Regular graphs with girth at least 5 and small second largest eigenvalue

In this paper we present an upper bound on the degree of a regular graph with girth at least 5 in terms of its second largest eigenvalue (usually denoted by λ2). Next, we consider bipartite r-regular graphs with girth at least 6 whose second largest eigenvalue satisfies and prove that such graphs are the incidence graphs of two-class partially balanced incomplete block designs and also balanced incomplete block designs. Upper bound on the order of such graphs is also given. We also consider the relations between the incidence graphs of two-class partially balanced incomplete block designs and distance-regular graphs. In particular, we determine all regular reflexive graphs of degree 3 with girth at least 5, all regular graphs with girth at least 5 satisfying and all bipartite regular reflexive graphs with girth at least 6.