arXiv subjects

Kavita Samant

Publications and source records attributed to Kavita Samant.

5 recordsLinked to original sources

arXiv2026

On the Generating Graph of Finite Abelian Groups

The generating graph $\Gamma(G)$ of a group $G$ is the graph whose vertex set is $G$, where two distinct vertices are adjacent if and only if they generate $G$. In this paper, we systematically study the structure of generating graphs of finite abelian groups (non-cyclic) and determine the set of all generating pairs. Moreover, we give some structural characterizations, in particular, we determine conditions under which $\Gamma(G)$ is regular, characterize when the isolated vertices form a subgroup, and establish necessary and sufficient conditions for two non-isomorphic finite abelian groups $G$ and $H$ to satisfy $\Gamma(G)\cong \Gamma(H)$. Furthermore, we compute the spectra of the adjacency and Laplacian matrices of these graphs.

math.CO↗

arXiv2025

An Arithmetic Characterization of 2-Generated Numbers

A group $G$ is said to be $k$-generated if it has a generating set with $k$ elements. A positive integer $n$ is called a \emph{2-generated number} if every group of order $n$ is 2-generated. In this article, we establish an arithmetic characterization of 2-generated numbers expressed in terms of the prime factorization of $n$.

math.GR↗

arXiv2025

Spectral Bounds of the Generating Graph of $\mathbb{Z}_n.$

Let $G$ be a group. A group is said to be $k$-generated if it can be generated by its $k$ elements. A generating set of $G$ is called a minimal generating set if no proper subset of it generates $G.$ A minimal generating set of a group can have different sizes. The generating graph $Γ(G)$ of a group $G$ is defined as a graph with the vertex set $G$, where two distinct vertices are adjacent if they together generate $G.$ This graph is particularly useful when studying 2-generated groups. In this context, consider the group $G = \mathbb{Z}_n$, the integers modulo $n.$ In this paper, we explore various graph-theoretic properties of the generating graph $Γ(\mathbb{Z}_n)$ and investigate the spectra of its adjacency and Laplacian matrices. Additionally, we explicitly determine the set of all possible minimal generating sets of $\mathbb{Z}_n$ of size $k.$

math.CO↗

arXiv2024

The Generating graph of Dicyclic Groups

For a group $G,$ the generating graph of $G,$ denoted by $Γ(G).$ We define $Q_n=\langle x,y: x^{2n}=y^4=1, x^n=y^2,y^{-1}xy=x^{-1}\rangle,$ the dicyclic group of order $4n.$ This paper primarily delves into exploring the graph characteristics and spectral properties of various matrices associated with $Γ(Q_n)$. Specifically, we determine the complete spectrum of the adjacency, Laplacian, distance, and eccentricity matrices. Additionally, we completely determine the spectrum pertaining to the distance and eccentricity matrices of the dihedral group of order $2n$, denoted as $D_n$.

math.CO↗

arXiv2023

Generating Graphs of Finite Dihedral Groups

For a group $G$, the generating graph $Γ(G)$ is defined as the graph with the vertex set $G$, and any two distinct vertices of $Γ(G)$ are adjacent if they generate $G$. In this paper, we study the generating graph of $D_n,$ where $D_n$ is a Dihedral group of order $2n$. We explore various graph theoretic properties, and determine complete spectrum of the adjacency and the Laplacian matrix of $Γ(D_n)$. Moreover, we compute some distance and degree based topological indices of $Γ(D_n)$.

math.CO↗