Total Vertex Irregularity Strength of the Disjoint Union of Sun Graphs

A vertex irregular total -labeling of a graph with vertex set and edge set is an assignment of positive integer labels to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of , denoted by is the minimum value of the largest label over all such irregular assignment. In this paper, we consider the total vertex irregularity strengths of disjoint union of isomorphic sun graphs, , disjoint union of consecutive nonisomorphic sun graphs, , and disjoint union of any two nonisomorphic sun graphs . 1. Introduction Let be a finite, simple, and undirected graph with vertex set and edge set . A vertex irregular total -labeling on a graph is an assignment of integer labels to both vertices and edges such that the weights calculated at vertices are distinct. The weight of a vertex in is defined as the sum of the label of and the labels of all the edges incident with , that is, The notion of the vertex irregular total -labeling was introduced by Ba？a et al. [1]. The total vertex irregularity strength of , denoted by , is the minimum value of the largest label over all such irregular assignments. The total vertex irregular strengths for various classes of graphs have been determined. For instances, Ba？a et al. [1] proved that if a tree with pendant vertices and no vertices of degree 2, then . Additionally, they gave a lower bound and an upper bound on total vertex irregular strength for any graph with vertices and edges, minimum degree and maximum degree , . In the same paper, they gave the total vertex irregular strengths of cycles, stars, and complete graphs, that is, , and . Furthermore, the total vertex irregularity strength of complete bipartite graphs for some and had been found by Wijaya et al. [2], namely, for , for , for , for , and for all and . Besides, they gave the lower bound on for , that is, . Wijaya and Slamin [3] found the values of total vertex irregularity strength of wheels , fans , suns and friendship graphs by showing that , , , . Ahmad et al. [4] had determined total vertex irregularity strength of Halin graph. Whereas the total vertex irregularity strength of trees, several types of trees and disjoint union of copies of path had been determined by Nurdin et al. [5–7]. Ahmad and Ba？a [8] investigated the total vertex irregularity strength of Jahangir graphs and proved that , for and conjectured that for and , They also proved that for the circulant graph, , and conjectured that for the circulant graph with degree at least 5, , . A sun graph is defined as the graph obtained from