Polynomial Properties on Large Symmetric Association Schemes

Abstract

In this paper we characterize “large” regular graphs using certain entries in the projection matrices onto the eigenspaces of the graph. As a corollary of this result, we show that “large” association schemes become P-polynomial association schemes. Our results are summarized as follows. Let G = (V, E) be a connected k-regular graph with d +1 distinct eigenvalues ${k = \theta_{0} > \theta_{1} > \cdots > \theta_{d}}$. Since the diameter of G is at most d, we have the Moore bound

$$|V| \leq M(k,d) = 1 + k \sum^{d-1}_{i=0} (k-1)^{i}.$$

Note that if |V| > M(k, d − 1) holds, the diameter of G is equal to d. Let E _i be the orthogonal projection matrix onto the eigenspace corresponding to θ _i . Let ∂(u, v) be the path distance of u, v ∈V.

Theorem. Assume ${|V| > M(k, d - 1)}$ holds. Then for x, y ∈V with ${\partial (x, y) = d}$, the (x, y) -entry of E _i is equal to

$$-\frac{1}{|V|} \prod _{j=1,2,...,d, j \neq i} \frac{\theta_{0}-\theta_{j}}{\theta_{i}-\theta_{j}}.$$

If a symmetric association scheme ${\mathfrak{X} = (X, \{R_{i}\}^{d}_{i=0})}$ has a relation R _i such that the graph (X, R _i ) satisfies the above condition, then ${\mathfrak{X}}$ is P-polynomial. Moreover we show the “dual” version of this theorem for spherical sets and Q-polynomial association schemes.