Polynomials on graphs |
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Institution: | Department of Mathematics University of Illinois 1409 West Green Street Urbana, Illinois 61801, USA |
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Abstract: | Let G be a simple graph with adjacency matrix A, and p(x) a polynomial with rational coefficients. If p(A) is the adjacency matrix of a graph, we denote that graph by p(G). We consider the question: Given a graph G, which polynomials p(x) give rise to a graph p(G) and what are those graphs? We give a complete answer if G is a distance-regular graph. We then derive some general relations between the polynomials p(x), the spectrum of A, and the automorphism group of G. |
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