Tight distance-regular graphs
| dc.creator | Jurisic, Aleksandar | |
| dc.creator | Koolen, Jack | |
| dc.creator | Terwilliger, Paul | |
| dc.date | 2001-08-29 | |
| dc.date.accessioned | 2026-07-07T04:43:10Z | |
| dc.date.available | 2026-07-07T04:43:10Z | |
| dc.description | 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. | |
| dc.description | 35 pages | |
| dc.identifier | https://arxiv.org/abs/math/0108196 | |
| dc.identifier | http://arxiv.org/abs/math/0108196 | |
| dc.identifier | J. Alg. Combin. 12 (2000) 163-197 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62098 | |
| dc.subject | Combinatorics | |
| dc.subject | Rings and Algebras | |
| dc.subject | 05E (primary), 15A (secondary) | |
| dc.title | Tight distance-regular graphs | |
| dc.type | text |