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Locally pseudo-distance-regular graphs. (English) Zbl 0861.05064

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 \(\Gamma\) 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 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 graph \(\Gamma\), with adjacency matrix \({\mathbf A}\), is distance-regular if and only if \(\Gamma\) has spectrally maximum diameter \(D\), all its vertices have eccentricity \(D\), and the distance matrix \({\mathbf A}_D\) is a polynomial of degree \(D\) in \({\mathbf A}\).

MSC:

05E30 Association schemes, strongly regular graphs
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
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