Some properties of the complement of intersection graph derived from topological space using intersection of open sets

In our most recent article, we construct a graph structure known as an intersection graph γ(τd) on a topological space(χ, τd). Now, in this article, we’ll look at some of the features of the complement of γ(τd) such as diameter, girth, connectivity, maximal independent sets. It is demonstrated that if r is the discrete topology on χ and |χ| > 2, then γ(τd)c is connected graph and we also determine its diameter and girth. The main finding of this study is, if τd is the discrete topology on χ, & |χ| ≥ 3, then γ(τd)c satisfies Beal’s conjecture. Moreover, different variants of domination number, degree and connectivity of γ(τd)c and neighborhood of open sets in the graph γ(τd)c are studied.