| Summary: | 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.
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