Total Vertex Irregular Labelings of Wheels, Fans, Suns and Friendship Graphs

Abstract

A total vertex irregular labeling of a graph G with v vertices and e edges is an assignment of integer labels to both vertices and edges so that the weights calculated at vertices are distinct.The total vertex irregularity strength of $G$, denoted by $tvs(G)$, is the minimum value of the largest label over all such irregular assignments.In this paper, we consider the total vertex irregular labelings of wheels W_n, fans $F_n$, suns $S_n$ and friendship graphs $f_n$.We show that $tvs(W_n)=\lceil \frac{n+3}{4}\rceil \text{ for }n\ge 3$,$tvs(F_n)=\lceil \frac{n+2}{4}\rceil \text{ for }n\ge 3$,$tvs(S_n)=\lceil \frac{n+1}{2}\rceil \text{ for }n\ge 3$,$tvs(f_n)=\lceil \frac{2n+2}{3}\rceil \text{ for all }n$.