1,134 research outputs found

    Some applications of the proper and adjacency polynomials in the theory of graph spectra

    Get PDF
    Given a vertex u\inV of a graph Γ=(V,E)\Gamma=(V,E), the (local) proper polynomials constitute a sequence of orthogonal polynomials, constructed from the so-called uu-local spectrum of Γ\Gamma. These polynomials can be thought of as a generalization, for all graphs, of the distance polynomials for te distance-regular graphs. The (local) adjacency polynomials, which are basically sums of proper polynomials, were recently used to study a new concept of distance-regularity for non-regular graphs, and also to give bounds on some distance-related parameters such as the diameter. Here we develop the subject of these polynomials and gave a survey of some known results involving them. For instance, distance-regular graphs are characterized from their spectra and the number of vertices at ``extremal distance'' from each of their vertices. Afterwards, some new applications of both, the proper and adjacency polynomials, are derived, such as bounds for the radius of Γ\Gamma and the weight kk-excess of a vertex. Given the integers k,μ≥0k,\mu\ge 0, let Γkμ(u)\Gamma_k^{\mu}(u) denote the set of vertices which are at distance at least kk from a vertex u∈Vu\in V, and there exist exactly μ\mu (shortest) kk-paths from uu to each each of such vertices. As a main result, an upper bound for the cardinality of Γkμ(u)\Gamma_k^{\mu}(u) is derived, showing that ∣Γkμ(u)∣|\Gamma_k^{\mu}(u)| decreases at least as O(μ−2)O(\mu^{-2}), and the cases in which the bound is attained are characterized. When these results are particularized to regular graphs with four distinct eigenvalues, we reobtain a result of Van Dam about 33-class association schemes, and prove some conjectures of Haemers and Van Dam about the number of vertices at distane three from every vertex of a regular graph with four distinct eigenvalues---setting k=2k=2 and μ=0\mu=0---and, more generally, the number of non-adjacent vertices to every vertex u∈Vu\in V, which have μ\mu common neighbours with it.Peer Reviewe

    Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)

    Full text link
    Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w=1, the only periodic orbits which contribute are the non back- scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure

    Graphs and networks theory

    Get PDF
    This chapter discusses graphs and networks theory
    • …
    corecore