On Total Irregularity Strength of Double-Star and Related Graphs

Let G = (V, E) be a simple and undirected graph with a vertex set V and an edge set E. A totally irregular total k-labeling f: V ∩ E → {1, 2,⋯, k} is a labeling of vertices and edges of G in such a way that for any two different vertices x and x1, their weights wt_f(x) = f(x) + ∑_xyεE f(xy) and wt_f(x′) = f(x′) + ∑_xyεE f(x′y′)are distinct, and for any two different edges xy and x1y1 their weights f (x) + f (xy) + f (y) and f (x1) + f (x1y1) + f (y1) are also distinct. A total irregularity strength of graph G, denoted byts(G), is defined as the minimum k for which G has a totally irregular total k-labeling. In this paper, we determine the exact value of the total irregularity strength for double-star S_n,m, n, m ≥ 3 and graph related to it, that is a caterpillar S_n,2,n, n ≥ 3. The results are and ts(S_n,2,n) = [n+m-1/2] ts(S_n,2,n)n.