Distance-regular graphs with complete multipartite $\mu$-graphs and AT4 family

Jurišić, Aleksandar ; Koolen, Jack

Journal of Algebraic Combinatorics, Tome 25 (2007) no. 4, pp. 459-471.

Résumé

Summary: Let $\Gamma $be an antipodal distance-regular graph of diameter 4, with eigenvalues $\theta_0>\theta_1>\theta_2>\theta_3>\theta_4$ theta_0>theta_1>theta_2>theta_3>theta_4. Then its Krein parameter $q _{11} ^{4}$ q_11^4 vanishes precisely when $\Gamma $is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when $\Gamma $is locally strongly regular with nontrivial eigenvalues $p$:= q $_{2}$ p:=$\theta_2 $and - $q$:= q $_{3}$ -q:=theta_3. When this is the case, the intersection parameters of $\Gamma $can be parametrized by $p, q$ and the size of the antipodal classes $r$ of $\Gamma $. Let $\Gamma $be an antipodal tight graph of diameter 4, denoted by AT4 (p, q, r), and let the $\mu $-graph be a graph that is induced by the common neighbours of two vertices at distance 2. Then we show that all the $\mu $-graphs of $\Gamma $are complete multipartite if and only if $\Gamma $is $AT4( sq, q, q)$ for some natural number $s$. As a consequence, we derive new existence conditions for graphs of the AT4 family whose $\mu $-graphs are not complete multipartite. Another interesting application of our results is also that we were able to show that the $\mu $-graphs of a distance-regular graph with the same intersection array as the Patterson graph are the complete bipartite graph $K _{4,4}$.

Détail

Keywords: keywords distance-regular graphs, antipodal, tight, locally strongly regular, $\mu $-graphs, AT4 family