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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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