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Generating Graphs of Finite Dihedral Groups

A. Satyanarayana Reddy, Kavita Samant

TL;DR

The paper studies the generating graph $\Gamma(D_n)$ of the dihedral group $D_n$ by partitioning the group into rotations and reflections and removing isolated vertices to form $\Delta_n$. It derives exact spectra for the adjacency and Laplacian matrices, showing $\Gamma_n$ is Laplacian-integral and providing explicit eigenvalues involving Ramanujan sums and the square-free part of $n$, with an additional adjacency-energy formula $AE(n)=\varphi(n)(2^k-1)+\sqrt{\varphi(n)^2+4n\varphi(n)}$. An alternative Kronecker-product-based method corroborates the spectrum, reinforcing integrality results and illustrating how prime factors shape the spectrum. The work also computes distance- and degree-based topological indices for $\Delta_n$, delivering closed-form expressions and special-case tables for prime and power-of-two sizes, bridging group-theoretic graphs with chemical graph indices and spectral graph theory.

Abstract

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 $Γ(D_n)$.

Generating Graphs of Finite Dihedral Groups

TL;DR

The paper studies the generating graph of the dihedral group by partitioning the group into rotations and reflections and removing isolated vertices to form . It derives exact spectra for the adjacency and Laplacian matrices, showing is Laplacian-integral and providing explicit eigenvalues involving Ramanujan sums and the square-free part of , with an additional adjacency-energy formula . An alternative Kronecker-product-based method corroborates the spectrum, reinforcing integrality results and illustrating how prime factors shape the spectrum. The work also computes distance- and degree-based topological indices for , delivering closed-form expressions and special-case tables for prime and power-of-two sizes, bridging group-theoretic graphs with chemical graph indices and spectral graph theory.

Abstract

For a group , the generating graph is defined as the graph with the vertex set , and any two distinct vertices of are adjacent if they generate . In this paper, we study the generating graph of where is a Dihedral group of order . We explore various graph theoretic properties, and determine complete spectrum of the adjacency and the Laplacian matrix of . Moreover, we compute some distance and degree based topological indices of .
Paper Structure (8 sections, 10 theorems, 51 equations, 1 figure, 4 tables)

This paper contains 8 sections, 10 theorems, 51 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Few properties of $\Delta_n$
  4. Spectrum of the generating graph of $D_n$
  5. Spectrum of the Adjacency Matrix
  6. An alternative approach
  7. Spectrum of the Laplacian Matrix
  8. Topological Indices of $\Delta_n$

Key Result

Theorem 4.3

If $\Pi$ is an equitable partition of a graph $\mathcal{G},$ then the characteristic polynomial of $A(\mathcal{G}/\Pi)$ divides the characteristic polynomial of $A(\mathcal{G}).$

Figures (1)

  • Figure 1: Generating graph of $D_3$, Generating graph of $D_{4},$ and Generating graph of $D_{5}.$

Theorems & Definitions (21)

  • Example 2.1
  • Example 4.1
  • Example 4.2
  • Theorem 4.3: Godsil and Royle, godsil
  • Theorem 4.4
  • proof
  • Theorem 4.5
  • Corollary 4.6
  • proof
  • Example 4.7
  • ...and 11 more