arXiv math/01081962001-08-29math.COmath.RA

紧距离正则图

Tight distance-regular graphs

Aleksandar Jurisic, Jack Koolen, Paul Terwilliger

备注：: 35 pages
发表信息：: J. Alg. Combin. 12 (2000) 163-197

摘要：

本文考虑一个距离正则图$\G$，其直径为$d \ge 3$，特征值为$k=θ_0>θ_1>... >θ_d$ .我们证明了交数$a_1, b_1$满足$$ (θ_1 + {k \over a_1+1}) (θ_d + {k \over a_1+1}) \ge - {ka_1b_1 \over (a_1+1)^2}. $$我们说$\G$是{ \it紧}只要$\G$不是二分的，等式在上面成立。我们用几种方法刻画紧性质。例如，我们证明了$\G$是紧的当且仅当相交数由包含$\G$独立参数的某些有理表达式给出。我们证明了$\G$是紧的当且仅当$a_1\not=0$，$a_d=0$和$\G$是Nomura意义下的1-齐性.我们证明了$\G$是紧的当且仅当每个局部图都是强正则连通的，且具有非平凡的特征值$-1-b_1(1+θ_1)^{-1}$和$-1-b_1(1+θ_d)^{-1}$。给出了紧距离正则图的三个无限族和九个零星例子。

英文摘要：

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.

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