Summary: 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 \(n \times n\) symmetric \(0 - 1\) matrices, \(J\) is an \(n \times n\) matrix with every entry being 1, and \(I\) is the identity \(n \times n\) 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.