Comments on “Extremal Cayley Digraphs of Finite Abelian Groups” [Intercon. Networks 12 (2011), no. 1-2, 125–135]

We comment on the paper “Extremal Cayley digraphs of finite Abelian groups” [Intercon. Networks 12 (2011), no. 1-2, 125–135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.Peer ReviewedPostprint (published version

Some results on the structure of multipoles in the study of snarks

Multipoles are the pieces we obtain by cutting some edges of a cubic graph in one or more points. As a result of the cut, a multipole M has vertices attached to a dangling edge with one free end, and isolated edges with two free ends. We refer to such free ends as semiedges, and to isolated edges as free edges. Every 3-edge-coloring of a multipole induces a coloring or state of its semiedges, which satisfies the Parity Lemma. Multipoles have been extensively used in the study of snarks, that is, cubic graphs which are not 3-edge-colorable. Some results on the states and structure of the so-called color complete and color closed multipoles are presented. In particular, we give lower and upper linear bounds on the minimum order of a color complete multipole, and compute its exact number of states. Given two multipoles M1 and M2 with the same number of semiedges, we say that M1 is reducible to M2 if the state set of M2 is a non-empty subset of the state set of M1 and M2 has less vertices than M1. The function v(m) is defined as the maximum number of vertices of an irreducible multipole with rn semiedges. The exact values of v(m) are only known for m <= 5. We prove that tree and cycle multipoles are irreducible and, as a byproduct, that v(m) has a linear lower bound.Peer ReviewedPostprint (published version

The Laplacian spectral excess theorem for distance-regular graphs

The spectral excess theorem states that, in a regular graph G, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess (a number that is computed by using the adjacency spectrum of G), and G is distance-regular if and only if equality holds. In this note we prove the corresponding result by using the Laplacian spectrum without requiring regularity of G

A short proof of the odd-girth theorem

Recently, it has been shown that a connected graph $\Gamma$ with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular. The proof of this result was based on the spectral excess theorem. In this note we present an alternative and more direct proof which does not rely on the spectral excess theorem, but on a known characterization of distance-regular graphs in terms of the predistance polynomial of degree $d$

The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues

Let G be a distance-regular graph with diameter d and Kneser graph K=Gd, the distance-d graph of G. We say that G is partially antipodal when K has fewer distinct eigenvalues than G. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with d distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.Peer ReviewedPostprint (author's final draft

Algebraic characterizations of bipartite distance-regular graphs

Bipartite graphs are combinatorial objects bearing some interesting symmetries. Thus, their spectra—eigenvalues of its adjacency matrix—are symmetric about zero, as the corresponding eigenvectors come into pairs. Moreover, vertices in the same (respectively, different) independent set are always at even (respectively, odd) distance. Both properties have well-known consequences in most properties and parameters of such graphs. Roughly speaking, we could say that the conditions for a given property to hold in a general graph can be somehow relaxed to guaranty the same property for a bipartite graph. In this paper we comment upon this phenomenon in the framework of distance-regular graphs for which several characterizations, both of combinatorial or algebraic nature, are known. Thus, the presented characterizations of bipartite distance-regular graphs involve such parameters as the numbers of walks between vertices (entries of the powers of the adjacency matrix A), the crossed local multiplicities (entries of the idempotents $E_i$ or eigenprojectors), the predistance polynomials, etc. For instance, it is known that a graph G, with eigenvalues $λ_0$ > $λ_1$ > · · · > $λ_d$ and diameter D = d, is distance-regular if and only if its idempotents $E_1$ and $E_d$ belong to the vector space D spanned by its distance matrices I,A,$A_2$, . . .$A_d$. In contrast with this, for the same result to be true in the case of bipartite graphs, only $E_1$ ∈ D need to be required.Peer Reviewe

Coloración de grafos

Estudiamos algunos resultados sobre rama-coloración de grafos y su relación con el teorema del mapa de cuatro colores. A tal fin, se introduce la noción de “coloración” de una conjunto de ramas y se estudian sus propiedades relacionadas con el álgebra de Boole

On congruence in ZnZ^n and the dimension of a multidimensional circulant

From a generalization to $Z^n$ of the concept of congruence we define a family of regular digraphs or graphs called multidimensional circulants, which turn to be Cayley (di)graphs of Abelian groups. This paper is mainly devoted to show the relationship between the Smith normal form for integral matrices and the dimension of such (di)graphs, that is the minimum ranks of the groups they can arise from. In particular, those 2-step multidimensional circulant which are circulants, that is Cayley (di)graphs of cyclic groups, are fully characterized. In addition, a reasoning due to Lawrence is used to prove that the cartesian product of $n$ circulants with equal number of vertice $p>2$, $p$ a prime, has dimension $n$.Peer Reviewe

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

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