A Study on Set-Graphs
Johan Kok, K. P. Chithra, N. K. Sudev, C. Susanth

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 is a cycle of length in . The number of primitive holes in a given graph is called the primitive hole number of that graph . The primitive degree of a vertex of a given graph is the number of primitive holes incident on the vertex . 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.
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