A sharp lower bound on the signless Laplacian index of graphs with (k,t)-regular sets

A new lower bound on the largest eigenvalue of the signless Laplacian spectra for graphs with at least one $(\kappa,\tau)$-regular set is introduced and applied to the recognition of non-Hamiltonian graphs or graphs without a perfect matching. Furthermore, computational experiments revealed that the introduced lower bound is better than the known ones. The paper also gives sufficient condition for a graph to be non Hamiltonian (or without a perfect matching)

On the automorphism groups of strongly regular graphs II

We derive strong constraints on the automorphism groups of strongly regular (SR) graphs, resolving old problems motivated by Peter Cameron's 1981 description of large primitive groups.Trivial SR graphs are the disjoint unions of cliques of equal size and their complements. Graphic SR graphs are the line-graphs of cliques and of regular bipartite cliques (complete bipartite graphs with equal parts) and their complements.We conjecture that the order of the automorphism group of a non-trivial, non-graphic SR graph is quasi-polynomially bounded, i.e., it is at most exp((logn)C) for some constant C, where n is the number of vertices.While the conjecture remains open, we find surprisingly strong bounds on important parameters of the automorphism group. In particular, we show that the order of every automorphism is O(n8), and in fact O(n) if we exclude the line-graphs of certain geometries. We prove the conjecture for the case when the automorphism group is primitive; in this case we obtain a nearly tight n1+log2n bound.We obtain these bounds by bounding the fixicity of the automorphism group, i.e., the maximum number of fixed points of non-identity automorphisms, in terms of the second largest (in magnitude) eigenvalue and the maximum number of pairwise common neighbors of a regular graph. We connect the order of the automorphisms to the fixicity through an old lemma by Ákos Seress and the author.We propose to extend these investigations to primitive coherent configurations and offer problems and conjectures in this direction. Part of the motivation comes from the complexity of the Graph Isomorphism problem

Diameter, Covering Index, Covering Radius and Eigenvalues

AbstractFan Chung has recently derived an upper bound on the diameter of a regular graph as a function of the second largest eigenvalue in absolute value. We generalize this bound to the case of bipartite biregular graphs, and regular directed graphs.We also observe the connection with the primitivity exponent of the adjacency matrix. This applies directly to the covering number of Finite Non Abelian Simple Groups (FINASIG). We generalize this latter problem to primitive association schemes, such as the conjugacy scheme of Paige's simple loop.By noticing that the covering radius of a linear code is the diameter of a Cayley graph on the cosets, we derive an upper bound on the covering radius of a code as a function of the scattering of the weights of the dual code. When the code has even weights, we obtain a bound on the covering radius as a function of the dual distance dl which is tighter, for d⊥ large enough, than the recent bounds of Tietäväinen

Resultados espectrais relacionados com a estrutura dos grafos

Doutoramento em MatemáticaNesta tese são estabelecidas novas propriedades espectrais de grafos com estruturas específicas, como sejam os grafos separados em cliques e independentes e grafos duplamente separados em independentes, ou ainda grafos com conjuntos (κ,τ)-regulares. Alguns invariantes dos grafos separados em cliques e independentes são estudados, tendo como objectivo limitar o maior valor próprio do espectro Laplaciano sem sinal. A técnica do valor próprio é aplicada para obter alguns majorantes e minorantes do índice do espectro Laplaciano sem sinal dos grafos separados em cliques e independentes bem como sobre o índice dos grafos duplamente separados em independentes. São fornecidos alguns resultados computacionais de modo a obter uma melhor percepção da qualidade desses mesmos extremos. Estudamos igualmente os grafos com um conjunto (κ,τ)-regular que induz uma estrela complementar para um valor próprio não-principal $. Além disso, é mostrado que$=κ-τ. Usando uma abordagem baseada nos grafos estrela complementares construímos, em alguns casos, os respectivos grafos maximais. Uma caracterização dos grafos separados em cliques e independentes que envolve o índice e as entradas do vector principal é apresentada tal como um majorante do número da estabilidade dum grafo conexo.In this thesis new spectral properties of graphs with a specific structure (as split graphs, nested split and double split graphs as well as graphs with (κ,τ)-regular sets) are deduced. Some invariants of nested split graphs are studied in order to bound the largest eigenvalue of signless Laplacian spectra. The eigenvalue technique is applied to obtain some lower and upper bounds on the index of signless Laplacian spectra of nested split graphs as well as on the index of double nested graphs. Computational results are provided in order to gain a better insight of quality of these bounds. The graphs having a (κ,τ)-regular set which induces a star complement for a non-main eigenvalue $are studied. Furthermore, it is shown that$= κ-τ. By the star complement technique, in some cases, maximal graphs with desired properties are constructed. A spectral characterization of families of split graphs involving its index and the entries of the principal eigenvector is given as well as an upper bound on the stability number of a connected graph