Special Intersection Graph in The Topological Graphs

Abstract

In this paper, new graphs $G_{\tau }=\left(V,E\right)$ are constructed from the discrete topological space $(X,\tau )$ . Several properties of this type of graphs are given such that: the clique number equals the number of elements in X also the number of pendants vertices, $G_{\tau }$ has no isolated vertices, the minimum degree in $G_{\tau }$ is one and maximum degree equal $n-1+{\sum }_{i=2}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{i}\right)$ , the minimum dominating set is determined and $\gamma (G_{\tau })$ is evaluated for $G_{\tau }$ and for corona and join operations between to discrete topological graphs. At what matter $\beta \left(G_{\tau }\right)=\gamma (G_{\tau })$ is discussed for $G_{\tau }$. Also that $G_{\tau }$ is proved a connected graph of order $2^n-2$ and it has no isolated vertex. Then, rad $G_{\tau }$ and diam $(G_{\tau })$ are evaluated.