Subgraphs of maximum matching graphs (Q1769200)

Let \(G\) be a graph. Define \({\mathcal M}(G)\) to be a graph whose vertices are matchings of the maximum size in \(G\) and edges are pairs of matchings that differ on exactly one edge. The author shows that any two nonadjacent vertices in \({\mathcal M}(G)\) may have at most \(2\) common neighbors. If they have exactly \(2\) common neighbors then the neighbors are nonadjacent. Moreover, for any vertex \(M\) in \({\mathcal M}(G)\), each component of the subgraph induced in \({\mathcal M}(G)\) by the set of neighbors of \(M\) is a complete graph.