Common Divisor Graph of Finite Groups

The interaction between groups and graphs is the most popular and gainful area of algebraic graph theory. We introduce and scrutinize a graph whose vertex set is a group $H$ and the vertices $a$ and $b$ with different orders are adjacent if \(c\notin \{e,a,b\}\) exists such that \(|c| | (|a|, |b|)\) or only one of \(a\) or \(b\) is the identity. We investigate this graph by presenting various examples and demonstrating some important properties and results.