Exploring modular multiplicative divisor labeling to expand graph families.

The graph K 1 is a trivial graph consisting of a single vertex with no edges. In contrast, the graph K 1 , λ is a complete bipartite graph with one internal node and λ leaf vertices. The join of two graphs K 1 and K 1 , λ (with central vertex ρ 0 ), represented as K 1 + K 1 , λ , is a graph including vertices V (K 1 + K 1 , λ) = V (K 1) ∪ V (K 1 , λ) and edges V (K 1 + K 1 , λ) = V (K 1) ∪ V (K 1 , λ) ∪ { σ 1 ρ 0 : σ 1 ∈ V (K 1) , ρ 0 ∈ V (K 1 , λ) }. When two graphs, K 1 and K 1 , λ are joined, the result is a graph in which vertex in graph K 1 is linked to every vertex in graph K 1 , λ . 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 T and the natural numbers from 1 to | v |. 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 n. We demonstrate the modular multiplication labeling when combining two graphs through their join (assuming λ 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. [ABSTRACT FROM AUTHOR]