2019/03/05 by Parthajit Bhowal, Bhowal, Parthajit, Deiborlang Nongsiang +3 · 1 citation
Chemistry · Mathematics · #FOS: Mathematics #Ferrocene Chemistry and Applications #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1903.01755
openalex publication_date 2019/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a finite non-solvable group with solvable radical Sol(G). The solvable graph Γs(G) of G is a graph with vertex set G∖ Sol(G) and two distinct vertices u and v are adjacent if and only if ⟨ u, v ⟩ is solvable. We show that Γs (G) is not a star graph, a tree, an n-partite graph for any positive integer n ≥ 2 and not a regular graph for any non-solvable finite group G. We compute the girth of Γs (G) and derive a lower bound of the clique number of Γs (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 Γs (G) and the solvability degree of G.