Locally Pseudo-Distance-Regular Graphs

Abstract

The concept of local pseudo-distance-regularity, introduced in this paper, can be thought of as a natural generalization of distance-regularity for non-regular graphs.Intuitively speaking, such a concept is related to the regularity of graph 1 when it is seen from a given vertex. The price to be paid for speaking about a kind of distance-regularity in the non-regular case seems to be locality. Thus, we find out that there are no genuine``global'' pseudo-distance-regular graphs: when pseudodistance-regularity is shared by all the vertices, the graph turns out to be distanceregular. Our main result is a characterization of locally pseudo-distance-regular graphs, in terms of the existence of the highest-degree member of a sequence of orthogonal polynomials. As a particular case, we obtain the following new characterization of distance-regular graphs: A graph 1, with adjacency matrix A, is distance-regular if and only if 1 has spectrally maximum diameter D, all its vertices have eccentricity D, and the distance matrix A D is a polynomial of degree D in A.

From Local Adjacency Polynomials to Locally Pseudo-Distance-Regular Graphs

“…Locally pseudo-distance-regular graphs, see [17] or Section 3, generalize, for the nonregular case, the concept of distance-regular graphs, extensively studied in the literature. See, for instance, the basic books of Biggs [3], Brouwer et al [6], Cvetkovic et al [8], and Godsil [18].…”

“…The values cos ; ij , 1 i n, 0 j d, were formally introduced by Cvetkovic as the``angles'' of 1 (see, for instance, [7].) As was shown in [17], when the graph is seen from a vertex, its local multiplicities play a role similar to the standard multiplicities. Thus, for any vertex e i , m i (* l ) 0 and d l=0 m i (* l )=1.…”

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

“…, d, in which case they turn out to be the distance polynomials. In fact, we have the following strongest proposition, which is a combination of results in [14,7]. …”

Portfolio Symmetry and Momentum

“…We know that if the graph G is 2-antipodal distance-regular 4 , then it can be characterized by the eigenvalues of its adjacency matrix, and their multiplicities, Fiol et al (1996) . It means that the properties of an investment strategy represented by such a graph can also be resumed by its eigevalues.…”