On a bipartite graph defined on groups

TL;DR

This paper introduces a bipartite graph based on a group's subgroups and generators, exploring its properties, relations to group probabilities, and specific structures for certain finite groups.

Contribution

It defines a new bipartite graph on groups, analyzes its parameters, and characterizes its structure for specific finite groups, linking group theory and graph theory.

Findings

Relations between the bipartite graph and group generating probabilities

Explicit structures of the graph for dihedral and dicyclic groups

Calculations of various graph parameters and topological indices

Abstract

Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. We introduce a bipartite graph $\mathcal{B}(G)$ on $G$ whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. We establish connections between $\mathcal{B}(G)$ and the generating graph of $G$. We also discuss about various graph parameters such as independence number, domination number, girth, diameter, matching number, clique number, irredundance number, domatic number and minimum size of a vertex cover of $\mathcal{B}(G)$. We obtain relations between $\mathcal{B}(G)$ and certain probabilities associated to finite groups. We also obtain expressions for various topological indices of $\mathcal{B}(G)$. Finally, we realize the structures of $\mathcal{B}(G)$ for the dihedral groups of order $2p$ and $2p^2$ and dicyclic groups of order $4p$ and $4p^2$ (where $p$ is any prime) including certain other small order groups.