Finite symmetric graphs with two‐arc transitive quotients II

Abstract

AbstractLet $\Gamma$ be a finite G‐symmetric graph whose vertex set admits a nontrivial G‐invariant partition $\cal B$. It was observed that the quotient graph $\Gamma_{\cal B}$ of $\Gamma$ relative to $\cal B$ can be (G, 2)‐arc transitive even if $\Gamma$ itself is not necessarily (G, 2)‐arc transitive. In a previous article of Iranmanesh et al., this observation motivated a study of G‐symmetric graphs ($\Gamma, \cal B$) such that $\Gamma_{\cal B}$ is (G, 2)‐arc transitive and, for blocks B, C ∈ $\cal B$ adjacent in $\Gamma_{\cal B}$, there are exactly |B| − 2(≥1) vertices in B which have neighbors in C. In the present article we investigate the general case where $\Gamma_{\cal B}$ is (G, 2)‐arc transitive and is not multicovered by $\Gamma$ (i.e., at least one vertex in B has no neighbor in C for adjacent B, C ∈ $\cal B$) by analyzing the dual $\cal D^\ast$(B) of the 1‐design $\cal D$(B) ≔ (B, $\Gamma_{\cal B}$(B), I), where $\Gamma_{\cal B}$(B) is the neighborhood of B in $\Gamma_{\cal B}$ and αIC 〈α ∈ B, C ∈ $\Gamma_{\cal B}$(B)〉 in $\cal D$(B) if and only if α has at least one neighbor in C. In this case, a crucial feature is that $\cal D^\ast$(B) admits G as a group of automorphisms acting 2‐transitively on points and transitively on blocks and flags. It is proved that the case when no point of $\cal D$(B) is incident with two blocks can be reduced to multicovers, and the case when no point of $\bar {\cal D}(B)$ is incident with two blocks can be partially reduced to the 3‐arc graph construction, where $\bar {\cal D}(B)$ is the complement of $\cal D$(B). In the general situation, both $\cal D^\ast$(B) and its complement $\bar {{\cal D}^\ast}(B)$ are (G, 2)‐point‐transitive and G‐block‐transitive 2‐designs, and exploring relationships between them and $\Gamma$ is an attractive research direction. In the article we investigate the degenerate case where $\cal D^\ast$(B) or $\bar {{\cal D}^\ast}(B)$ is a trivial Steiner system with block size 2, that is, a complete graph. In each of these cases, we give a construction which produces symmetric graphs with the corresponding properties, and we prove further that every such graph $\Gamma$ can be constructed from $\Gamma_{\cal B}$ by using the construction. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 167–193, 2007