Computation of hosaya polynomial, wiener and hyper wiener index of jahangir graph \(j_{6,m}\)

Let \(G = (V, E)\) be a simple connected graph with vertex set \(V\) and edge set \(E\). For two vertices \(u\) and \(v\) in a graph \(G\), the distance \(d(u, v)\) is the shortest path between \(u\) and \(v\) in \(G\). Graph theory has much advancements in the field of theoretical chemistry. Recently, chemical graph theory is becoming very popular among researchers because of its wide applications of mathematics in chemistry. One of the important distance based topological index is the Wiener index, defined as the sum of distances between all pairs of vertices of \(G\), defined as \(W(G) = \sum_{ u,v∈V (G)} d(u, v)\). The Hosaya polynomial is defined as \(H(G, x) =\sum _{u,v∈V (G)} x ^{d(u,v)}\). The hyper Wiener index is defined as \(WW(G) =\sum_{u,v∈V (G)} d(u, v) + \frac{1}{2}\sum_{u,v∈V (G)}d^{2}(u, v)\). In this paper, we study and compute Hosaya polynomial, Wiener index and hyper Wiener index for Jahangir graph \(J_{6,m}\), \(m ≥ 3\). Furthermore, we give exact values of these topological indices.