs, called the general symplectic graphs Sp(2v, q), associated with nonsingular alternate matrices is introduced, and the parameters of these graphs, their chromatic numbers as well as their groups of graph automorphisms are determined.
Abstract: A new family of strongly regular graphs, called the general symplectic graphs Sp(2v, q), associated with nonsingular alternate matrices is introduced. Their parameters as strongly regular graphs, their chromatic numbers as well as their groups of graph automorphisms are determined.
47 citations
TL;DR: New upper bounds on the bisection width of graphs which have a regular vertex degree are derived and are shown to be at most ( 1 6 + e) | V | , e > 0.
46 citations
01 Jan 2006
TL;DR: It is shown that the size of an excessive factorization of a regular graph can exceed the degree of the graph by an arbitrarily large quantity.
Abstract: An excessive factorization of a graph G is a minimum set F of 1-factors of G whose union is E(G) In this paper we study excessive factorizations of regular graphs We introduce two graph parameters related to excessive factorizations and show that their computation is NP-hard We pose a number of questions regarding these parameters We show that the size of an excessive factorization of a regular graph can exceed the degree of the graph by an arbitrarily large quantity We conclude with a conjecture on the excessive factorizations of r-graphs
TL;DR: Using two backtrack algorithms based on different techniques, designed and implemented independently, they were able to determine up to isomorphism all strongly regular graphs with parameters $v=45, k=12, $\lambda=\mu=3$.
Abstract: Using two backtrack algorithms based on different techniques, designed and implemented independently, we were able to determine up to isomorphism all strongly regular graphs with parameters $v=45$, $k=12$, $\lambda=\mu=3$. It turns out that there are $78$ such graphs, having automorphism groups with sizes ranging from $1$ to $51840$.
TL;DR: In this article, the authors enumerated two-weight codes with relatively small parameters up to equivalence, and some properties of these codes and strongly-regular graphs were presented, including properties of strongly regular graphs.
Abstract: Projective two-weight codes with relatively small parameters are enumerated up to equivalence. Some properties of codes and related strongly-regular graphs are presented.
TL;DR: In this paper, the quasi-strongly regular graphs of grade 2 were studied and a spectral gap-type result for the eigenvalues of a strongly regular graph was derived.
Abstract: We study the quasi-strongly regular graphs, which are a combinatorial generalization of the strongly regular and the distance regular graphs. Our main focus is on quasi-strongly regular graphs of grade 2. We prove a “spectral gap”-type result for them which generalizes Seidel's well-known formula for the eigenvalues of a strongly regular graph. We also obtain a number of necessary conditions for the feasibility of parameter sets and some structural results. We propose the heuristic principle that the quasi-strongly regular graphs can be viewed as a “lower-order approximation” to the distance regular graphs. This idea is illustrated by extending a known result from the distance-regular case to the quasi-strongly regular case. Along these lines, we propose a number of conjectures and open problems. Finally, we list the all the proper connected quasi-strongly graphs of grade 2 with up to 12 vertices.
TL;DR: This paper begins the determination of all primitive strongly regular graphs with chromatic number equal to 5 using eigenvalue techniques and computer enumerations, showing that there are at most 43 possible parameter sets for such a graph.
TL;DR: In this article, it was shown that G(n,ρ/n) contains a 3-regular subgraph with high probability whenever ρ > λ ≈ 5.1494.
Abstract: In this paper, we prove that there exists a function ρk = (4 + o(1))k such that G(n,ρ/n) contains a k-regular graph with high probability whenever ρ > ρk. In the case of k = 3, it is also shown that G(n,ρ/n) contains a 3-regular graph with high probability whenever ρ > λ ≈ 5.1494. These are the first constant bounds on the average degree in G(n,p) for the existence of a k-regular subgraph. We also discuss the appearance of 3-regular subgraphs in cores of random graphs. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006Research supported by NSF Grant DSM 9971788 and DARPA Grant F33615-01-C-1900. Research undertaken while visiting Microsoft Research.Research undertaken during a postdoc at Microsoft Research.
TL;DR: This paper investigates, for a 2-edge-connected graph G with diameter at most 2, the group connectivity number Λ g(G) = min{n: G is A-connected for every abelian group A with |A| ≥ n}, and shows that any such graph G satisfies Κ g (G) ≤ 6.
Abstract: Let G be an undirected graph, A be an (additive) abelian group and A* = A - {0}. A graph G is A-connected if G has an orientation D(G) such that for every function b : V(G) ↦A satisfying Σv ∈ V(G) b(v) = 0, there is a function f: E(G)↦A* such that at each vertex v ∈ V(G), the amount of f values on the edges directed out from v minus the amount of f values on the edges directed into v equals b(v). In this paper, we investigate, for a 2-edge-connected graph G with diameter at most 2, the group connectivity number Λ g(G) = min{n: G is A-connected for every abelian group A with |A| ≥ n}, and show that any such graph G satisfies Λ g (G) ≤ 6. Furthermore, we show that if G is such a 2-edge-connected diameter 2 graph, then Λ g(G) = 6 if and only if G is the 5-cycle; and when G is not the 5-cycle, then Λ g (G) = 5 if and only if G is the Petersen graph or G belongs to two infinite families of well characterized graphs.
TL;DR: It is shown that if 2e>=10n^2-6n+1, then the minimum degree of a strongly vertex-magic graph is at least three, and the upper and lower bounds of any vertex degree in terms of n and e are obtained.
TL;DR: It is shown by exhaustive computer search that the list of five known biplanes with k = 11 is complete, which implies that there exists no 3-(57, 12, 2) design, no 11211 symmetric configuration, and no (324, 57, 0, 12) strongly regular graph.
TL;DR: In this paper, it was shown that the strongly regular graph with parameters $v=105, $k=32, $\lambda=4$ and $\mu=12$ is uniquely determined by its parameters (upto isomorphism).
Abstract: We show that the strongly regular graph with parameters $v=105$, $k=32$, $\lambda=4$ and $\mu=12$ is uniquely determined by its parameters (upto isomorphism).
TL;DR: New ( 96,20,4,4) and (96,19,2-4) regular partial difference sets are constructed, together with the corresponding strongly regular graphs.
Abstract: New (96,20,4,4) and (96,19,2,4) regular partial difference sets are constructed, together with the corresponding strongly regular graphs. Our source are (96,20,4) regular symmetric designs.
TL;DR: In this paper, the complexity of the line, middle and total graph of a regular graph G is analyzed in terms of the characteristic polynomial of the covering of the graph.
TL;DR: In this article, the authors introduced a class of decompositions of complete graphs into certain strongly regular graphs which share a common spread, and they later came to realize that these strongly regular graph are GQ(s,t)-graphs, and thus, due to a famous resultof A.E. Brouwer, deletion of the spread leads in each case to an antipodal distance regular graph which is an (s+1)-fold cover of the complete graph K
Abstract: Our motivation stems from the paper [9], in which the authors introduce a classof decompositions of complete graphs into certain strongly regular graphs whichshare a common spread. Analyzing their construction, we soon came to realize thatthese strongly regular graphs are GQ(s,t)-graphs, and thus, due to a famous resultof A.E. Brouwer, deletion of the spread leads in each case to an antipodal distanceregular graph which is an (s+1)-fold cover of the complete graph K
TL;DR: It is shown that for any graph G in the collection of graphs that can be obtained from K7 and K3,3,1,1 by a series of ΔY- and YΔ-transformations, c2(G) cannot be embedded into 4-space.
TL;DR: In this article, it was shown that if G is a connected graph, then G 3 and T(G) (the total graph of G ) are ID-factor-critical, and G 4 (when ∣V(G)) is even) is strongly IM-extendable.
TL;DR: In this article, the zeta functions of a graph bundle and of any (regular or irregular) covering graph are derived for abelian or dihedral groups and its fibre is a regular graph.
Abstract: As a continuation of computing the zeta function of a regular covering graph by Mizuno and Sato in [9], we derive in this paper computational formulae for the zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the voltages to derive them lie in an abelian or dihedral group and its fibre is a regular graph, those formulae can be simplified. As a by-product, the zeta function of the cartesian product of a graph and a regular graph is obtained. The same work is also done for a discrete torus and for a discrete Klein bottle.
TL;DR: The set of canonical equivalence relations on [G]n, where g is a random graph, is determined, extending the result of Erdos and Rado for the integers to random graphs.
TL;DR: In this article, the authors construct (k±1)-regular graphs which provide sequences of expanders by adding or substracting appropriate 1- factors from given sequences of k-regular graphs.
Abstract: We construct (k±1)-regular graphs which provide sequences of expanders by adding or substracting appropriate 1- factors from given sequences of k- regular graphs. We compute numerical examples in a few cases for which the given sequences are from the work of Lubotzky, Phillips, and Sarnak (with k − 1 the order of a finite field). If k + 1 = 7, our construction results in a sequence of 7- regular expanders with all spectral gaps at least 6−2√5 ≈ 1.52; the corresponding minoration for a sequence of Ramanujan 7- regular graphs (which is not known to exist) would be 7 − 2√6 ≈ 2.10.
TL;DR: In this paper, a polynomial P(n, r, t) of degree n in t, which depends only on n, r is given for a strongly regular graph G(m, n) where n ≡ −1 (mod 4) and m is sufficiently large.
TL;DR: In this article, the authors study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a vertex is the sets of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set edges at distance n-1 from a.
Abstract: We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ3(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.