4 research outputs found
On symmetric association schemes and associated quotient-polynomial graphs
Let denote an undirected, connected, regular graph with vertex set
, adjacency matrix , and distinct eigenvalues. Let denote the subalgebra of Mat
generated by . We refer to as the {\it adjacency algebra} of
. In this paper we investigate algebraic and combinatorial structure of
for which the adjacency algebra is closed under
Hadamard multiplication. In particular, under this simple assumption, we show
the following: (i) has a standard basis ;
(ii) for every vertex there exists identical distance-faithful intersection
diagram of with cells; (iii) the graph is
quotient-polynomial; and (iv) if we pick then
has distinct eigenvalues if and only if
spanspan. We describe the
combinatorial structure of quotient-polynomial graphs with diameter and
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 , a simple variation of the algorithm allow us to
decide wheter is distance-regular or not. In this context, we also
propose an algorithm to find which distance- matrices are polynomial in ,
giving also these polynomials.Comment: 22 pages plus 4 pages of reference