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

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.

tvs(W_n) = \lceil \frac{n+3}{4} \rceil \text{ for } n \geq 3,

tvs(F_n) = \lceil \frac{n+2}{4} \rceil \text{ for } n \geq 3,

tvs(S_n) = \lceil \frac{n+1}{2} \rceil \text{ for } n \geq 3,

tvs(f_n) = \lceil \frac{2n+2}{3} \rceil \text{ for all } n.