New Examples of Graphs without Small Cycles and of Large Size

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