Characterizing distance-regularity of graphs by the spectrum

D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs

The D-eigenvalues {µ1,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=1p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated lin...

Bounds for the Regularity of Edge Ideal of Vertex Decomposable and Shellable Graphs

Bounding cochordal cover number of graphs via vertex stretching

It is shown that when a special vertex stretching is applied to a graph, the cochordal cover number of the graph increases exactly by one, as it happens to its induced matching number and (Castelnuovo-Mumford) regularity. As a consequence, it is shown that the induced matching number and cochordal cover number of a special vertex stretching of a graph G are equal provided G is well-covered bipa...

Some spectral and quasi-spectral characterizations of distance-regular graphs

This is a new contribution to the question: Can we see from the spectrum of a graph whether it is distance-regular? By generalizing some results of Van Dam and Haemers, among others, we give some new spectral and quasi-spectral characterizations of distance-regularity. In this area of research, typical results concluding that a graph is distance regular require that G is cospectral with a dista...

The Spectral Excess Theorem for Distance-Regular Graphs: A Global (Over)view

Distance-regularity of a graph is in general not determined by the spectrum of the graph. The spectral excess theorem states that a connected regular graph is distance-regular if for every vertex, the number of vertices at extremal distance (the excess) equals some given expression in terms of the spectrum of the graph. This result was proved by Fiol and Garriga [From local adjacency polynomial...