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 $n-1+\sum^{n-1}_{i=2}\binom{n-1}{i}$ , 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.