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 Ei be the orthogonal projection matrix onto the eigenspace corresponding to θi. Let ∂(u, v) be the path distance of u, v ∈V.