Showing papers on "Strongly regular graph published in 2007"

Classical approach to the graph isomorphism problem using quantum walks

Abstract: Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.

Pseudo-Paley graphs and skew Hadamard difference sets from presemifields

Abstract: Let (K, + ,*) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, *) is a skew Hadamard difference set or a Paley type partial difference set in (K, +) according as q is congruent to 3 modulo 4 or q is congruent to 1 modulo 4. Applying this result to the Coulter---Matthews presemifield and the Ding---Yuan variation of it, we recover a recent construction of skew Hadamard difference sets by Ding and Yuan [7]. On the other hand, applying this result to the known presemifields with commutative multiplication and having order q congruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute the p-ranks of these pseudo-Paley graphs when q = 34, 36, 38, 310, 54, and 74. The p-rank results indicate that these graphs seem to be new. Along the way, we also disprove a conjecture of Rene Peeters [17, p. 47] which says that the Paley graphs of nonprime order are uniquely determined by their parameters and the minimality of their relevant p-ranks.

Comparing the Zagreb Indices

Abstract: Let G = (V, E) be a simple graph with n = vertical bar V vertical bar vertices and m = vertical bar E vertical bar edges ; let d(1), d(2), ..., d(n) denote the degrees of the vertices of G. If Delta = max d(i) <= 4, G is a chemical graph. The first and second Zagreb indices are defined as [GRAPHICS] We show that for all chemical graphs M-1/n <= M-2/m. This does not hold for all general graphs, connected or not.

Tight Lower Bounds on the Size of a Maximum Matching in a Regular Graph

Abstract: In this paper we study tight lower bounds on the size of a maximum matching in a regular graph For k ≥ 3, let G be a connected k-regular graph of order n and let a'(G) be the size of a maximum matching in G We show that if k is even, then α'(G) ≥ min {(k2+4 / k2+k+2)} x n/2 , n-1/2}, while if k is odd, then α(G) ≤ (k3-k2-2)n-2k+2 / 2(k3-3k) We show that both bounds are tight

Principal eigenvectors of irregular graphs

Abstract: Let G be a connected graph. Thispaper s extreme entriesof the principal eigenvector x of G, the unique positive unit eigenvector corresponding to the greatest eigenvalue λ1 of the adjacency matrix of G.I fG hasmaximum degree ∆, the greates t entry xmax of x isat mos t 1/ 1+ λ 2 /∆. This improves a result of Papendieck and Recht. The least entry xmin of x aswell asthe principal ratio xmax/xmin are studied. It is conjectured that for connected graphs of order n ≥ 3, the principal ratio isalwaysattained by one of the lollipop graphsobtained by attaching a path graph to a vertex of a complete graph.

Non-Backtracking Random Walks and Cogrowth of Graphs

Abstract: Let X be a locally finite, connected graph without vertices of degree 1. Non-backtracking random walk moves at each step with equal probability to one of the "forward" neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of X. Thus we obtain for infinite X that the n-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when X is finite. This provides a short proof of an old result concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when X is non-regular, but small cycles are dense in X, we show that the graph X is non-amenable if and only if the non-backtracking n-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen.

The Spectral Radius and the Maximum Degree of Irregular Graphs

Abstract: Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that $$ \Delta-\lambda_1>{1\over nD}, $$ where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.

Strongly Regular Graphs with Maximal Energy

Abstract: The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Koolen and Moulton have proved that the energy of a graph on n vertices is at most n(1 + √n)/2, and that equality holds if and only if the graph is strongly regular with parameters (n, (n+√n)/2, (n+2√n)/4, (n+2√n)/4). Such graphs are equivalent to a certain type of Hadamard matrices. Here we survey constructions of these Hadamard matrices and the related strongly regular graphs.

Vertex-Magic Total Labelings of Regular Graphs

Abstract: In this paper it is shown that if a graph $G$ possesses a spanning subgraph $H$ with a strong vertex magic total labeling (VMTL) and $G-E(H)$ is even-regular, then $G$ also has a strong VMTL. Among other things, this is used to conclude that all Hamiltonian regular graphs of odd order possess strong VMTLs. A relationship is then demonstrated between regular graphs of even degree and sparse magic squares. We next consider cubic graphs of order $2n$ consisting of two 2-factors of order $n$, connected by a 1-factor (quasi-prisms). Based on McQuillan’s construction of VMTLs of such 3-regular graphs, VMTLs are derived for similar regular graphs of any odd degree. Finally, a construction is given for VMTLs of quartic graphs of order $4n+2$ consisting of two cycles of odd order $n$ connected by a 2-factor (simple quasi-anti-prisms), and based on this construction VMTLs are derived for similar regular graphs of any even degree.

Double circulant codes from two class association schemes

Abstract: Two class association schemes consist of either strongly regular graphs (SRG) or doubly regular tournaments (DRT). We construct self-dual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and Gaborit Quadratic Double Circulant codes (QDC). As new examples SRG's give 4 (resp. 5) new Type I (resp. Type II) [72, 36, 12] codes. We construct a [200, 100, 12] Type II code invariant under the Higman-Sims group, a [200, 100, 16] Type II code invariant under the Hall-Janko group, and more generally self-dual binary codes attached to rank three groups.

Ring geometries, Two-Weight Codes and Strongly Regular Graphs

Abstract: It is known that a linear two-weight code $C$ over a finite field $\F_q$ corresponds both to a multiset in a projective space over $\F_q$ that meets every hyperplane in either $a$ or $b$ points for some integers $a

Representations of directed strongly regular graphs

Abstract: We develop a theory of representations in R^m for directed strongly regular graphs, which gives a new proof of a nonexistence condition of Jorgensen [L.K. Jorgensen, Non-existence of directed strongly regular graphs, Discrete Math. 264 (2003) 111-126]. We also describe some new constructions.

On strongly regular bicirculants

Abstract: An n-bicirculant is a graph having an automorphism with two orbits of length n and no other orbits. This article deals with strongly regular bicirculants. It is known that for a nontrivial strongly regular n-bicirculant, n odd, there exists a positive integer m such that n=2m^2+2m+1. Only three nontrivial examples have been known previously, namely, for m=1,2 and 4. Case m=1 gives rise to the Petersen graph and its complement, while the graphs arising from cases m=2 and m=4 are associated with certain Steiner systems. Similarly, if n is even, then n=2m^2 for some m>=2. Apart from a pair of complementary strongly regular 8-bicirculants, no other example seems to be known. A necessary condition for the existence of a strongly regular vertex-transitive p-bicirculant, p a prime, is obtained here. In addition, three new strongly regular bicirculants having 50, 82 and 122 vertices corresponding, respectively, to m=3,4 and 5 above, are presented. These graphs are not associated with any Steiner system, and together with their complements form the first known pairs of complementary strongly regular bicirculants which are vertex-transitive but not edge-transitive.

Efficient matrix rank computation with application to the study of strongly regular graphs

Abstract: We present algorithms for computing the p-rank of integer matrices. They are designed to be particularly effective when p is a small prime, the rank is relatively low, and the matrix itself is large and dense and may exceed virtual memory space. Our motivation comes from the study of difference sets and partial difference sets in algebraic design theory. The p-rank of the adjacency matrix of an associated strongly regular graph is a key tool for distinguishing difference set constructions and thus answering various existence questions and conjectures. For the p-rank computation, we review several memory efficient methods, and present refinements suitable to the small prime, small rank case. We give a new heuristic approach that is notably effective in practice as applied to the strongly regular graph adjacency matrices. It involves projection to a matrix of order slightly above the rank. The projection is extremely sparse, is chosen according to one of several heuristics, and is combined with a small dense certifying component. Our algorithms and heuristics are implemented in the LinBox library. We also briefly discuss some of the software design issues and we present results of experiments for the Paley and Dickson sequences of strongly regular graphs.

Colouring Random 4-Regular Graphs

Abstract: We show that a random 4-regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. The proof uses an efficient algorithm which a.a.s. 3-colours a random 4-regular graph. The analysis includes use of the differential equation method, and exponential bounds on the tail of random variables associated with branching processes. A substantial part of the analysis applies to random $d$-regular graphs in general.

A generalization of Wallis-Fon-Der-Flaass construction of strongly regular graphs

Abstract: In this paper the Wallis-Fon-Der-Flaass construction of strongly regular graphs is generalized. As a result new prolific series of strongly regular graphs are obtained. Some of them have new parameters.

On graphs with three distinct Laplacian eigenvalues

Abstract: In this paper, an equivalent condition of a graph G with t (2 ≤ t ≤ n) distinct Laplacian eigenvalues is established. By applying this condition to t = 3, if G is regular (necessarily be strongly regular), an equivalent condition of G being Laplacian integral is given. Also for the case of t = 3, if G is non-regular, it is found that G has diameter 2 and girth at most 5 if G is not a tree. Graph G is characterized in the case of its being triangle-free, bipartite and pentagon-free. In both cases, G is Laplacian integral.

Directed strongly regular graphs

Abstract: Some families of directed strongly regular graphs with t = μ are constructed by using antiflags of 1 1 2 -designs.

On Small World Semiplanes with Generalised Schubert Cells

Abstract: The well known “real-life examples” of small world graphs, including the graph of binary relation: “two persons on the earth know each other” contains cliques, so they have cycles of order 3 and 4. Some problems of Computer Science require explicit construction of regular algebraic graphs with small diameter but without small cycles. The well known examples here are generalised polygons, which are small world algebraic graphs i.e. graphs with the diameter d≤clog k−1(v), where v is order, k is the degree and c is the independent constant, semiplanes (regular bipartite graphs without cycles of order 4); graphs that can be homomorphically mapped onto the ordinary polygons. The problem of the existence of regular graphs satisfying these conditions with the degree ≥k and the diameter ≥d for each pair k≥3 and d≥3 is addressed in the paper. This problem is positively solved via the explicit construction. Generalised Schubert cells are defined in the spirit of Gelfand-Macpherson theorem for the Grassmanian. Constructed graph, induced on the generalised largest Schubert cells, is isomorphic to the well-known Wenger’s graph. We prove that the family of edge-transitive q-regular Wenger graphs of order 2q n , where integer n≥2 and q is prime power, q≥n, q>2 is a family of small world semiplanes. We observe the applications of some classes of small world graphs without small cycles to Cryptography and Coding Theory.

Small alliances in graphs

Abstract: Let G = (V,E) be a graph. A nonempty subset S ⊆ V is a (strong defensive) alliance of G if every node in S has at least as many neighbors in S than in V \S. This work is motivated by the following observation: when G is a locally structured graph its nodes typically belong to small alliances. Despite the fact that finding the smallest alliance in a graph is NP-hard, we can at least compute in polynomial time depthG(v), the minimum distance one has to move away from an arbitrary node v in order to find an alliance containing v. We define depth(G) as the sum of depthG(v) taken over v ⊆ V. We prove that depth(G) can be at most 1/4(3n2 - 2n + 3) and it can be computed in time O(n3). Intuitively, the value depth(G) should be small for clustered graphs. This is the case for the plane grid, which has a depth of 2n. We generalize the previous for bridgeless planar regular graphs of degree 3 and 4. The idea that clustered graphs are those having a lot of small alliances leads us to analyze the value of rp(G) = IP{S contains an alliance}, with S ⊆ V randomly chosen. This probability goes to 1 for planar regular graphs of degree 3 and 4. Finally, we generalize an already known result by proving that if the minimum degree of the graph is logarithmically lower bounded and if S is a large random set (roughly |S| > n/2), then also rp(G) → 1 as n→8.