Distance-regular graphs with light tails

Abstract

Let $\Gamma$ be a distance-regular graph with valency $k\ge 3$ and diameter $d\ge 2$. It is well known that the Schur product $E\circ F$ of any two minimal idempotents of $\Gamma$ is a linear combination of minimal idempotents of $\Gamma$. Situations where there is a small number of minimal idempotents in the above linear combination can be very interesting, since they usually imply strong structural properties, see for example $Q$-polynomial graphs, tight graphs in the sense of Jurišić, Koolen and Terwilliger, and 1- or 2-homogeneous graphs in the sense of Nomura. In the case when $E=F$, the rank one minimal idempotent $E_0$ is always present in this linear combination and can be the only one only if $E=E_0$ or $E=E_d$ and $\Gamma$ is bipartite. We study the case when $E\circ E\in \mathrm{span}\{E_0,H\}\setminus \mathrm{span}\left\{E_0\right\}$ for some minimal idempotent $H$ of $\Gamma$. We call a minimal idempotent $E$ with this property a light tail. Let $\theta$ be an eigenvalue of $\Gamma$ not equal to $\pm k$ and with multiplicity $m$. We show that

$$\frac{m-k}{k}\ge -\frac{{(\theta +1)}^2{a}_1(a_1+1)}{{((a_1+1)\theta +k)}^2+k{a}_1{b}_1}\text{.}$$

Let $E$ be the minimal idempotent corresponding to $\theta$. The equality case is equivalent to $E$ being a light tail. Two additional characterizations of the case when $E$ is a light tail are given. One involves a connection between two cosine sequences and the other one a parameterization of the intersection numbers of $\Gamma$ with $a_1$ and the cosine sequence corresponding to $E$. We also study distance partitions of vertices with respect to two vertices and show that the distance-regular graphs with light tails are very close to being 1-homogeneous. In particular, their local graphs are strongly regular.