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Tight distance-regular graphs. (English) Zbl 0959.05121
Let $$\Gamma$$ be a distance-regular graph with diameter $$d\geq 3$$ and eigenvalues $$k=\theta_0>\cdots{}>\theta_d$$. Then the fundamental bound is proved in Theorem 6.2: $\Biggl(\theta_1+\frac{k}{a_1+1}\Biggr) \Biggl(\theta_d+\frac{k}{a_1+1}\Biggr)\geq -\frac{ka_1b_1}{(a_1+1)^2}.$ We say $$\Gamma$$ is tight, if $$a_1\neq 0$$ and the equality holds in the fundamental bound. For eigenvalue $$\theta$$ of $$\Gamma$$ define the cosine sequence $$\sigma_0=1,\dots{},\sigma_d$$ associated with $$\theta$$ by the equalities $$c_i\sigma_{i-1}+a_i\sigma_i+b_i\sigma_{i+1}=\theta\sigma_i$$ $$(0\leq i\leq d)$$, where $$\sigma_{-1}$$ and $$\sigma_{d+1}$$ are indeterminates. For each edge $$\{x,y\}$$ define $$f(x,y)=a_1^{-1}|\{(z,w)\mid z,w\in \Gamma(x)\cap \Gamma(y)$$ and $$\partial(z,w)=2\}|$$.
Theorem 3.5. Let $$\Gamma$$ be a nonbipartite distance-regular graph with diameter $$d\geq 3$$ and eigenvalues $$k=\theta_0>\cdots{}>\theta_d$$. Then for each edge $$\{x,y\}$$, $b_1\frac{k+\theta_d(a_1+1)}{(k+\theta_d)(1+\theta_d)}\leq f(x,y)\leq b_1\frac{k+\theta_1(a_1+1)}{(k+\theta_1)(1+\theta_1)}.$ Some characterizations of tight graphs are obtained in terms of cosine sequences of two eigenvalues (Theorem 7.2) and auxiliary parameters (Theorem 8.3) or feasibility conditions (Theorem 9.3).
Theorem 12.6. Let $$\Gamma$$ be a distance-regular graph with diameter $$d\geq 3$$ and eigenvalues $$k=\theta_0>\cdots{}>\theta_d$$ and let $$b^-=-1-b_1/(\theta_1+1)$$, $$b^+=-1-b_1/(\theta_d+1)$$. Then the following are equivalent: $$(1)$$ $$\Gamma$$ is tight; $$(2)$$ for each vertex $$x$$ the local graph $$\Gamma(x)$$ is connected strongly regular with eigenvalues $$a_1,b^+,b^-$$; $$(3)$$ for some vertex $$x$$ the local graph $$\Gamma(x)$$ is connected strongly regular with eigenvalues $$a_1,b^+,b^-$$.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
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##### References:
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