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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:
05E30 Association schemes, strongly regular graphs
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