A new inequality for distance-regular graphs.(English)Zbl 0814.05074

Let $$\Gamma$$ be a distance-regular graph with $$n$$ vertices and diameter $$d\geq 3$$. Let $$i$$ be fixed and $$d(x,y)= i$$. If $$d(x, z)= 1$$ then the possible values for $$d(y, z)$$ are $$i$$, $$i+1$$, $$i-1$$. The intersection numbers $$a_ i$$, $$b_ i$$, $$c_ i$$ denote the corresponding number of possible vertices $$z$$ and are independent of the choice of $$x$$ and $$y$$.
Let $$A$$ be an $$n\times n$$ distance matrix and $$M$$ a commutative (semi- simple) Bose-Mesner algebra, i.e. $$\mathbb{R}$$-algebra, generated by $$A$$. Let $$E$$ be a primitive idempotent of $$M$$, $$AE= EA= \theta E$$ and $$nE= \sum^ d_ 0\theta^*_ k A^ k$$.
An inequality, involving intersection numbers, $$\theta \leq \theta^*_ k$$ is proved for each $$3\leq i\leq d$$. It appears that equality is attained for $$i= 3$$ if and only if it is attained for all $$i$$ and this happens if and only if $$\Gamma$$ is a $$Q$$-polynomials with respect to $$E$$. The inequalities looks simpler if for some $$q$$ values $$qc_ i- b_ i q(qc_{i-1}- b_{i-1})$$ are independent of $$i$$. For $$q\neq 0, 1, -1$$ they can be writen as $$c_ i\frac {q^{2-i}- 1}{1- q^ i}\geq c_{i- 1}(q^{2- i}- 1)$$.

MSC:

 5e+30 Association schemes, strongly regular graphs
Full Text:

References:

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