Abstract
The edge-Hosoya polynomial of a graph is the edge version of the famous Hosoya polynomial. Therefore, the edge-Hosoya polynomial counts the number of (unordered) pairs of edges at distance \(k \geq 0\) in a given graph. It is well known that this polynomial is closely related to the edge-Wiener index and the edge-hyper-Wiener index. As the main result of this paper, we greatly generalize an earlier result by providing a method for calculating the edge-Hosoya polynomial of a graph \(G\) which is obtained by identifying two edges of connected bipartite graphs \(G_1\) and \(G_2\). To show how the main theorem can be used, we apply it to phenylene chains. In particular, we present the recurrence relations and a linear time algorithm for calculating the edge-Hosoya polynomial of any phenylene chain. As a consequence, closed formula for the edge-Hosoya polynomial of linear phenylene chains is derived.
