| Abstract: | Let  be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition  with block size v. Let be the quotient graph of relative to and [B,C] the bipartite subgraph of induced by adjacent blocks B,C of . In this paper we study such graphs for which is connected, (G, 2)-arc transitive and is almost covered by in the sense that [B,C] is a matching of v-1  2 edges. Such graphs arose as a natural extremal case in a previous study by the author with Li and Praeger. The case  K v+1 is covered by results of Gardiner and Praeger. We consider here the general case where  K v+1, and prove that, for some even integer n 4, is a near n-gonal graph with respect to a certain G-orbit on n-cycles of . Moreover, we prove that every (G, 2)-arc transitive near n-gonal graph with respect to a G-orbit on n-cycles arises as a quotient of a graph with these properties. (A near n-gonal graph is a connected graph  of girth at least 4 together with a set  of n-cycles of such that each 2-arc of is contained in a unique member of .) |
