Tight distance-regular graphs

Details Tight distance-regular graphs Tight distance-regular graphs

2001-08-29

arxiv.org/abs/math/0108196v1

J. Alg. Combin. 12 (2000) 163-197

Mathematics

Combinatorics

35 pages

Scientific paper

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 $$ (\theta_1 + {k \over a_1+1}) (\theta_d + {k \over a_1+1}) \ge - {ka_1b_1 \over (a_1+1)^2}. $$ 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 $-1-b_1(1+\theta_1)^{-1}$ and $-1-b_1(1+\theta_d)^{-1}$. Three infinite families and nine sporadic examples of tight distance-regular graphs are given.

Affiliated with

Also associated with

No associations

Rating

Tight distance-regular graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Tight distance-regular graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Tight distance-regular graphs will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-595043