Vulnerability measures and topological indices are crucial in solving various problems such as the stability of the communication networks and development of mathematical models for chemical compounds. In 1947, Harry Wiener introduced a topological index related to molecular branching. Since then, more than 100 topological indices for graphs were introduced. Many graph polynomials play important roles in measuring such indices. Hosoya polynomial is among many of them. Introduced by Hosoya in 1988, the Hosoya polynomial of a given graph $G$ is a polynomial with the coefficients being the numbers of pairs of vertices in $G$ with all possible distances. For a given graph $G$, an extension graph is called Mycielskian graph of $G$, defined by Mycielski in 1955. In this paper, we investigate relationships between the Hosoya polynomial of any graph and that of its Mycielskian graph. The results are applied to compute the vulnerability measures, closeness and betweenness centrality, and the extended Wiener indices of selected graphs and their Mycielskian graphs. In the network science, these measures are commonly used to describe certain connectivity properties of a network. It is fascinating to see how graph polynomials are useful in other scientific fields.