Modular total vertex irregularity strength of graphs

Abstract

<abstract><p>A (modular) vertex irregular total labeling of a graph $ G $ of order $ n $ is an assignment of positive integers from $ 1 $ to $ k $ to the vertices and edges of $ G $ with the property that all vertex weights are distinct. The vertex weight of a vertex $ v $ is defined as the sum of numbers assigned to the vertex $ v $ itself and to the edge's incident, while the modular vertex weight is defined as the remainder of the division of the vertex weight by $ n $. The (modular) total vertex irregularity strength of $ G $ is the minimum $ k $ for which such labeling exists. In this paper, we obtain estimations on the modular total vertex irregularity strength, and we evaluate the precise values of this invariant for certain graphs.</p></abstract>