Showing papers on "Strongly regular graph published in 2009"
TL;DR: A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix which is in a prescribed way defined for any graph as discussed by the authors, which is called �� -theory.
Abstract: A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix �� which is in a prescribed way defined for any graph. This theory is called �� -theory. We outline a spectral theory of graphs based on the signless Laplacians �� and compare it with other spectral theories, in particular with those based on the adjacency matrix �� and the Laplacian �� . The �� -theory can be composed using various connections to other theories: equivalency with �� -theory and �� -theory for regular graphs, or with �� -theory for bipartite graphs, general analogies with �� -theory and analogies with �� -theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.
302 citations
TL;DR: In this article, it was shown that if R is a commutative ring and S is a finite ring, then T ( Γ ( R ) is a Hamiltonian graph.
117 citations
TL;DR: A spectral technique suggested by coined quantum walks can be used to distinguish between graphs that are cospectral with respect to standard matrix representations and is provided, far less prone to the problems ofcospectrality.
68 citations
TL;DR: In this paper, it was shown that if m⩾9 and G has no 4-cycle, then μ2⩽m, with equality if G is a star, fails for 4 ⩾m ⩽8.
54 citations
TL;DR: In this paper, it was shown that if G is r-regular, with diam(G) 2, and Fi, i = 1, 2, 3, 4, are not induced subgraphs of G, then the k-th iterated line graph Lk(G), for k 1, ED(L k (G)) depends solely on n and r.
Abstract: The distance or D-eigenvalues of a graph G are the eigenvalues of its distance matrix. The distance or D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. Two graphs G1 and G2 are said to be D-equienergetic if ED(G1 )= ED(G2). Let F1 be the 5-vertex path, F2 the graph obtained by identifying one vertex of a triangle with one end vertex of the 3-vertex path, F3 the graph obtained by identifying a vertex of a triangle with a vertex of another triangle and F4 be the graph obtained by identifying one end vertex of a 4-vertex star with a middle vertex of a 3-vertex path. In this paper we show that if G is r-regular, with diam(G) 2, and Fi, i =1 , 2, 3, 4, are not induced subgraphs of G, then the k-th iterated line graph Lk(G) has exactly one positive D-eigenvalue. Further, if G is r-regular, of order n ,d iam(G) 2, and G does not have Fi, i =1 , 2, 3, 4, as an induced subgraph, then for k 1, ED(L k (G)) depends solely on n and r. This result leads to the construction of non D-cospectral, D-equienergetic graphs having same number of vertices and same number of edges.
44 citations
TL;DR: It is proved that if @k(G)>=max{(a+1)b+2k2,(a-1)^[email protected](G)+4bk4b}, then G is an (a,b,k)-critical graph.
40 citations
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TL;DR: In this paper, it is known that the maximum value of P2(G) for G ∈ G(v,e) occurs at one or both of two special graphs in G (v, e), i.e., the quasi-star graph or the complete graph.
Abstract: Let G(v,e) be the set of all simple graphs with v vertices and e edges and let P2(G) = P d 2 denote the sum of the squares of the degrees, d1,...,dv, of the vertices of G. It is known that the maximum value of P2(G) for G ∈ G(v,e) occurs at one or both of two special graphs in G(v,e)—the quasi-star graph or the quasi-complete graph. For each pair (v,e), we determine which of these two graphs has the larger value of P2(G). We also determine all pairs (v,e) for which the values of P2(G) are the same for the quasi-star and the quasi-complete graph. In addition to the quasi-star and quasi-complete graphs, we find all other graphs in G(v,e) for which the maximum value of P2(G) is attained. Density questions posed by previous authors are examined.
39 citations
TL;DR: An entropy based method is used to study two graph maximization problems and upper bound the number of matchings of fixed size in a d-regular graph on N vertices, giving asymptotic evidence for a conjecture of S. Friedland et al.
39 citations
01 Jan 2009
TL;DR: A graph G with n vertices is said to be "hypoenergetic" if E(G) < n and if G ≥ n, then G G is greater or equal to n as mentioned in this paper.
Abstract: A graph G with n vertices is said to be "hypoenergetic" if E(G)
37 citations
TL;DR: This paper argues, using the theory of spin glasses in physics, that in random regular graphs the maximum cut size asymptotically equals the number of edges in the graph minus the minimum bisection size.
Abstract: Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of edges in the graph minus the minimum bisection size. Maximum cut and minimal bisection are two famous NP-complete problems with no known general relation between them, hence our conjecture is a surprising property of random regular graphs. We further support the conjecture with numerical simulations. A rigorous proof of this relation is obviously a challenge.
35 citations
TL;DR: It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are stronglydistance-balanced.
Abstract: A graph G is strongly distance-balanced if for every edge uv of G and every i>=0 the number of vertices x with d(x,u)=d(x,v)-1=i equals the number of vertices y with d(y,v)=d(y,u)-1=i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distance-balanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O(mn) for their recognition, where m is the number of edges and n the number of vertices of the graph in question, are given.
TL;DR: In this paper, the spectral properties of clique-inserted-graphs were studied and the spectral dynamics of iterations of the cliqueinsertion were analyzed in terms of the characteristic polynomial of the regular graph.
01 Jan 2009
TL;DR: It is shown, following a suggestion of Peter Cameron (1996), that while strongly regular graphs provide some interesting examples, one must look beyond this class in general for the desired approximations.
Abstract: We take up Peter Cameron's problem of the classification of count- ably infinite graphs which are homogeneous as metric spaces in the graph metric (Cam98). We give an explicit catalog of the known examples, together with results supporting the conjecture that the catalog may be complete, or nearly so. We begin in Part I with a presentation of Fra¨osse's theory of amalgamation classes and the classification of homogeneous structures, with emphasis on the case of homogeneous metric spaces, from the discovery of the Urysohn space to the connection with topological dynamics developed in (KPT05). We then turn to a discussion of the known metrically homogeneous graphs in Part II. This includes a 5-parameter family of homogeneous metric spaces whose connections with topological dynamics remain to be worked out. In the case of diameter 4, we find a variety of examples buried in the tables at the end of (Che98), which we decode and correlate with our catalog. In the final Part we revisit an old chestnut from the theory of homoge- neous structures, namely the problem of approximating the generic triangle free graph by finite graphs. Little is known about this, but we rephrase the problem more explicitly in terms of finite geometries. In that form it leads to questions that seem appropriate for design theorists, as well as some ques- tions that involve structures small enough to be explored computationally. We also show, following a suggestion of Peter Cameron (1996), that while strongly regular graphs provide some interesting examples, one must look beyond this class in general for the desired approximations.
TL;DR: In this paper, the authors give necessary and sufficient conditions for a bipartite graph K"m,n, [email protected]?n to be a sigma labelled graph.
TL;DR: This paper is able to show that for any r>=4, every r-regular graph of odd order [email protected]?17 has a strong VMTL, and introduces 'mirror' labelings which provide a suitable starting point from which the construction can proceed.
TL;DR: It is proven that for any integers k ≥ 2 and n ≥ k 2 + 4 k + 1, the generalized Petersen graph GP( n, k ) is not strongly distance-balanced.
Abstract: A graph X is said to be strongly distance-balanced whenever for any edge uv of X and any positive integer i , the number of vertices at distance i from u and at distance i + 1 from v is equal to the number of vertices at distance i + 1 from u and at distance i from v . It is proven that for any integers k ≥ 2 and n ≥ k 2 + 4 k + 1, the generalized Petersen graph GP( n , k ) is not strongly distance-balanced.
TL;DR: In this article, a complete classification of tetravalent one-regular graphs of order twice a product of two primes is given, and it is shown that with the exception of four graphs of orders 12 and 30, all such graphs are Cayley graphs on Abelian, dihedral, or generalized dihedral groups.
Abstract: A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this article a complete classification of tetravalent one-regular graphs of order twice a product of two primes is given. It follows from this classification that with the exception of four graphs of orders 12 and 30, all such graphs are Cayley graphs on Abelian, dihedral, or generalized dihedral groups.
TL;DR: It is proved that R(G) = 2 − 1/k for every graph G with the following properties: (A) every vertex has at most one neighbour of degree one; (B) vertices of degree at least 3 are not connected by an edge; (C) the girth of the graph is at least 6.
Abstract: In this paper we consider the secret sharing problem on special access structures with minimal qualified subsets of size two, i.e. secret sharing on graphs. This means that the participants are the vertices of the graph and the qualified subsets are the subsets of V(G) spanning at least one edge. The information ratio of a graph G is denoted by R(G) and is defined as the ratio of the greatest size of the shares a vertex has to remember and of the size of the secret. Since the determination of the exact information ratio is a non-trivial problem even for small graphs (i.e. for V(G) = 6), every construction can be of particular interest. Let k be the maximal degree in G. In this paper we prove that R(G) = 2 − 1/k for every graph G with the following properties: (A) every vertex has at most one neighbour of degree one; (B) vertices of degree at least 3 are not connected by an edge; (C) the girth of the graph is at least 6. We prove this by using polyhedral combinatorics arguments and the entropy method.
TL;DR: In this paper, the authors consider graphs with three distinct eigenvalues and characterize those with the largest eigenvalue less than 8, and give an upper bound on the number of vertices of graphs with a given number of distinct Eigenvalues in terms of the largest Eigenvalue.
TL;DR: In this article, it is shown how to construct 3-valent Cayley graphs that are 5-arc-transitive (in answer to a question by Cai Heng Li).
TL;DR: A structural description of strongly regular tri-Cayley graphs of cyclic groups is given.
Abstract: A graph is called tri-Cayley if it admits a semiregular subgroup of automorphisms having three orbits of equal length. In this paper, the structure of strongly regular tri-Cayley graphs is investigated. A structural description of strongly regular tri-Cayley graphs of cyclic groups is given.
TL;DR: It is proved that the n-dimensional burnt pancake graph Bn is super spanning connected if and only if n ≠ 2.
Abstract: Let u and v be any two distinct vertices of an undirected graph G, which is k-connected. For 1 ≤ w ≤ k, a w-container C(u, v) of a k-connected graph G is a set of w-disjoint paths joining u and v. A w-container C(u, v) of G is a w*-container if it contains all the vertices of G. A graph G is w*-connected if there exists a w*-container between any two distinct vertices. Let κ(G) be the connectivity of G. A graph G is super spanning connected if G is i*-connected for 1 ≤ i ≤ κ(G). In this paper, we prove that the n-dimensional burnt pancake graph Bn is super spanning connected if and only if n ≠ 2.
TL;DR: By the Principle of Inclusion and Exclusion it is shown that if G is a regular graph or a semi-regular bipartite graph, then the closed formulae of the matching polynomial and Hosoya index of S(G) are obtained.
Journal Article•
TL;DR: Ore proved that in general μ(G) ≤ max{1,n − σ2(G), which is the minimum number of vertex disjoint paths required to cover the vertices of G.
Abstract: Let G be a simple graph of order n. The path cover number μ(G )i s defined to be the minimum number of vertex disjoint paths required to cover the vertices of G. Ore proved that in general μ(G) ≤ max{1 ,n − σ2(G)}. We conjecture that if G is k-regular, then μ(G) ≤ n k+1 and we prove this for k ≤ 5.
TL;DR: It is proved that the sum of k largest eigenvalues of G is at most 12(k+1)n and this bound is shown to be best possible in the sense that for every k there exist graphs whose sum is 12( k+12)n-o(k^-^2^/^5)n.
01 Jan 2009
TL;DR: A sufficient condition that the graph G be non-hypoenergetic (i.e., E(G ≥ n )i s � [M2(G)]3/M4(G) ≥ n) is given in this article.
Abstract: Let G be an n-vertex graph, with eigenvalues λ1 ,λ 2 ,...,λ n .T henE(G )= � n=1 |λi| is its energy and Mk(G )= � n=1 (λi) k its k-th spectral moment. A sufficient condition that the graph G be non-hypoenergetic (i. e., E(G) ≥ n )i s � [M2(G)]3/M4(G) ≥ n . In a recent �
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TL;DR: In this paper, a simplified version of the theory of strongly regular graphs is developed for the case in which the graphs have no triangles, which leads to direct proofs of the Krein conditions, and the characterization of strong regular graphs with no triangles such that the second subconstituent is also strongly regular.
Abstract: A simplified version of the theory of strongly regular graphs is developed for the case in which the graphs have no triangles. This leads to (i) direct proofs of the Krein conditions, and (ii) the characterization of strongly regular graphs with no triangles such that the second subconstituent is also strongly regular. The method also provides an effective means of listing feasible parameters for such graphs.
TL;DR: It is shown that the irreducible graphs in this family have quasiprimitive automorphism groups, and it is proved (using the Classification of Finite Simple Groups) that no graph in thisfamily has a holomorphic simple automorphisms group.
Abstract: In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parameters of the graphs in this family that reduce to complete graphs.
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TL;DR: In this paper, the authors examined the structure of vertex-and edge-transitive strongly regular graphs, using normal quotient reduction, and showed that the irreducible graphs in this family have quasiprimitive automorphism groups.
Abstract: In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parameters of the graphs in this family that reduce to complete graphs.