Symmetric graphs with complete quotients

Abstract

Let $\Gamma$ be a $G$-symmetric graph with vertex set $V$. We suppose that $V$ admits a $G$-partition $\mathcal{B}=\{B_0,...,B_b\}$, with parts of size $v$, and that the quotient graph induced on $\mathcal{B}$ is a complete graph of order $b+1$. Then, for each pair of distinct suffices $i,j$, the graph induced on the union $B_i\cup B_j$ is bipartite with each vertex of valency $0$ or $t$ (a constant). When $t=1$, it was shown earlier how a flag-transitive $1$-design $D(B_i)$ induced on a part $B_i$ can sometimes be used to classify possible triples $(\Gamma ,G,\mathcal{B})$. Here we extend these ideas to $t>1$ and prove that, if the group induced by $G$ on a part $B_i$ is $2$-transitive and the "blocks" of $D(B_i)$ have size less than $v$, then either (i) $v<b$, or (ii) the triple $(\Gamma ,G,\mathcal{B})$ is known explicitly.

arXiv commentThe first version of this manuscript dates from 2000. It was uploaded to the arXiv since several people wished to have a copy. This new version is updated with a literature review up to 2017. It is submitted for publication and is currently under review (September 2017)

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