Triangle-degree and triangle-distinct graphs

TL;DR

This paper explores graphs where each vertex has a unique triangle-degree, constructing an infinite family of such graphs and analyzing their degree and size properties.

Contribution

It introduces the concept of triangle-degree-distinct graphs and provides a construction for an infinite family of such graphs, expanding understanding of their structure.

Findings

Constructed an infinite family of triangle-degree-distinct graphs.

Analyzed the vertex degrees and sizes of these graphs.

Abstract

Let $G$ be a simple graph and $v$ be a vertex of $G$. The triangle-degree of $v$ in $G$ is the number of triangles that contain $v$. While every graph has at least two vertices with the same degree, there are graphs in which every vertex has a distinct triangle-degree. In this paper, we construct an infinite family of graphs with this property. We also study the vertex degrees and size of graphs with this property.