Fibonacci Edge Irregular Labeling for Graphs Obtained through Vertex Identification with Python Coding
M. Uma Devi, M. Kamaraj, S. Arockiaraj · Mathematics and Statistics · 2026
Graph labeling is an active research area in graph theory, with applications in coding theory, cryptography, and network design. Among the several labeling schemes, edge irregular labelings are most interesting due to their irregularity strength. In this paper, a new labeling scheme called the Fibonacci edge irregular labeling (FEIL) is introduced. In an FEIL, the vertices of a graph are assigned distinct labels from the set , where is a positive integer. Each edge is assigned the sum of the labels of its end vertices, and the resulting edge labels are to be distinct Fibonacci numbers, namely . A graph that admits a FEIL is called a Fibonacci edge irregular graph (FEIG). The Fibonacci edge irregularity strength of , denoted by , is the minimum positive integer for which admits a FEIL. In this paper, we determine the Fibonacci edge irregularity strength for the star , the friendship graph , the bistar , and the graph obtained by vertex identification of the friendship graph with the star . We also establish the non-existence of a FEIL for the path , the cycle , and the complete graph . Additionally, we provide Python code to computationally verify the existence of FEIL for these graph families. The paper also highlights possible future research directions for FEIL.