Modular edge irregularity strength of graphs

Abstract

<abstract><p>For a simple graph $ G = (V, E) $ with the vertex set $ V(G) $ and the edge set $ E(G) $, a vertex labeling $ \varphi: V(G) \to \{1, 2, \dots, k\} $ is called a $ k $-labeling. The weight of an edge under the vertex labeling $ \varphi $ 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.</p></abstract>