On symmetric association schemes and associated quotient-polynomial graphs

Abstract

Let Γ denote an undirected, connected, regular graph with vertex set X, adjacency matrix A, and d + 1 distinct eigenvalues. Let A = A(Γ) denote the subalgebra of Mat X (C) generated by A. We refer to A as the adjacency algebra of Γ. In this paper we investigate algebraic and combinatorial structure of Γ for which the adjacency algebra A is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) A has a standard basis {I, F 1 , . . . , F d }; (ii) for every vertex there exists identical distance-faithful intersection diagram of Γ with d + 1 cells; (iii) the graph Γ is quotient-polynomial; and (iv) if we pick F ∈ {I, F 1 , . . . , F d } then F has d + 1 distinct eigenvalues if and only if span{I, F 1 , . . . , F d } = span{I, F, . . . , F d }. We describe the combinatorial structure of quotient-polynomial graphs with diameter 2 and 4 distinct eigenvalues. As a consequence of the technique from the paper we give an algorithm which computes the number of distinct eigenvalues of any Hermitian matrix using only elementary operations. When such a matrix is the adjacency matrix of a graph Γ, a simple variation of the algorithm allow us to decide wheter Γ is distance-regular or not. In this context, we also propose an algorithm to find which distance-i matrices are polynomial in A, giving also these polynomials.