The modular irregularity strength of middle graphs of path graphs, triangular snake graphs, and diamond graphs

Source abstract

Given a simple undirected graph, a modular irregularity labeling is an edge labeling that maps each edge to a positive integer such that the weight of all vertices is distinct. The weight of a vertex is defined as the sum of the label values of the incident edges modulo a certain number, and it is bijective. The minimum value from the range of labelings that allows the graph to have a modular irregularity labeling is known as the graph's modular irregularity strength. This study explicitly establishes the modular irregularity strength for three specific graph classes: the middle graph of a path graph, the triangular snake graph, and the diamond graph, all of which have not been previously studied. The method used in this study involves explicit construction and analysis of the upper and lower bounds of the modular irregularity strength for the graphs. Research on modular irregularity labeling is important theoretically because it can deepen the understanding of more complex graph structures and optimize labeling. Received: 30 November 2025 | Revised: 10 June 2026 | Accepted: 01 September 2026