Modular edge irregularity strength of graphs

For a simple graph $G=(V,E)$ with the vertex set $V(G)$ and the edge set $E(G)$, a vertex labeling $\phi :V(G)\to \{1,2,\dots ,k\}$ is called a $k$-labeling. The weight of an edge under the vertex labeling $\phi$ is the sum of the labels of its end vertices and the modular edge-weight is the remainder of the division of this sum by $|E(G)|$. A vertex $k$-labeling is called a modular edge irregular if for every two different edges their modular edge-weights are different. The maximal integer $k$ minimized over all modular edge irregular $k$-labelings is called the modular edge irregularity strength of $G$. In the paper we estimate the bounds on the modular edge irregularity strength and for caterpillar, cycle, friendship graph and $n$-sun we determine the precise values of this parameter that prove the sharpness of the lower bound.