6 research outputs found
Bounding the diameter and the mean distance of a graph from its eigenvalues: Laplacian versus adjacency matrix methods
AbstractRecently, several results bounding above the diameter and/or the mean distance of a graph from its eigenvalues have been presented. They use the eigenvalues of either the adjacency or the Laplacian matrix of the graph. The main object of this paper is to compare both methods. As expected, they are equivalent for regular graphs. However, the situation is different for nonregular graphs: While no method has a definite advantage when bounding above the diameter, the use of the Laplacian matrix seems better when dealing with the mean distance. This last statement follows from improved bounds on the mean distance obtained in the paper
An eigenvalue characterization of antipodal distance-regular graphs
Let be a regular (connected) graph with vertices and distinct eigenvalues. As a main result, it is shown that is an -antipodal distance-regular graph if and only if the distance graph is constituted by disjoint coies of the complete graph , with satisfying an expression in terms of and the distinct eigenvalues.Peer Reviewe
On middle cube graphs
We study a family of graphs related to the -cube. The middle cube graph of parameter k is the subgraph of induced by the set of vertices whose binary representation has either or number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors).Postprint (author's final draft
Algebraic characterizations of distance-regular graphs
We survey some old and some new characterizations of distance-regular graphs, which depend on information retrieved from their adjacency matrix. In particular, it is shown that a regular graph with d+1 distinct eigenvalues is distance-regular if and only if a numeric equality, involving only the spectrum of the graph and the numbers of vertices at distance d from each vertex, is satisfied.Peer Reviewe
Some families of orthogonal polynomials of a discrete variable and their applications to graphs and codes
We present some related families of orthogonal polynomials of a discrete variable
and survey some of their applications in the study of (distance-regular) graphs and
(completely regular) codes. One of the main peculiarities of such orthogonal systems
is their non-standard normalization condition, requiring that the square norm of each
polynomial must equal its value at a given point of the mesh. For instance, when they
are de¯ned from the spectrum of a graph, one of these families is the system of the pre-
distance polinomials which, in the case of distance-regular graphs, turns out to be the
sequence of distance polinomials. The applications range from (quasi-spectral) char-
acterizations of distance-regular graphs, walk-regular graphs, local distance-regularity
and completely regular codes, to some results on representation theory
Contribució a la teoria espectral de grafs problemes mètrics i distància-regularitat
Postprint (published version