Asian-European Journal of Mathematics, Volume (16), No (10), Year (2023-8)

Title : ( A bipartite graph associated to a finite group )

Authors: Mansoureh Mahtabi , Ahmad Erfanian , Robabeh Mahtabi ,

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Abstract

Let G be a finite group and AG be the automorphism group of G (i.e. Aut(G)). We associated a bipartite graph, denoted by ΓG, to G and its automorphism group Aut(G) as follows: two parts of the vertex set are G\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\L(G, Z(G)) and AG\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Autc(G), where L(G, Z(G)) is the set of elements g ∈ G such that g−1α(g) ∈ Z(G) for all α ∈ AG and Autc(G) is the set of automorphisms β ∈ AG such that g−1β(g) ∈ Z(G) for all g ∈ G. Two vertices g ∈ G\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\L(G, Z(G)) and α ∈ AG\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Autc(G) are adjacent if and only if g−1α(g) ∈ Z(G). In this paper, we investigate some fundamental properties of ΓG such as connectivity, diameter, girth, Hamiltonian, independence and dominating numbers. Moreover, planarity and outer planarity of the graph are studied.

Keywords

Bipartite graph; automorphism group; diameter; independence number; planer; outer planar.
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@article{paperid:1098204,
author = {Mahtabi, Mansoureh and Erfanian, Ahmad and ربابه مهتابی},
title = {A bipartite graph associated to a finite group},
journal = {Asian-European Journal of Mathematics},
year = {2023},
volume = {16},
number = {10},
month = {August},
issn = {1793-5571},
keywords = {Bipartite graph; automorphism group; diameter; independence number; planer; outer planar.},
}

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%0 Journal Article
%T A bipartite graph associated to a finite group
%A Mahtabi, Mansoureh
%A Erfanian, Ahmad
%A ربابه مهتابی
%J Asian-European Journal of Mathematics
%@ 1793-5571
%D 2023

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