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.

Consider a simple graph $G=(V,E)$ of size $m$ with the vertex set $V$ and the edge set $E$. A modular edge-irregular total $k$-labeling of a graph $G$ is a labeling scheme for the vertices and edges with the labels $1,2,\dots ,k$ that allows the modular weights of any two different edges to be distinct, where the modular weight of an edge is the remainder of the division of the weight (i.e., the sum of the label of the edge itself and the labels of its two end vertices) by $m$. The maximal integer $k$, minimized over all modular edge-irregular total $k$-labelings of the graph $G$ is called the modular total edge-irregularity strength. In the paper, we generalize the approach to edge-irregular evaluations, introduce the notion of the modular total edge-irregularity strength and obtain its boundary estimation. For certain families of graphs, we investigate the existence of modular edge-irregular total labelings and determine the precise values of the modular total edge-irregularity strength in order to prove the sharpness of the lower bound.