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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.

This paper introduces a new parameter / = / ( G ) for a loopless digraph G, which can be thought of as a generalization of the girth of a graph. Let K, A, 6, and D denote respectively the connectivity, arc-connectivity, minimum degree, and diameter of G. Then it is proved that A = 6 if D s 21 and K = 6 if D I 21 -1. Analogous results involving upper bounds for K and A are given for the more general class of digraphs with loops. Sufficient conditions for a digraph to be super-A and super-rc are also given. As a corollary, maximally connected and superconnected iterated line digraphs and (undirected) graphs are characterized. DIGRAPHS AND LINE DIGRAPHSThis paper concentrates on the connectivity of digraphs. More precisely, we study the relation between the connectivity, the diameter, and a new parameter that, in the case of graphs, is closely related with the girth. The results obtained have some interesting corollaries. For example, it is shown that iterated line digraphs are maximally connected if the iteration order is large enough.Let us first recall the notation used throughout the paper. Let G = ( K A ) denote a digraph with (finite) set of vertices V = V(G) and set of arcs A = A(G), which are ordered pairs of (not necessarily different) vertices of V. So, loops are allowed but parallel arcs are not. If e = (x, y ) E A, we say that x is adjacent to y and that y is ad'ucenrfrom x . Let r-(x) and I'+ (x) denote respectively the sets of vertices adjacent to and from x , i.e., the sets of in-neighbors and out-neighbors of x . Their cardinalities are the in-degree of x , K ( x ) = lI'-(x)l, and the out-degree of x , 6'(x) = Ir'(x)[. The minimum degree of G, 6 = 6(G), is the minimum over all the in-degrees and out-degrees of the vertices of G.

We survey some old and some new characterizations of distance-regular graphs, which depend on information retrieved from their adjacency matrix. In particular, it is shown that a regular graph with d + 1 distinct eigenvalues is distance-regular if and only if a numeric equality, involving only the spectrum of the graph and the numbers of vertices at distance d from each vertex, is satisfied.

A new operation on graphs is introduced and some of its properties are studied. We call it hierarchical product, because of the strong (connectedness) hierarchy of the vertices in the resulting graphs. In fact, the obtained graphs turn out to be subgraphs of the cartesian product of the corresponding factors. Some well-known properties of the cartesian product, such as a reduced mean distance and diameter, simple routing algorithms and some optimal communication protocols are inherited by the hierarchical product. We also address the study of some algebraic properties of the hierarchical product of two or more graphs. In particular, the spectrum of the binary hypertree T m (which is the hierarchical product of several copies of the complete graph on two vertices) is fully characterized; turning out to be an interesting example of graph with all its eigenvalues distinct. Finally, some natural generalizations of the hierarchic product are proposed.