Let G be a graph with p vertices and q edges and let = 0,1, 2, , . 2 q A              A vertex labeling   : f V G A  i e a vertex equitable labeling of G if it induces an edge labeling s said to b given by      * =  f uv f u f v  * f such that     1 v a v b   and     * = 1,2,3, , f f f E q  , where   f v a is the ber of verti num ces v with   = f v a for . ...

A total vertex irregular k-labeling φ of a graph G is a labeling of the vertices and edges of G with labels from the set {1, 2, . . . , k} in such a way that for any two different vertices x and y their weights wt(x) and wt(y) are distinct. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x. The minimum k for which the graph G...

Nowadays, the problem of finding families graphs for which one may ensure existence a vertex-labeling and/or an edge-labeling based on certain class integers, constitutes challenge researchers in both number and graph theory. In this paper, we focus those vertex-labelings whose induced multiplicative assigns hyper totient numbers to edges graph. way, introduce characterize notions re...