On symmetric association schemes and associated quotient-polynomial graphs

Let $Γ$ denote an undirected, connected, regular graph with vertex set $X$, adjacency matrix $A$, and ${d+1}$ distinct eigenvalues. Let ${\mathcal A}={\mathcal A}(Γ)$ denote the subalgebra of Mat$_X({\mathbb C})$ generated by $A$. We refer to ${\mathcal A}$ as the {\it adjacency algebra} of $Γ$. In this paper we investigate algebraic and combinatorial structure of $Γ$ for which the adjacency algebra ${\mathcal A}$ is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) ${\mathcal A}$ has a standard basis $\{I,F_1,\ldots,F_d\}$; (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 $F\in \{I,F_1,\ldots,F_d\}$ then $F$ has $d+1$ distinct eigenvalues if and only if span$\{I,F_1,\ldots,F_d\}=$span$\{I,F,\ldots,F^d\}$. 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.

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