Total vertex irregularity strength of certain classes of unicyclic graphs

A total vertex irregular k-labeling phi 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 has a vertex irregular total k-labeling is called the total vertex irregularity strength of G. We have determined an exact value of the total vertex irregularity strength of certain classes of unicyclic graphs.