On Vertex Irregular Total k-labeling and Total Vertex Irregularity Strength of Lollipop Graphs

Let G be a connected graph with vertex set V ( G ) and edge set E ( G ). A vertex irregular total k -labeling λ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , k } of a graph G is a labeling of vertices and edges of G in such a way that the weights of any two different vertices x and y are distinct. The weight of a vertex x in G , denoted by wt ( x ), is defined as the sum of the label of x and the labels of all edges incident with the vertex x . The total vertex irregularity strength of G , denoted by tvs ( G ), is the smallest positive integer k for which the graph G has a vertex irregular total k -labeling. The ( m, n )-lollipop graphs denoted by L m,n is a graph obtained by joining a complete graph K m to a path graph P n with a bridge. In this research, we investigate tvs of lollipop graphs L m,n for m ≥ 3 and n ≥ 1, denoted by tvs ( L m,n ).