### 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.**Published:**2022-03-14

**How to Cite this Article:**Reeta Madan, Soni Pathak, R. A. Muneshwar, K. L. Bondar, Study of some properties of complement of open subset inclusion graph of a topological space, J. Math. Comput. Sci., 12 (2022), Article ID 108 Copyright © 2022 Reeta Madan, Soni Pathak, R. A. Muneshwar, K. L. Bondar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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