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 graphΓwhen 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 pseudo-distance-regularity is shared by all the vertices, the graph turns out to be distance-regular. 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 graphΓ, with adjacency matrixA, is distance-regular if and only ifΓhas spectrally maximum diameterD, all its vertices have eccentricityD, and the distance matrixADis a polynomial of degreeDinA.