Annals of Combinatorics

, Volume 20, Issue 2, pp 379–386 | Cite as

Polynomial Properties on Large Symmetric Association Schemes

  • Hiroshi Nozaki


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, vV.
Theorem. Assume \({|V| > M(k, d - 1)}\) holds. Then for x, yV 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.


polynomial association scheme Moore bound graph spectrum s-distance set absolute bound 

Mathematics Subject Classification

05E30 05B20 


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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsAichi University of EducationKariyaJapan

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