ON $(\triangle, 1)$-GRAPHS

Abstract

Let $G = (V, E)$ be a graph and $\lambda $ a non-negative integer. A graph $G$ is called a $(\lambda, 1)$-{\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $\lambda$ triangles, and $(c2)$ every two non-adjacent vertices in $G$ are the end-vertices of exactly one two-edge path in $G$. It turns out that there are infinitely many feasible 4-tuples $(v, d, \lambda, 1)$ with $\lambda \ge 1$. On the other hand (and this is our main result), there is no $(v, d, \lambda, 1)$-graphs with $\lambda \ge 1$. As a byproduct, we obtain a generalization of the classical Friendship Theorem.