On graphs associated to topological spaces

Abstract

Let 𝑋 be a set and (𝑋,𝑇) be a topological space. A new type of graph on 𝑃(𝑋), namely the closure graph of 𝑇 is introduced. The closure graph denoted by Γ^c whose vertex set is 𝑃(𝑋) in which two distinct vertices 𝐴 and 𝐵 are adjacent if A'∩B' ⊆ (A∩B)'. In this paper, the closure graph is shown as a simple, connected graph with diameter at most two. Furthermore, the girth of the closure graph Γ^cof 𝑇 is three if 𝑋 contains more than one point. Also, several graph properties are studied.