From Local Adjacency Polynomials to Locally Pseudo-Distance-Regular Graphs

The local adjacency polynomials can be thought of as a generalization, for all graphs, of (the sums of ) the distance polynomials of distance-regular graphs. The term ``local'' here means that we ``see'' the graph from a given vertex, and it is the price we must pay for speaking of a kind of distance-regularity when the graph is not regular. It is shown that when the value at * (the maximum eigenvalue of the graph) of the local adjacency polynomials is large enough, then the eccentricity of the base vertex tends to be small. Moreover, when such a vertex is ``tight'' (that is, the value of a certain polynomial just fails to satisfy the condition) and fulfils certain additional extremality conditions, then all the polynomials attain their maximum possible values at *, and the graph turns out to be pseudo-distance-regular around the vertex. As a consequence of the above results, some new characterizations of distance-regular graphs are derived. For example, it is shown that a regular graph 1 with d+1 distinct eigenvalues is distance-regular if, and only if, the number of vertices at distance d from any given vertex is the value at * of the highest degree member of an orthogonal system of polynomials, which depend only on the spectrum of the graph. 1997 Academic Press