Complete (2,2) Bipartite Graphs

A bipartite graph G can be treated as a (1, 1) bipartite graph in the sense that, no two vertices in the same part are at distance one from each other. A (2, 2) bipartite graph is an extension of the above concept in which no two vertices in the same part are at distance two from each other. In this article, analogous to complete (1, 1) bipartite graphs which have the maximum number of pairs of vertices having distance one between them, a complete (2, 2) bipartite graph is defined as follows. A complete (2, 2) bipartite graph is a graph which is (2, 2) bipartite and has the maximum number of pairs of vertices (u, v) such that d(u, v) = 2. Such graphs are characterized and their properties are studied. The expressions are derived for the determinant, the permanent and spectral properties of some classes of complete (2, 2) bipartite graphs. A class of graphs among complete (2, 2) bipartite graphs having golden ratio in their spectrum is obtained.