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.

be points on the real line. For every k=1, 2, ..., d, the k-alternating polynomial P k is the polynomial of degree k and normBecause of this optimality property, these polynomials may be thought of as the discrete version of the Chebychev polynomials T k and, for particular values of the given points, P k coincides in fact with the``shifted'' T k . In general, however, those polynomials seem to bear a much more involved structure than Chebychev ones. Some basic properties of the P k are studied, and it is shown how to compute them in general. The results are then applied to the study of the relationship between the (standard or Laplacian) spectrum of a (not necessarily regular) graph or bipartite graph and its diameter, improving previous results.

This paper considers the (A, 0 ) problem: to maximize the order of graphs with given maximum degree A and diameter 0, of importance for its implications in the design of interconnection networks. Two cubic graphs of diameters 5 and 8 and orders 70 and 286, respectively, and a graph of degree 5, diameter 3 and order 66 are presented, which improve the previously known orders for these values of A and D. By its construction, these graphs have a large automorphism group.

We consider in this paper the (d,k) problem for directed graphs: to maximize the number of vertices in a digraph of degree d and diameter k. For any values of d and k, we construct a graph with a number of vertices larger than (d 2 -1)/d 2 times the (non-attainable) Moore bound. In particular, this solves the (d,k) digraph problem for k=2. We also show that these graphs can be obtained as line digraph iterations and that this technique provides us with a simple local routing algorithm for the corresponding networks.

No abstract