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Structural and Spectral Properties of Prime Order Element Graph of Finite Abelian Groups

Tapa Manna, Supriyo Dutta, Baby Bhattacharya

Abstract

Given a finite group $G$, the \emph{Prime Order Element (POE) Graph} $Γ(G)$ consists of the group elements as the vertices, and two vertices $x$ and $y$ are adjacent if and only if $o(xy)$ is prime. This paper presents a thorough structural and spectral analysis of the POE graphs associated with the finite Abelian groups of different types. The order of a finite Abelian group may be a prime or a product of primes, which influences the structure of POE graphs. The POE graph is connected when the order of the Abelian group is a square-free integer. The POE graphs of the other Abelian groups have multiple connected components. Some of these components are isomorphic to the POE graph of a lower-order group. We study various graph-theoretic properties of the components, including regularity and bipartiteness. Arranging the elements of the group in a number of particular orders, we observe the block structure in the adjacency matrix of POE graphs. It assists us in investigating the spectral properties of POE graphs. We explicitly derive the characteristic polynomials governing both integral and irrational eigenvalues, and compute the eigenvalues with multiplicity in terms of the structure of the graphs.

Structural and Spectral Properties of Prime Order Element Graph of Finite Abelian Groups

Abstract

Given a finite group , the \emph{Prime Order Element (POE) Graph} consists of the group elements as the vertices, and two vertices and are adjacent if and only if is prime. This paper presents a thorough structural and spectral analysis of the POE graphs associated with the finite Abelian groups of different types. The order of a finite Abelian group may be a prime or a product of primes, which influences the structure of POE graphs. The POE graph is connected when the order of the Abelian group is a square-free integer. The POE graphs of the other Abelian groups have multiple connected components. Some of these components are isomorphic to the POE graph of a lower-order group. We study various graph-theoretic properties of the components, including regularity and bipartiteness. Arranging the elements of the group in a number of particular orders, we observe the block structure in the adjacency matrix of POE graphs. It assists us in investigating the spectral properties of POE graphs. We explicitly derive the characteristic polynomials governing both integral and irrational eigenvalues, and compute the eigenvalues with multiplicity in terms of the structure of the graphs.
Paper Structure (8 sections, 49 theorems, 45 equations, 6 figures, 1 table)

This paper contains 8 sections, 49 theorems, 45 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Properties of $\Gamma({\mathbb{Z}_p}^{(n)})$
  4. Properties of $\Gamma(\mathbb{Z}_{p^n})$
  5. Properties of $\Gamma(\mathbb{Z}_{2^n})$
  6. Properties of ${\Gamma(\mathbb{Z}_{2^{n_1}} \times \mathbb{Z}_{2^{n_2}}\times \dots \times \mathbb{Z}_{2^{n_k}}})$
  7. Properties of $\Gamma(\mathbb{Z}_{2^{n_1}{p_2}^{n_2}})$
  8. Properties of $\Gamma(\mathbb{Z}_{{p_1^{n_1}}{p_2^{n_2}}})$

Key Result

Theorem 1.1

Every finitely generated abelian group $G$ is isomorphic to a direct sum of groups where $r \ge 0$ is a non-negative integer and $n_{1}, n_{2}, \dots, n_{k} \in \mathbb{N}$ satisfy $n_{1} \mid n_{2} \mid \cdots \mid n_{k}$.

Figures (6)

  • Figure 1: POE graphs of $\mathbb{Z}_p^{(n)}$ for $n = 2$ and $p = 3$ and $5$. In both the graphs the vertex $(0, 0)$ represents the identity element of the graph. It is connected to all other vertices.
  • Figure 2: The POE graph $\Gamma({\mathbb{Z}_{5^2}})$, which has three components. The leftmost component is isomorphic to $\Gamma({\mathbb{Z}})$. The other two components are are isomorphic to each other.
  • Figure 3: POE graphs associated to $\mathbb{Z}_4$ and $\mathbb{Z}_8$. They are union of $K_2$ graphs and isolated vertices.
  • Figure 4: The POE graph $\Gamma({\mathbb{Z}_2} \times {\mathbb{Z}_8})$. The graph is disconnected. Note that none of the connected components are isomorphic to $\Gamma(\mathbb{Z}_2)$ or $\Gamma(\mathbb{Z}_2)$, which are depicted in \ref{['Z_2_n_POE_graphs']}.
  • Figure 5: The POE graph $\Gamma({\mathbb{Z}_{2^23^2}})$ is a graph with $4$ connected components.
  • ...and 1 more figures

Theorems & Definitions (95)

  • Definition 1.1
  • Theorem 1.1: Fundamental Theorem of Finitely Generated Abelian Groups:
  • Lemma 2.1
  • Definition 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • Lemma 4.1
  • proof
  • ...and 85 more