A Study of Fibonacci Cordial Labeling in Structured Graph Families

A \emph{Fibonacci cordial labeling} of a graph $G$ is an injective function $f: V(G) \rightarrow \{F_0, F_1, \dots, F_n\}$, where $F_i$ denotes the $i^{\text{th}}$ Fibonacci number, such that the induced edge labeling $f^*: E(G) \rightarrow \{0,1\}$, given by $f^*(uv) = (f(u) + f(v))$ $(\bmod\ 2)$, satisfies the balance condition $|e_f(0) - e_f(1)| \le 1$. Here, $e_f(0)$ and $e_f(1)$ represent the number of edges labeled 0 and 1, respectively. A graph that admits such a labeling is termed a \emph{Fibonacci cordial graph}. In this paper, we investigate the existence and construction of Fibonacci cordial labelings for several families of graphs, including \emph{Generalized Petersen graphs}, \emph{open and closed helm graphs}, \emph{joint sum graphs}, and \emph{circulant graphs of small order}. New results and examples are presented, contributing to the growing body of knowledge on graph labelings inspired by numerical sequences.