Locally Pseudo-Distance-Regular Graphs

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 AD is a polynomial of degree D in A. 1996 Academic Press, Inc.