A Study on Set-Graphs

TL;DR

This paper introduces set-graphs, explores their properties, and investigates the primitive hole number and degree, providing formulas and results on their structure and characteristics.

Contribution

It defines set-graphs and analyzes their properties, including primitive hole number and degree, with new formulas and insights into their structural features.

Findings

Primitive hole number of set-graphs derived recursively.

Degree analysis of vertices based on subset cardinality.

Number of largest complete subgraphs in set-graphs.

Abstract

A \textit{primitive hole} of a graph $G$ is a cycle of length $3$ in $G$. The number of primitive holes in a given graph $G$ is called the primitive hole number of that graph $G$. The primitive degree of a vertex $v$ of a given graph $G$ is the number of primitive holes incident on the vertex $v$. In this paper, we introduce the notion of set-graphs and study the properties and characteristics of set-graphs. We also check the primitive hole number and primitive degree of set-graphs. Interesting introductory results on the nature of order of set-graphs, degree of the vertices corresponding to subsets of equal cardinality, the number of largest complete subgraphs in a set-graph etc. are discussed in this study. A recursive formula to determine the primitive hole number of a set-graph is also derived in this paper.