Tight distance-regular graphs

Abstract

We consider a distance-regular graph $\G$ with diameter $d\ge 3$ and eigenvalues $k={\theta }_0>{\theta }_1>...>{\theta }_d$. We show the intersection numbers $a_1,b_1$ satisfy $$({\theta }_1+\frac{k}{a_1+1})({\theta }_d+\frac{k}{a_1+1})\ge -\frac{k{a}_1{b}_1}{(a_1+1{)}^2}.$$ We say $\G$ is {\it tight} whenever $\G$ is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show $\G$ is tight if and only if the intersection numbers are given by certain rational expressions involving $d$ independent parameters. We show $\G$ is tight if and only if $a_1\ne 0$, $a_d=0$, and $\G$ is 1-homogeneous in the sense of Nomura. We show $\G$ is tight if and only if each local graph is connected strongly-regular, with nontrivial eigenvalues $-1-b_1(1+{\theta }_1{)}^{-1}$ and $-1-b_1(1+{\theta }_d{)}^{-1}$. Three infinite families and nine sporadic examples of tight distance-regular graphs are given.

35 pages