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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_{i=0}^{d-1}(k-1)^i. \\] Note that if $|V|> M(k,d-1)$ holds, the diameter of $G$ is equal to $d$. 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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_{i=0}^{d-1}(k-1)^i. \\] Note that if $|V|> M(k,d-1)$ holds, the diameter of $G$ is equal to $d$. 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