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An inequality involving two eigenvalues of a bipartite distance-regular graph. (English) Zbl 1001.05124
Summary: Let \(\Gamma\) denote a bipartite distance-regular graph with diameter \(D\geq 4\) and valency \(k\geq 3\). Let \(\theta\), \(\theta'\) denote eigenvalues of \(\Gamma\) other than \(k\) and \(-k\). We obtain an inequality involving \(\theta\), \(\theta'\) and the intersection numbers of \(\Gamma\), which we refer to as the bipartite fundamental bound (BFB). Let \(E\), \(F\) denote the primitive idempotents of \(\Gamma\) associated with \(\theta\), \(\theta'\), respectively. We show that the following are equivalent: (i) \(\theta\), \(\theta'\) satisfy equality in the BFB; (ii) the entry-wise product \(E\circ F\) is a linear combination of at most two primitive idempotents of \(\Gamma\); (iii) \(E\circ F\) is a linear combination of exactly two primitive idempotents of \(\Gamma\). Let \(\Phi\) denote the set of pairs \(\theta\), \(\theta'\), where \(\theta\) and \(\theta'\) are eigenvalues of \(\Gamma\) other than \(k\) and \(-k\) that satisfy equality in the BFB. We determine \(\Phi\). The answer depends on a certain expression \(\Delta\) involving the intersection numbers of \(\Gamma\). We show \(\Phi\neq \emptyset\) and \(\Delta= 0\) if and only if \(\Gamma\) is 2-homogeneous in the sense of Curtin and Nomura. We show that if \(D\) is even and at least 6, then \(\Phi\neq\emptyset\) if and only if the halved graph \({1\over 2}\Gamma\) is tight in the sense of Jurisic, Koolen, and Terwilliger. We show that for \(D= 4\) or \(D=5\), \(\Phi\neq \emptyset\) if and only if \(\Gamma\) is antipodal.

MSC:
05E30 Association schemes, strongly regular graphs
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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