Special Intersection Graph in The Topological Graphs

TL;DR

This paper introduces a new class of graphs derived from discrete topological spaces, analyzing their properties such as clique number, degrees, domination, connectivity, and diameter, with specific focus on their structural characteristics.

Contribution

It constructs and investigates properties of special intersection graphs from topological spaces, including degree, domination, connectivity, and their behavior under graph operations.

Findings

Clique number equals the number of elements in X.

Graph has no isolated vertices and is connected.

Minimum dominating set and graph parameters are explicitly determined.

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.