Modular Irregularity Strength of Generalized Book Graph

Consider a graph G with a nonempty set of vertices V(G) and a set of edges E(G). Let Z_n represent the group of integers modulo n, and let k be a positive integer. A modular irregular labeling of a graph G with order n is a labeling of its edges, ϕ : E(G) → {1,2,...,k}, such that a weight function σ : V(G) → Z_n is induced. The weight function is defined as follows: σ(v) = ∑_u∈N(v₎ ϕ(uv) for all vertices v in V(G), where the summation is taken over all vertices u that adjacent with vertex v in G, and this weight function σ must be bijective. The minimum value of k such that a labeling exists in a graph G is called the modular irregularity strength of G, denoted as ms(G). In this research, we have determined the exact values of the modular irregularity strength for generalized book graphs.