Solvable graphs of finite groups

Temel Bulgular & Veri:We show that $\Gamma_s (G)$ 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 $G$.

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Sonuç & Katkı:We conclude the paper by obtaining a relation between $\Gamma_s (G)$ and the solvability degree of $G$.

🔬Tür:Araştırma Makalesi

📍Kapsam:Genişletilmiş Akademik Örneklem

⚙️Değişkenler:solvable graph · genus · solvability degree

Abstract

Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph $\Gamma_s(G)$ of $G$ is a graph with vertex set $G\setminus Sol(G)$ and two distinct vertices $u$ and $v$ are adjacent if and only if $\langle u, v \rangle$ is solvable. We show that $\Gamma_s (G)$ 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 $\Gamma_s (G)$ and derive a lower bound of the clique number of $\Gamma_s (G)$. 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 $\Gamma_s (G)$ and the solvability degree of $G$.

Anahtar kelimeler

solvable graph genus solvability degree finite group

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Bu Makaleye Atıf Yap

Nath, R. K., Bhowal, P., Nongsiang, D. (2022). Solvable graphs of finite groups. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 6, 1955–1964. https://ajindex.com/articles/529691

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