Abstract

For any prime power q 3, we consider two infinite series of bipartite q-regular edge-transitive graphs of orders 2q³ and 2q⁵ which are induced subgraphs of regular generalized 4-gons and 6-gons, respectively. We compare these two series with two families of graphs, H₃(p) and H₅ (p), p is a prime, constructed recently by Wenger ([26]), which are new examples of extremal graphs without 6- and 10-cycles respectively. We prove that the first series contains the family h₃ (p) for q = p 3. Then we show that no member of the second family H₅(p) is a subgraph of a generalized 6-gon.

Then, for infinitely many values of q, we construct a new series of bipartite q-regular edge-transitive graphs of order 2q⁵ and girth 10.

Finally, for any prime power q 3, we cosntruct a new infinite series of bipartite q-regular edge-transitive graphs of order 2q⁹ and girth g 14.

Our construction were motivated by some results on embeddings of Chevalley group geometries in the corresponding Lie algebras and a construction of a blow-up for an incident system and a graph.