Further results on the join graph of a finite group

Authors : Zahra BAHRAMI, Bijan TAERI

Pages : 2097-2113

View : 14 | Download : 6

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $G$ be a finite group which is not cyclic of prime power order. The join graph $\Deltainsert ignore into journalissuearticles values(G);$ is an undirected simple whose vertices are the proper subgroups of $G$, which are not contained in the Frattini subgroup $\Phiinsert ignore into journalissuearticles values(G);$ of $G$ and two vertices $H$ and $K$ are joined by an edge if and only if $G=\langle H,K\rangle$. We classify finite groups whose join graphs have domination number $\leq 2$ and independence number $\leq 3$. We show that $\Deltainsert ignore into journalissuearticles values(G);\cong \Deltainsert ignore into journalissuearticles values(A_4);$ if and only if $G\cong A_4$. We also show that if the independence number of $\Deltainsert ignore into journalissuearticles values(G);$ is less than $15$, then $G$ is solvable; moreover, if the equality holds and $G$ is nonsolvable, then $G/\Phiinsert ignore into journalissuearticles values(G);\cong A_5$.
Keywords : Finite group, join graph, domination number, independence number, alternating group