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 $p^2q^2$, 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.