Study of some properties of complement of open subset inclusion graph of a topological space

Abstract

In the recent paper, authors introduced a graph topological structure, called as open subset inclusion graph 𝚥(𝜏) of a topological space (𝑋, 𝜏) on a finite set 𝑋 and discussed some important properties of this graph. In this paper, we discuss some properties of the graph 𝚥(𝜏)^𝑐. It is shown that, if 𝜏 is a discrete topology defined on nonempty set 𝑋 with |𝑋| ≤ 3, then the graph 𝚥(𝜏)^𝑐 is bipartite, and if |𝑋| = 2, then the graph 𝚥(𝜏)^𝑐 is regular & complete bipartite. Moreover, if 𝜏 is a discrete topology defined on nonempty set 𝑋 with |𝑋| = 2 or |𝑋| = 3 then it is shown that the graph 𝚥(𝜏)^𝑐 is Hamiltonian, vertex-transitive, edge-transitive and has a perfect matching. We also provide exact value of the independence number, vertex connectivity and edge connectivity of the graph 𝚥(𝜏)^𝑐 of a discrete topology defined on nonempty set𝑋 with |𝑋| = 2 or |𝑋| = 3. Main finding of this work is that, if (𝑋, 𝜏) is a discrete topological space with |𝑋| = 2 or |𝑋| = 3 then it is shown that 𝚥(𝜏)^𝑐 is distance-transitive graph and distance regular graph.