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From local adjacency polynomials to locally pseudo-distance-regular graphs. (English) Zbl 0888.05056
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 \(\lambda\) (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 \(\lambda\), 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 \(\Gamma\) 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 \(\lambda\) of the highest degree member of an orthogonal system of polynomials, which depends only on the spectrum of the graph.

MSC:
05E25 Group actions on posets, etc. (MSC2000)
05E20 Group actions on designs, etc. (MSC2000)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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