Friendship Two-Graphs

A friendship graph is a graph in which every two distinct vertices have exactly one common neighbor. All finite friendship graphs are known, each of them consists of triangles having a common vertex. We extend friendship graphs to two-graphs, a two-graph being an ordered pair G = ( G 0 , G 1 ) of edge-disjoint graphs G 0 and G 1 on the same vertex-set V ( G 0 ) = V ( G 1 ). One may think that the edges of G are colored with colors 0 and 1. In a friendship two-graph , every unordered pair of distinct vertices u , v is connected by a unique bicolored 2-path. The pairs of adjacency matrices of friendship two-graphs are solutions to the matrix equation AB + BA = J − I , where A and B are n × n symmetric 0 − 1 matrices, J is an n × n matrix with every entry being 1, and I is the identity n × n matrix. We show that there is no finite friendship two-graph with minimum vertex degree at most two. However, we construct an infinite such graph, and this construction can be extended to an infinite (uncountable) family. Also, we find a finite friendship two-graph, conjecture that it is unique, and prove this conjecture for the two-graphs that have a dominating vertex.