Aleksandar Jurisic | Searcharxiv

arXiv2009

A characterization of Q-polynomial distance-regular graphs

We obtain the following characterization of $Q$-polynomial distance-regular graphs. Let $\G$ denote a distance-regular graph with diameter $d\ge 3$. Let $E$ denote a minimal idempotent of $\G$ which is not the trivial idempotent $E_0$. Let $\{θ_i^*\}_{i=0}^d$ denote the dual eigenvalue sequence for $E$. We show that $E$ is $Q$-polynomial if and only if (i) the entry-wise product $E \circ E$ is a linear combination of $E_0$, $E$, and at most one other minimal idempotent of $\G$; (ii) there exists a complex scalar $β$ such that $θ^*_{i-1}-βθ^*_i + θ^*_{i+1}$ is independent of $i$ for $1 \le i \le d-1$; (iii) $θ^*_i \ne θ^*_0$ for $1 \le i \le d$.

math.CO↗

arXiv2001

Tight distance-regular graphs

We consider a distance-regular graph $\G$ with diameter $d \ge 3$ and eigenvalues $k=θ_0>θ_1>... >θ_d$. We show the intersection numbers $a_1, b_1$ satisfy $$ (θ_1 + {k \over a_1+1}) (θ_d + {k \over a_1+1}) \ge - {ka_1b_1 \over (a_1+1)^2}. $$ 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 $a_1\not=0$, $a_d=0$, 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 $-1-b_1(1+θ_1)^{-1}$ and $-1-b_1(1+θ_d)^{-1}$. Three infinite families and nine sporadic examples of tight distance-regular graphs are given.

math.CO↗