A Study On Set-Graphs

Abstract

A 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 setgraphs. We also check the primitive hole number and primitive degree of set-graphs. Interesting introductory results on the nature of order of setgraphs, 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.

Theb-chromatic number of certain graphs and digraphs

“…Recently, the families of graphs namely, Jaco graphs, linear Jaco graphs, ornated graphs, Rasta graphs, set-graphs, edge-set graphs and edge-joint graphs were introduced in different studies in [6,7,8,10,11,12].…”

A Study on Edge-Set Graphs of Certain Graphs

“…Let N denotes the set of all positive integers. Then, the notion of a setgraph has been introduced in [6] associated with a given non-empty set, A (n) = {a 1 , a 2 , a 3 , . .…”

“…, a n }, n ∈ N be a non-empty set and the i-th s-element subset of A (n) be denoted by A s,i = ∅}. The set-graph corresponding to set A (n) , denoted G A (n) , is defined to be the graph with V (G A (n) ) = {v s,i : A A detailed study on set-graphs has also been done in [6]. Some of the major results proved in that paper, which are relevant in our present study, are mentioned in this section.…”

“…Theorem 1.2. [6] Let G A (n) be a set-graph. Then, the vertices v s,i , v s,j of G A (n) , corresponding to subsets A (n) s,i and A (n) s,j in S of equal cardinality, have the same degree in G A (n) .…”

“…The degree of a vertex in a set-graph has been determined in [6] using the principle of inclusion and exclusion as follows. Theorem 1.3.…”