ON $(\triangle, 1)$-GRAPHS

TL;DR

This paper investigates a special class of graphs called $( riangle, 1)$-graphs, proving their non-existence for certain parameters and generalizing the Friendship Theorem, with implications for graph theory structures.

Contribution

The paper introduces the concept of $( riangle, 1)$-graphs, proves their non-existence for $ ext{lambda} \\ge 1$, and generalizes the classical Friendship Theorem.

Findings

No $(v, d, \\lambda, 1)$-graphs exist for $\\lambda \\ge 1$

Infinitely many feasible 4-tuples $(v, d, \\lambda, 1)$ with $\\lambda \\ge 1$ are identified

Generalization of the Friendship Theorem derived from the main results

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.