Showing papers on "Strongly regular graph published in 2017"
TL;DR: Based on a generic construction of linear codes from mappings and by employing weakly regular bent functions, a new class of linear p-ary codes with three weights given with its weight distribution is provided.
Abstract: We contribute to the knowledge of linear codes with few weights from special polynomials and functions. Substantial efforts (especially due to C. Ding) have been directed towards their study in the past few years. Such codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. Based on a generic construction of linear codes from mappings and by employing weakly regular bent functions, we provide a new class of linear p-ary codes with three weights given with its weight distribution. The class of codes presented in this paper is different from those known in literature.
98 citations
TL;DR: It is shown that if G is an (n, d, λ)-network and λ = O ( d ) , the average clustering coefficient c ¯ ( G ) of G satisfies c ¯ ∼ d / n for large d and this description also holds for strongly regular graphs and Erdős–Renyi graphs.
47 citations
TL;DR: This paper constructed linear codes with few weights are constructed from inhomogeneous quadratic functions over the finite fieldGF(p), where p is an odd prime.
Abstract: Linear codes with few weights have been an interesting subject of study for many years, as these codes have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, linear codes with few weights are constructed from inhomogeneous quadratic functions over the finite field $${\mathrm {GF}}(p)$$GF(p), where p is an odd prime. They include some earlier linear codes as special cases. The weight distributions of these linear codes are also determined.
43 citations
TL;DR: In this article, the authors give three different proofs of the main result of Anantharaman and Le Masson (Duke Math J 164(4):723-765, 2015), establishing quantum ergodicity for eigenfunctions of the laplacian on large regular graphs of fixed degree.
Abstract: We give three different proofs of the main result of Anantharaman and Le Masson (Duke Math J 164(4):723–765, 2015), establishing quantum ergodicity—a form of delocalization—for eigenfunctions of the laplacian on large regular graphs of fixed degree. These three proofs are much shorter than the original one, quite different from one another, and we feel that each of the four proofs sheds a different light on the problem. The goal of this exploration is to find a proof that could be adapted for other models of interest in mathematical physics, such as the Anderson model on large regular graphs, regular graphs with weighted edges, or possibly certain models of non-regular graphs. A source of optimism in this direction is that we are able to extend the last proof to the case of anisotropic random walks on large regular graphs.
38 citations
TL;DR: In this article, a hemisystem on the Hermitian surface ℋ(3, q2), q ≥ 7 odd, admitting a subgroup of PΩ-(4, q) of order q2(q +1) is constructed.
Abstract: Abstract A hemisystem on the Hermitian surface ℋ(3, q2), q ≥ 7 odd, admitting a subgroup of PΩ-(4, q) of order q2(q +1) is constructed. Also, a new family of Cameron-Liebler line classes of PG(3, q), q ≥ 5 odd, with parameter (q2 + 1)/2 is provided.
36 citations
TL;DR: In this paper, exact expressions for the hyper-Zagreb index of graph operations containing cartesian product and join of n graphs, splice, link and chain of graphs are presented.
Abstract: Let G be a simple connected graph. The Hyper-Zagreb index is defined as $$\textit{HM}(G)=\sum _{uv\in E_{G}}(d_{G}(u)+d_{G}(v))^2$$
. In this paper some exact expressions for the hyper-Zagreb index of graph operations containing cartesian product and join of n graphs, splice, link and chain of graphs will be presented. We also apply these results to some graphs to chemical and general interest, such as $$C_4$$
nanotube, rectangular grid, prism, complete n-partite graph.
34 citations
31 citations
TL;DR: It is proved that the (adjacency) energy of any graph (bipartite or not) is a weighted sum of the traces of even powers of the adjacency matrix, and is used to find bounds for the energy in terms of subgraphs contributing to it.
30 citations
TL;DR: The spectral radius ź1 and the energy E e x of the Aex-matrix are examined, and the respective extremal graphs characterized.
23 citations
TL;DR: In this article, a construction of q-ary linear codes from trace and norm functions over finite fields is presented, where the weight distributions of the linear codes are determined in some cases based on Gauss sums.
23 citations
TL;DR: This paper classifies the connected graphs with precisely three distinct eigenvalues and second largest eigenvalue at most 1.
TL;DR: This paper proposes a construction of q-ary linear codes with few weights employing general quadratic forms over the finite field Fq, where q is an odd prime power.
Abstract: Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, a construction of q-ary linear codes with few weights employing general quadratic forms over the finite field Fq${\mathbb {F}}_{q}$ is proposed, where q is an odd prime power. This generalizes some earlier constructions of p-ary linear codes from quadratic bent functions over the prime field Fp${\mathbb {F}}_{p}$, where p is an odd prime. The complete weight enumerators of the resultant q-ary linear codes are also determined.
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TL;DR: It is shown that, for every strongly regular graph, there is some perturbation which results in pretty good state transfer, and for any strongly regular graphs, that $\phi(X\setminus e)$ does not depend on the choice of e.
Abstract: Quantum walks, an important tool in quantum computing, have been very successfully investigated using techniques in algebraic graph theory. We are motivated by the study of state transfer in continuous-time quantum walks, which is understood to be a rare and interesting phenomenon. We consider a perturbation on an edge $uv$ of a graph where we add a weight $\beta$ to the edge and a loop of weight $\gamma$ to each of $u$ and $v$. We characterize when for this perturbation results in strongly cospectral vertices $u$ and $v$. Applying this to strongly regular graphs, we give infinite families of strongly regular graphs where some perturbation results in perfect state transfer. Further, we show that, for every strongly regular graph, there is some perturbation which results in pretty good state transfer. We also show for any strongly regular graph $X$ and edge $e \in E(X)$, that $\phi(X\setminus e)$ does not depend on the choice of $e$.
TL;DR: It is proved that the two problems of partitioning the vertex set of a graph into two parts V A and V B are NP-complete with any nonnegative integers a and b except a = b = 0 .
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TL;DR: This paper presents a new class of binary linear codes with three weights from plateaued Boolean functions and their weight distributions and introduces the notion of (weakly) regular plateaued functions in odd characteristic.
Abstract: Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do this, we generalize the recent contribution of Mesnager given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present a new class of binary linear codes with three weights from plateaued Boolean functions and their weight distributions. We next introduce the notion of (weakly) regular plateaued functions in odd characteristic $p$ and give concrete examples of these functions. Moreover, we construct a new class of three-weight linear $p$-ary codes from weakly regular plateaued functions and determine their weight distributions. We finally analyse the constructed linear codes for secret sharing schemes.
TL;DR: In this paper, it was shown that no two non-isomorphic butterfly-like graphs with the same girth are A-cospectral, and then presented a new upper and lower bound for the i-th largest eigenvalue of L (G) and Q (G ), respectively.
TL;DR: It is proved that, for graphs with girth at least l − 1, statement (i) holds for every m ≥ 1 ; and it is observed that, statements (ii) and (i) also holds forevery m ≤ 1.
TL;DR: A new infinite family of distance-regular antipodal r -covers of a complete graph on q 3 + 1 vertices is found, where q is odd and r is any divisor of q + 1 such that (q + 1 ) ∕ r is odd.
TL;DR: In this paper, the authors consider spectral characterization on the second largest distance eigenvalue of graphs, and prove that the graphs with are determined by their D-spectra, if any graph with the same distance spectrum as G is isomorphic to G.
Abstract: Let G be a simple connected graph of order n and D(G) be the distance matrix of G. Suppose that is the distance spectrum of G. A graph G is said to be determined by its D-spectrum if any graph with the same distance spectrum as G is isomorphic to G. In this paper, we consider spectral characterization on the second largest distance eigenvalue of graphs, and prove that the graphs with are determined by their D-spectra.
TL;DR: In this article, the authors prove the non-existence of strongly regular graphs with parameters (76, 30, 8, 14, 14 ) and a lower bound on the number of 4-cliques, and then use these properties to show that the graph cannot exist.
TL;DR: It is proved that all distance-regular graphs with diameter D ≥ 3 are 2-extendable and several better lower bounds are obtained for the extendability of distance- regular graphs of valency k ≥ 3 that depend on k, λ and μ and the inequalities are incomparable with known eigenvalue bounds for these combinatorial parameters.
TL;DR: In this paper, a general construction of strongly regular graphs from the collinearity graph of a finite classical polar space of rank at least 3 over a finite field of order q is presented.
Abstract: We describe a general construction of strongly regular graphs from the collinearity graph of a finite classical polar spaces of rank at least 3 over a finite field of order q. We show that these graphs are non-isomorphic to the collinearity graphs and have the same parameters. For most of these parameters, the collinearity graphs were the only known examples, and so many of our examples are new.
TL;DR: It is proved that t(M(G)) can be expressed in terms of the summation of weights of spanning trees of G with some weights on its edges with some weighted edges.
TL;DR: The eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived and is derived as an application.
TL;DR: In this article, the authors studied the general spinless quadratic fermion Hamiltonian with interaction matrices given by the symmetric and antisymmetric parts of the adjacency matrix of a directed graph.
Abstract: In this paper, we study the general spinless quadratic fermion Hamiltonian with interaction matrices given by the symmetric and antisymmetric parts of the adjacency matrix of a directed graph. The correlation matrix and entanglement entropy are provided for the ground state of the Hamiltonian, analytically. We also show that a volume law scaling holds for some scalable sets of nonsymmetric association-scheme graphs. The scaling of the entanglement entropy is then used as a tool for studying the graph isomorphism problem, in particular to distinguish some nonisomorphic pairs of directed strongly regular graphs.
TL;DR: In this paper, the distance-regular Cayley line graphs with least eigenvalue ≥ 3 and diameter ≥ 3 were classified into lattice graphs, triangular graphs, and line graphs of incidence graphs of certain projective planes.
Abstract: We classify the distance-regular Cayley graphs with least eigenvalue $$-2$$-2 and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.
TL;DR: In this article, the authors extended the definition of regular dilation to graph products of N, which is an important class of quasi-lattice ordered semigroups, and showed that a representation of a graph product has an isometric Nica-covariant dilation if and only if it is ⁎-regular.
TL;DR: The anti-Ramsey number for matchings in regular bipartite graphs is considered and its value under several conditions is determined.
Abstract: Let 𝒢 be a family of graphs. The anti-Ramsey number AR(G,𝒢) for 𝒢 in the graph G is the maximum number of colors in an edge coloring of G that does not have any rainbow copy of any graph in 𝒢. In this paper, we consider the anti-Ramsey number for matchings in regular bipartite graphs and determine its value under several conditions.
TL;DR: In this paper, an exact upper bound on the distance between the normalized Laplacian spectra of two nonisomorphic graphs has been derived in terms of Randic energy.
TL;DR: This paper presents an overview of results, which completely answer the question of existence of regular handicap graphs of even order.