Solvable graphs of finite groups

Authors : Parthajit BHOWAL, Deiborlang NONGSİANG, Rajat NATH

Pages : 1955-1964

Doi:10.15672/hujms.573766

View : 14 | Download : 8

Publication Date : 2020-12-08

Article Type : Research Paper

Abstract :Let $G$ be a finite non-solvable group with solvable radical $Solinsert ignore into journalissuearticles values(G);$. The solvable graph $\Gamma_sinsert ignore into journalissuearticles values(G);$ of $G$ is a graph with vertex set $G\setminus Solinsert ignore into journalissuearticles values(G);$ and two distinct vertices $u$ and $v$ are adjacent if and only if $\langle u, v \rangle$ is solvable. We show that $\Gamma_s insert ignore into journalissuearticles values(G);$ is not a star graph, a tree, an $n$-partite graph for any positive integer $n \geq 2$ and not a regular graph for any non-solvable finite group $G$. We compute the girth of $\Gamma_s insert ignore into journalissuearticles values(G);$ and derive a lower bound of the clique number of $\Gamma_s insert ignore into journalissuearticles values(G);$. We prove the non-existence of finite non-solvable groups whose solvable graphs are planar, toroidal, double-toroidal, triple-toroidal or projective. We conclude the paper by obtaining a relation between $\Gamma_s insert ignore into journalissuearticles values(G);$ and the solvability degree of $G$.
Keywords : solvable graph, genus, solvability degree, finite group