Exploring modular multiplicative divisor labeling to expand graph families

The graph [Formula: see text] is a trivial graph consisting of a single vertex with no edges. In contrast, the graph [Formula: see text] is a complete bipartite graph with one internal node and [Formula: see text] leaf vertices. The join of two graphs [Formula: see text] and [Formula: see text] (with central vertex [Formula: see text]), represented as [Formula: see text], is a graph including vertices [Formula: see text] and edges [Formula: see text]. When two graphs, [Formula: see text] and [Formula: see text] are joined, the result is a graph in which vertex in graph [Formula: see text] is linked to every vertex in graph [Formula: see text]. Modular multiplicative divisor (MMD) labeling is a vertex and edge labeling scheme with the following key features: Vertex labeling: MMD labeling establishes a bijection between the vertices of the graph [Formula: see text] and the natural numbers from 1 to [Formula: see text]. This bijection ensures a one-to-one correspondence, providing a unique label for each vertex. Edge labeling: The labeling of edges follows a specific rule, the edge’s label is determined by calculating the result of multiplying the labels assigned to its connected vertices, with the outcome adjusted by modulo [Formula: see text]. We demonstrate the modular multiplication labeling when combining two graphs through their join (assuming [Formula: see text] is even) and also on the even arbitrary super subdivision (EASS) of the join of two graphs. Additionally, we explore a related research question that arises in this particular context. The findings have potential applications in network topology design and optimization.