85 research outputs found
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 with
distinct eigenvalues and odd-girth 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
An Odd Characterization of the Generalized Odd Graphs
2010 Mathematics Subject Classification: 05E30, 05C50;distance-regular graphs;generalized odd graphs;odd-girth;spectra of graphs;spectral excess theorem;spectral characterization
A characterization and an application of weight-regular partitions of graphs
A natural generalization of a regular (or equitable) partition of a graph,
which makes sense also for non-regular graphs, is the so-called weight-regular
partition, which gives to each vertex a weight that equals the
corresponding entry of the Perron eigenvector . This
paper contains three main results related to weight-regular partitions of a
graph. The first is a characterization of weight-regular partitions in terms of
double stochastic matrices. Inspired by a characterization of regular graphs by
Hoffman, we also provide a new characterization of weight-regularity by using a
Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular
graphs. In addition, we show an application of weight-regular partitions to
study graphs that attain equality in the classical Hoffman's lower bound for
the chromatic number of a graph, and we show that weight-regularity provides a
condition under which Hoffman's bound can be improved
The spectral excess theorem for distance-biregular graphs
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 graphPeer ReviewedPostprint (published version
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
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