Showing papers on "Strongly regular graph published in 2005"

Peer-to-peer networks based on random transformations of connected regular undirected graphs

Abstract: We present k-Flipper, a graph transformation algorithm that transforms regular undirected graphs. Given a path of k+2 edges it interchanges the end vertices of the path. By definition this operation preserves regularity and connectivity. We show that every regular connected graph can be reached by a series of these operations for all k iÝ 1. We use a randomized version, called Random k-Flipper, in order to create random regular connected undirected graphs that may serve as a backbone for peer-to-peer networks. We prove for degree diE ¦¸(log n) that a series of O(dn) Random k-Flipper operations with k ∈ ¦¨(d2n2 log 1/¦A) transforms any graph into an expander graph with high probability, i.e. 1-n-¦¨(1).The Random 1-Flipper is symmetric, i.e. the transformation probability from any labeled d-regular graph G to G' is equal to those from G' to G. From this and the reachability property we conclude that in the limit a series of Random 1-Flipper operations converges against an uniform probability distribution over all connected labeled d-regular graphs. For degree d ∈ ω(1) growing with the graph size this implies that iteratively applying Random 1-Flipper transforms any given graph into an expander asymptotically almost surely.We use these operations as a maintenance operation for a peer-to-peer network based on random regular connected graphs that provides high robustness and recovers from degenerate network structures by continuously applying these random graph transformations. For this, we describe how network operations for joining and leaving the network can be designed and how the concurrency of the graph transformations can be handled.

On bicyclic graphs with minimal energies

Abstract: The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let G(n) be the class of bicyclic graphs G on n vertices and containing no disjoint odd cycles of lengths k and l with k + l ≡ 2 (mod 4). In this paper, the graphs in G(n) with minimal, second-minimal and third-minimal energies are determined.

Physically-motivated dynamical algorithms for the graph isomorphism problem

Abstract: The graph isomorphism problem (GI) plays a central role in the theory of computational complexity and has importance in physics and chemistry as well [1, 2]. No polynomial-time algorithm for solving GI is knowm We investigate classical and quantum physics-based polynomial-time algorithms for solving the graph isomorphism problem in which the graph structure is reflected in the behavior of a dynamical system. We show that a classical dynamical algorithm proposed by Gudkov and Nussinov [25] as well as its simplest quantum generalization fail to distinguish pairs of non-isomorphlc strongly regular graphs. However, by combining the algorithm of Gudkov and Nussinov with a construction proposed by Rudolph [26] in which one examines a graph describing the dynamics of two particles on the original graph, we find an algorithm that successfully distinguishes all pairs of non-isomorphic strongly regular graphs that we tested with up to 29 vertices.

Regular orientable embeddings of complete bipartite graphs

Abstract: In this paper, it will be shown that the isomorphism classes of regular orientable embeddings of the complete bipartite graph Kn,n are in one-to-one correspondence with the permutations on n elements satisfying a given criterion, and the isomorphism classes of them are completely classified when n is a product of any two (not necessarily distinct) prime numbers. For other n, a lower bound of the number of those isomorphism classes of Kn,n is obtained. As a result, many new regular orientable embeddings of the complete bipartite graph are constructed giving an answer of Nedela-Skoviera's question raised in [12]. © 2005 Wiley Periodicals, Inc. J Graph Theory

Sampling regular graphs and a peer-to-peer network

Abstract: We consider a simple Markov chain for d-regular graphs on n vertices, and show that the mixing time of this Markov chain is bounded above by a polynomial in n and d. A related Markov chain for d-regular graphs on a varying number of vertices is introduced, for even degree d. We use this to model a certain peer-to-peer network structure. We prove that the related chain has mixing time which is bounded by a polynomial in N, the expected number of vertices, under reasonable assumptions about the arrival and departure process.

Equienergetic complement graphs

Abstract: The energy of a graph G is the sum of the absolute values of its eigenvalues. Two graphs are said to be equienergetic if their energies are equal. In this paper we show that if G is a regular graph on n vertices and of degree r ≥ 3, then E( )

A note on the spherical embeddings of strongly regular graphs

Abstract: A primitive symmetric association scheme of class 2 is naturally embedded as a two-distance set in the unit sphere of Euclidean space, with respect to the primitive idempotent E1 of the Bose-Mesner algebra of the association scheme. Then it is shown that the ratio of the two distances of the two-distance set is instantly read from the character table (i.e., the first eigen matrix P) of the association scheme.

Codes from Generalized Hexagons

Abstract: In this paper, we construct some codes that arise from generalized hexagons with small parameters. As our main result we discover two new projective two-weight codes constructed from two-character sets in PG(5,4) and PG(11,2). These in turn are constructed using a new distance-2-ovoid of the classical generalized hexagon H(4). Also the corresponding strongly regular graph is new. The two-character set is the union of two orbits in PG(5,4) under the action of L2(13).

Some relations between distance-based polynomials of trees

Abstract: The Hosoya polynomial H(G,λ) of a graph G has the property that its first derivative at λ = 1 is equal to the Wiener index. Sometime ago two distance-based graph invariants were studied - the Schultz index S and its modification S*. We construct distance-based graph polyno­mials H1(G,λ) and H2(C,λ), such that their first derivatives at λ = 1 are, respectively, equal to S and S*. In case of trees, H1(G,λ) and H2(G,λ) are related with H(G,λ).

A note on Ramsey numbers for books

Abstract: The book with n pages Bn is the graph consisting of n triangles sharing an edge. The book Ramsey number r(Bm,Bn) is the smallest integer r such that either Bm ⊂ G or Bn ⊂ G for every graph G of order r. We prove that there exists a positive constant c such that r(Bm,Bn) = 2n + 3 for all n ≥ cm. Our proof is based mainly on counting; we also use a result of Andrasfai, Erdos, and Sos stating that triangle-free graphs of order n and minimum degree greater than 2n-5 are bipartite. © 2005 Wiley Periodicals, Inc. J Graph Theory

Randomized greedy algorithms for finding small k-dominating sets of regular graphs

Abstract: A k-dominating set of a graph G is a subset D of the vertices of G such that every vertex of G is either in D or at distance at most k from a vertex in D. It is of interest to find k- dominating sets of small cardinality. In this paper we consider simple randomized greedy algorithms for finding small k-dominating sets of regular graphs. We analyze the average-case performance of the most efficient of these simple heuristics showing that it performs surprisingly well on average. The analysis is performed on random regular graphs using differential equations. This, in turn, proves upper bounds on the size of a minimum k-dominating set of random regular graphs.

Integer sets with prescribed pairwise differences being distinct

Abstract: We label the vertices of a given graph G with positive integers so that the pairwise differences over its edges are all distinct. Let D(G) be the smallest value that the largest label can have.For example, for the complete graph Kn, the labels must form a Sidon set. Hence, D(Kn) = (1 + o(1))n2. Rather surprisingly, we demonstrate that there are graphs with only n3/2 + o(1) edges achieving this bound.More generally, we study the maximum value of D(G) that a graph G of the given order n and size m can have. We obtain bounds which are sharp up to a logarithmic multiplicative factor. The analogous problem for pairwise sums is considered as well. Our results, in particular, disprove a conjecture of Wood.

Edge-local equivalence of graphs

Abstract: The local complement G*i of a simple graph G at one of its vertices i is obtained by complementing the subgraph induced by the neighborhood of i and leaving the rest of the graph unchanged. If e={i,j} is an edge of G then G*e=((G*i)*j)*i is called the edge-local complement of G along the edge e. We call two graphs edge-locally equivalent if they are related by a sequence of edge-local complementations. The main result of this paper is an algebraic description of edge-local equivalence of graphs in terms of linear fractional transformations of adjacency matrices. Applications of this result include (i) a polynomial algorithm to recognize whether two graphs are edge-locally equivalent, (ii) a formula to count the number of graphs in a class of edge-local equivalence, and (iii) a result concerning the coefficients of the interlace polynomial, where we show that these coefficients are all even for a class of graphs; this class contains, as a subset, all strongly regular graphs with parameters (n, k, a, c), where k is odd and a and c are even.

Two Classes of Integral Regular Graphs.

Abstract: A graph G is called integral or Laplacian integral if all the eigenvalues of the adjacency matrix A(G) or the Laplacian matrix Lap(G) = D(G)−A(G) of G are integers, where D(G) denote the diagonal matrix of the vertex degrees of G. Let Kn,n+1 ≡ Kn+1,n and K1,p[(p−1)Kp] denote the (n+1)-regular graph with 4n+2 vertices and the p-regular graph with p2 + 1 vertices, respectively. In this paper, we shall give the spectra and characteristic polynomials of Kn,n+1 ≡ Kn+1,n and K1,p[(p − 1)Kp] from the theory on matrices. We derive the characteristic polynomials for their complement graphs, their line graphs, the complement graphs of their line graphs and the line graphs of their complement graphs. We also obtain the numbers of spanning trees for such graphs. When p = n2 + n + 1, these graphs are not only integral but also Laplacian integral. The discovery of these integral graphs is a new contribution to the search of integral graphs.

Ovoids and spreads of finite classical generalized hexagons and applications

Abstract: One intuitively describes a generalized hexagon as a point-line geometry full of ordinary hexagons, but containing no ordinary n-gons for n<6. A generalized hexagon has order (s,t) if every point is on t+1 lines and every line contains s+1 points. The main result of my PhD Thesis is the construction of three new examples of distance-2 ovoids (a set of non-collinear points that is uniquely intersected by any chosen line) in H(3) and H(4), where H(q) belongs to a special class of order (q,q) generalized hexagons. One of these examples has lead to the construction of a new infinite class of two-character sets. These in turn give rise to new strongly regular graphs and new two-weight codes, which is why I dedicate a whole chapter on codes arising from small generalized hexagons. By considering the (0,1)-vector space of characteristic functions within H(q), one obtains a one-to-one correspondence between such a code and some substructure of the hexagon. A regular substructure can be viewed as the eigenvector of a certain (0,1)-matrix and the fact that eigenvectors of distinct eigenvalues have to be orthogonal often yields exact values for the intersection number of the according substructures. In my thesis I reveal some unexpected results to this particular technique. Furthermore I classify all distance-2 and -3 ovoids (a maximal set of points mutually at maximal distance) within H(3). As such we obtain a geometrical interpretation of all maximal subgroups of G2(3), a geometric construction of a GAB, the first sporadic examples of ovoid-spread pairings and a transitive 1-system of Q(6,3). Research on derivations of this 1-system was followed by an investigation of common point reguli of different hexagons on the same Q(6,q), with nice applications as a result. Of these, the most important is the alternative construction of the Holz design and a subdesign. Furthermore we theoretically prove that the Holz design on 28 points only contains Hermitian and Ree unitals (previously shown by Tonchev by computer). As these Holz designs are one-point extensions of generalized quadrangles, we dedicate a final chapter to the characterization of the affine extension of H(2) using a combinatorial property.

Algebraic Characterizations of Graph Regularity Conditions

Abstract: It is well-known that a connected finite simple graph is regular if and only if the all-ones matrix spans an ideal of its adjacency algebra. We show that several other graph regularity conditions involving pairs and triples of vertices also have ideal theoretic characterizations in some appropriate algebras.

Distance Regular Covers of Complete Graphs from Latin Squares

Abstract: We describe a new construction of distance regular covers of a complete graph Kq2t with fibres of size q2t-1, q a power of 2. When q=2, the construction coincides with the one found in [D. de Caen, R. Mathon, G.E. Moorhouse. J. Algeb. Combinatorics, Vol. 4 (1995) 317] and studied in [T. Bending, D. Fon-Der-Flaass, Elect. J. Combinatorics, Vol. 5 (1998) R34]. The construction uses, as one ingredient, an arbitrary symmetric Latin square of order q; so, for large q, it can produce many different covers.

A distance regular graph with intersection array (21, 16, 8; 1, 4, 14) does not exist

Abstract: We show that a distance regular graph Γ with intersection array (21, 16, 8; 1, 4, 14) does not exist The proof uses algebraic properties of a positive semidefinite matrix related to the neighbourhood of a vertex of Γ

One-Factorizations of Regular Graphs of Order 12

Abstract: Algorithms for classifying one-factorizations of regular graphs are studied. The smallest open case is currently graphs of order 12; one-factorizations of $r$-regular graphs of order 12 are here classified for $r\leq 6$ and $r=10,11$. Two different approaches are used for regular graphs of small degree; these proceed one-factor by one-factor and vertex by vertex, respectively. For degree $r=11$, we have one-factorizations of $K_{12}$. These have earlier been classified, but a new approach is presented which views these as certain triple systems on $4n-1$ points and utilizes an approach developed for classifying Steiner triple systems. Some properties of the classified one-factorizations are also tabulated.

Symmetric Squares of Graphs

Abstract: We consider symmetric powers of a graph. In particular, we show that the spectra of the symmetric square of strongly regular graphs with the same parameters are equal. We also provide some bounds on the spectra of the symmetric squares of more general graphs. The connection with generic exchange Hamiltonians in quantum mechanics is discussed in an appendix.