arXiv subjects
Publications and source records attributed to M. A. Fiol.
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.
Let $Γ$ be a distance-regular graph with diameter $d$ and Kneser graph $K=Γ_d$, the distance-$d$ graph of $Γ$. We say that $Γ$ is partially antipodal when $K$ has fewer distinct eigenvalues than $Γ$. 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.
Let the Kneser graph $K$ of a distance-regular graph $Γ$ be the graph on the same vertex set as $Γ$, where two vertices are adjacent when they have maximal distance in $Γ$. We study the situation where the Bose-Mesner algebra of $Γ$ is not generated by the adjacency matrix of $K$. In particular, we obtain strong results in the so-called `half antipodal' case.
A graph $G$ with $d+1$ distinct eigenvalues is called strongly distance-regular if $G$ itself is distance-regular, and its distance-$d$ graph $G_d$ is strongly-regular. In this note we provide a spectral characterization of those distance-regular graphs with diameter $d=4$ which are strongly distance-regular. As a byproduct, it is shown that all bipartite strongly distance-regular graphs with such a diameter are antipodal.
We apply eigenvalue interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs, and characterize equality. This leads to generalizations of, and variations on theorems by Grone, and Grone and Merris. As a consequence we obtain inequalities involving bounds for some well-known parameters of a graph, such as edge-connectivity, and the isoperimetric number.
Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph G is distance-regular if and only if its spectral excess (a number that can be computed from the spectrum) equals the average excess (the mean of the numbers of vertices at extremal distance from every vertex). The aim of this paper is to derive new characterizations of regularity and distance-regularity for the more restricted family of bipartite graphs. In this case, some characterizations of (bi)regular bipartite graphs are given in terms of the mean degrees in every partite set and the Hoffman polynomial. Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs. Finally, a new version of the spectral excess theorem for bipartite graphs is presented.
Multipoles are the pieces we obtain by cutting some edges of a cubic graph. As a result of the cut, a multipole $M$ has dangling edges with one free end, which we call semiedges. Then, 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 $M_1$ and $M_2$ with the same number of semiedges, we say that $M_1$ is reducible to $M_2$ if the state set of $M_2$ is a non-empty subset of the state set of $M_1$ and $M_2$ has less vertices than $M_1$. The function $v(m)$ is defined as the maximum number of vertices of an irreducible multipole with $m$ semiedges. The exact values of $v(m)$ are only known for $m\le 5$. We prove that tree and cycle multipoles are irreducible and, as a byproduct, that $v(m)$ has a linear lower bound.
The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the spectral excess theorem for bipartite distance-biregular graphs.
The local spectrum of a vertex set in a graph has been proven to be very useful to study some of its metric properties. It also has applications in the area of pseudo-distance-regularity around a set and can be used to obtain quasi-spectral characterizations of completely (pseudo-)regular codes. In this paper we study the relation between the local spectrum of a vertex set and the local spectrum of each of its subconstituents. Moreover, we obtain a new characterization for completely pseudo-regular codes, and consequently for completely regular codes, in terms of the relation between the local spectrum of an extremal set of vertices and the local spectrum of its antipodal set. We also present a new proof of the version of the Spectral Excess Theorem for extremal sets of vertices.
We give a quasi-complete solution of the (Δ,N) problem for two well-known families of digraphs used as good models for large interconnection networks. In our study we also relate both families, the New Amsterdam and Manhattan digraphs, with the double-step graphs (or circulant graphs with degree two).
A graph is edge-distance-regular when it is distance-regular around each of its edges and it has the same intersection numbers for any edge taken as a root. In this paper we give some (combinatorial and algebraic) proofs of the fact that every edge-distance-regular graph $\G$ is distance-regular and homogeneous. More precisely, $\G$ is edge-distance-regular if and only if it is bipartite distance-regular or a generalized odd graph. Also, we obtain the relationships between some of their corresponding parameters, mainly, the distance polynomials and the intersection numbers.
A simple but empirically efficient heuristic algorithm for the edge-coloring of graphs is presented. Its basic idea is the displacement of "conflicts" (repeated colors in the edges incident to a vertex) along paths of adjacent vertices whose incident edges are recolored by swapping alternating colors (that is, doing a Kempe interchange). The results of performance tests on random cubic and $Δ$-regular graphs are presented, and a full implementation of the algorithm is given to facilitate its use and the reproducibility of results.
This paper presents bounds for the variation of the spectral radius $λ(G)$ of a graph $G$ after some perturbations or local vertex/edge modifications of $G$. The perturbations considered here are the connection of a new vertex with, say, $g$ vertices of $G$, the addition of a pendant edge (the previous case with $g=1$) and the addition of an edge. The method proposed here is based on continuous perturbations and the study of their differential inequalities associated. Within rather economical information (namely, the degrees of the vertices involved in the perturbation), the best possible inequalities are obtained. In addition, the cases when equalities are attained are characterized. The asymptotic behavior of the bounds obtained is also discussed. For instance, if $G$ is a connected graph and $G_u$ denotes the graph obtained from $G$ by adding a pendant edge at vertex $u$ with degree $δ_u$, then, $$ λ(G_u)\le λ(G)+\frac{δ_u}{λ^3(G)}+\textrm{o}(\frac{1}{λ^3(G)}). $$
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 out 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 dimensions of such (di)graphs, that is the minimum ranks of the groups they can arise from. In particular, those 2-step multidimensional circulants 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 vertices $p>2$, $p$ a prime, has dimension $n$.
Let $G$ be a connected graph with vertex set $V$ and a {\em weight function} $ρ$ that assigns a nonnegative number to each of its vertices. Then, the {\em $ρ$-moment} of $G$ at vertex $u$ is defined to be $M_G^ρ(u)=\sum_{v\in V} ρ(v)\dist (u,v) $, where $\dist(\cdot,\cdot)$ stands for the distance function. Adding up all these numbers, we obtain the {\em $ρ$-moment of $G$}: $$ M_G^ρ=\sum_{u\in V}M_G^ρ(u)=1/2\sum_{u,v\in V}\dist(u,v)[ρ(u)+ρ(v)]. $$ This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the {\em Wiener index} $W(G)$, when $ρ(u)=1/2$ for every $u\in V$, and the {\em degree distance} $D'(G)$, obtained when $ρ(u)=δ(u)$, the degree of vertex $u$. In this paper we derive some exact formulas for computing the $ρ$-moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding $ρ$-moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same $ρ$-moment for every $ρ$ (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product.
This note deals with the relationship between the total number of $k$-walks in a graph, and the sum of the $k$-th powers of its vertex degrees. In particular, it is shown that the the number of all $k$-walks is upper bounded by the sum of the $k$-th powers of the degrees.
Let $v(n)$ be the minimum number of voters with transitive preferences which are needed to generate any strong preference pattern (ties not allowed) on $n$ candidates. Let $k=\lfloor \log_2 n\rfloor$. We show that $v(n)\le n-k$ if $n$ and $k$ have different parity, and $v(n)\le n-k+1$ otherwise.
Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some "weights" (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights "regularize" the graph, and hence allow us to define a kind of regular partition, called "pseudo-regular," intended for general graphs. Here we show how to use interlacing for proving results about some weight parameters and pseudo-regular partitions of a graph. For instance, generalizing a well-known result of Lovász, it is shown that the weight Shannon capacity $Θ^*$ of a connected graph $\G$, with $n$ vertices and (adjacency matrix) eigenvalues $λ_1>λ_2\ge\...\ge λ_n$, satisfies $$ Θ\le Θ^* \le \frac{\|\vecnu\|^2}{1-\frac{λ_1}{λ_n}} $$ where $Θ$ is the (standard) Shannon capacity and $\vecnu$ is the positive eigenvector normalized to have smallest entry 1. In the special case of regular graphs, the results obtained have some interesting corollaries, such as an upper bound for some of the multiplicities of the eigenvalues of a distance-regular graph. Finally, some results involving the Laplacian spectrum are derived. spectrum are derived.