Graphs with $r$-friendship property


For $r\geq 1$, a graph has $r$-friendship property if every pair of vertices has exactly $r$ common neighbours. The motivation for this definition is from the Friendship theorem, which is on the graphs with $1$-friendship property. The Friendship theorem, first proved by Erdős, Rényi, and Sós in 1966, states that if $G$ is a graph in which every pair of vertices has exactly one common neighbour, then $G$ has a universal vertex $v$ adjacent to all others, and the graph induced by $V(G)\setminus {v }$ is a matching. In this presentation, we study graphs with $r$-friendship property, where $r\geq 2$. We show all such graphs are strongly regular. Furthermore, we prove that for any $r\geq 2$, there are only finitely many graphs with $r$-friendship property. This is an ongoing joint work with Karen Gunderson.