(2, 1)-total labeling number of the join graph of path and fan

The (2,1)-total labeling number λ₂^T(G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two adjacent edges have the same label and the difference between the labels of a vertex and its incident edges is at least 2. In this paper, we studied the upper bound of λ₂^T(G) of P_m ⋁ F_n+1.