A construction of imprimitive symmetric graphs which are not multicovers of their quotients

This paper gives a sufficient and necessary condition for the existence of an (X, s)-arc-transitive imprimitive graph which is not a multicover of a given quotient graph.Comment: 16 pages with 1 figure, Published in Discrete Math 201

Symmetric graphs with 2-arc transitive quotients

A graph \Ga is $G$-symmetric if \Ga admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of \Ga, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is imprimitive on V(\Ga), namely when V(\Ga) admits a nontrivial $G$-invariant partition \BB, the quotient graph \Ga_{\BB} of \Ga with respect to \BB is always $G$-symmetric and sometimes even $(G, 2)$-arc transitive. (A $G$-symmetric graph is $(G, 2)$-arc transitive if $G$ is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for \Ga_{\BB} to be $(G, 2)$-arc transitive (regardless of whether \Ga is $(G, 2)$-arc transitive) in the case when $v-k$ is an odd prime $p$, where $v$ is the block size of \BB and $k$ is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of $v, k$ and two other parameters with respect to (\Ga, \BB) together with a certain 2-point transitive block design induced by (\Ga, \BB). We prove further that if $p=3$ or $5$ then these necessary conditions are essentially sufficient for \Ga_{\BB} to be $(G, 2)$-arc transitive.Comment: To appear in Journal of the Australian Mathematical Society. (The previous title of this paper was "Finite symmetric graphs with two-arc transitive quotients III"