Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Some Bi-Cayley Graphs over Cyclic Groups of Order $p^2 q^2$ and Related Extensions

Iqbal Atmaja, Yeni Susanti, Ahmad Erfanian

Abstract

We investigate structural and combinatorial properties of Bi-Cayley graphs defined over cyclic groups of order $p^2q^2$, where $p$ and $q$ are distinct primes. We begin by describing their fundamental group-theoretic underpinnings. The main focus is on analyzing their connectivity, girth, clique number, chromatic number, diameter, and independence number. It is shown that these Bi-Cayley graphs are connected, biregular with explicitly determined degrees, and possess girth three. Furthermore, we prove that their diameter is equal to five. We further extend several results to Bi-Cayley graphs over arbitrary finite groups under suitable restrictions on the connecting set, with particular emphasis on the case where the connecting set consists of all its involutions. These results clarify structural similarities and differences between Cayley graphs and their Bi-Cayley generalizations.

On Some Bi-Cayley Graphs over Cyclic Groups of Order $p^2 q^2$ and Related Extensions

Abstract

We investigate structural and combinatorial properties of Bi-Cayley graphs defined over cyclic groups of order , where and are distinct primes. We begin by describing their fundamental group-theoretic underpinnings. The main focus is on analyzing their connectivity, girth, clique number, chromatic number, diameter, and independence number. It is shown that these Bi-Cayley graphs are connected, biregular with explicitly determined degrees, and possess girth three. Furthermore, we prove that their diameter is equal to five. We further extend several results to Bi-Cayley graphs over arbitrary finite groups under suitable restrictions on the connecting set, with particular emphasis on the case where the connecting set consists of all its involutions. These results clarify structural similarities and differences between Cayley graphs and their Bi-Cayley generalizations.
Paper Structure (6 sections, 47 theorems, 32 equations, 4 figures)

This paper contains 6 sections, 47 theorems, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Results and Discussion
  4. Bi-Cayley Graph Over Cyclic Groups
  5. Extension to Arbitrary Finite Group
  6. Conclusion

Key Result

Proposition 2.5

The Bi-Cayley graph $\Gamma = \mathrm{BiCay}(G; S_1,S_2,S_3)$ consists of two Cayley subgraphs.

Figures (4)

  • Figure 1: Bi-Cayley graph $\mathrm{BiCay}(\mathrm{Sym}_3; S_1, S_2, S_3)$
  • Figure 2: Subgraph induced by $\{ 0\} \times \mathbb{Z}_{36}$
  • Figure 3: Subgraph induced by $\{ 1\} \times \mathbb{Z}_{36}$
  • Figure 4: Bi-Cayley over $G \cong \mathbb{Z}_{36}$

Theorems & Definitions (81)

  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • Proposition 2.8
  • proof
  • ...and 71 more