Modular irregularity strength of vertex amalgamation and comb product path with cycle related graphs

Consider a graph G = (V(G), E(G)) , where V(G) is a nonempty set of vertices and E(G) is a set of edges. Let Z n be the group of integers modulo n , and let k be a positive integer. A modular irregular labeling of a graph G of order n is a k -edge labeling ϕ : E(G) → {1, 2, … , k} , such that an induced weight function wt ϕ : V(G) → Z n is bijective. The weight function is defined as follows: wt ϕ (u) = Σ u ∈ N(v) ϕ(uv) (mod n) for all vertices v in V(G) . The minimum value of k is called the mo