  Let G be a simple connected graph having vertex set V and edge set E. The
vertex-set and edge-set of G denoted by V(G) and E(G), respectively. The length
of the smallest path between two vertices is called the distance. Mathematical
chemistry is the area of research engaged in new application of mathematics in
chemistry. In mathematics chemistry, we have many topological indices for any
molecular graph, that they are invariant on the graph automorphism. In this
research paper, we computing the Wiener index and the Hosoya polynomial of the
Jahangir graphs $J_5,m$ for all integer number $m \geq 3$.
