2012/05/05 by Guangjun Xu, Xu, Guangjun, Sanming Zhou +1
Computer Science · Engineering · Mathematics · #05C25 #05E18 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems #math.CO #msc:05C25 #msc:05E18
paper · pdf · doi:10.48550/arxiv.1205.1084
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")
openalex publication_date 2012/05/05 · arxiv created 2013/11/26 · arxiv updated 2013/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.