The structures of the A4−graphs in specific finite simple groups

The main idea of the thesis is to study the connections between two important fields of mathematics: Group theory and graph theory. The essential purpose is to analyze the algebraic characteristics of certain finite simple groups by building a graph with vertices that correspond to the elements of the groups. Furthermore, the computational approach was employed in order to attain the required outcome. Assume that $G$ is a group and that $X$ is a subset of $G$. The vertex set of the $A4-$graph indicated by $A4(G,X)$ is $X$, with vertices $x,y \in X$ linked by an edge if and only if $x\neq y$ and $xy^{-1}=yx^{-1}$. We investigated the $A4(G, X)$ when $G$ is a certain Janko sporadic simple group, such as $J_1, J_2$ or $J_3$, and $X$ is a $G-$conjugacy class of order 3. The purpose is to investigate the alternating group $A_4$ within these simple groups. The disc structures were studied. Moreover, the diameter, clique number, girth, the collapsed adjacency matrix for the $A4-$graph as well as the local clustering coefficient of the $A4-$graph were all determined. Furthermore, a computer technique was utilized to perform all of the operations related to studying the $A4-$graph.