The alternating and adjacency polynomials, and their relation with the spectra and diameters of graphs

Let Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ_1 > λ_2 > · · · > λ_d. For every k = 0,1, . . . ,d −1, the k-alternating polynomial P_k is defined to be the polynomial of degree k and norm |Peer Reviewe

An eigenvalue characterization of antipodal distance-regular graphs

Let $\Gamma$ be a regular (connected) graph with $n$ vertices and $d+1$ distinct eigenvalues. As a main result, it is shown that $\Gamma$ is an $r$-antipodal distance-regular graph if and only if the distance graph $\Gamma_d$ is constituted by disjoint coies of the complete graph $K_r$, with $r$ satisfying an expression in terms of $n$ and the distinct eigenvalues.Peer Reviewe

On k-Walk-Regular Graphs

Considering a connected graph $G$ with diameter $D$, we say that it is \emph{$k$-walk-regular}, for a given integer $k$ $(0\leq k \leq D)$, if the number of walks of length $\ell$ between vertices $u$ and $v$ only depends on the distance between them, provided that this distance does not exceed $k$. Thus, for $k=0$, this definition coincides with that of walk-regular graph, where the number of cycles of length $\ell$ rooted at a given vertex is a constant through all the graph. In the other extreme, for $k=D$, we get one of the possible definitions for a graph to be distance-regular. In this paper we present some algebraic characterizations of $k$-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of $G$. Moreover, some results, concerning some parameters of a geometric nature, such as the cosines, and the spectrum of walk-regular graphs are presented

On almost distance-regular graphs

Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study `almost distance-regular graphs'. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called $m$-walk-regularity. Another studied concept is that of $m$-partial distance-regularity or, informally, distance-regularity up to distance $m$. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of $(\ell,m)$-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem

On Almost Distance-Regular Graphs

2010 Mathematics Subject Classification: 05E30, 05C50;distance-regular graph;walk-regular graph;eigenvalues;predistance polynomial

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

Given a vertex u\inV of a graph $\Gamma=(V,E)$, the (local) proper polynomials constitute a sequence of orthogonal polynomials, constructed from the so-called $u$-local spectrum of $\Gamma$. 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 $\Gamma$ and the weight $k$-excess of a vertex. Given the integers $k,\mu\ge 0$, let $\Gamma_k^{\mu}(u)$ denote the set of vertices which are at distance at least $k$ from a vertex $u\in V$, and there exist exactly $\mu$ (shortest) $k$-paths from $u$ to each each of such vertices. As a main result, an upper bound for the cardinality of $\Gamma_k^{\mu}(u)$ is derived, showing that $|\Gamma_k^{\mu}(u)|$ decreases at least as $O(\mu^{-2})$, 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 $3$-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 $k=2$ and $\mu=0$---and, more generally, the number of non-adjacent vertices to every vertex $u\in V$, which have $\mu$ common neighbours with it.Peer Reviewe