Terminal Hosoya Polynomial of Line Graphs

The terminal Hosoya polynomial of a graph is defined as , where is the number of pairs of pendant vertices of that are at distance . In this paper we obtain terminal Hosoya polynomial of line graphs. 1. Introduction Let be a connected graph with vertex set and edge set . The degree of a vertex in is the number of edges incident to it and is denoted by . If , then is called a pendant vertex or terminal vertex. An edge of a graph is called a pendant edge if or . Two edges are said to be independent if they are not adjacent to each other. The distance between the vertices and in is equal to the length of a shortest path joining them and is denoted by . The Hosoya polynomial of a graph is a distance based polynomial introduced by Hosoya [1] in 1988 under the name “Wiener polynomial.” However today it is called the Hosoya polynomial [2–6]. For a connected graph , the Hosoya polynomial denoted by is defined as where is the number of pairs of vertices of that are at distance and is the parameter. Estrada et al. [7] studied the chemical applications of Hosoya polynomial. The interesting property of is that its first derivative at is equal to the well-known Wiener index of , the sum of the distances between all pairs of vertices of [8]. That is, Gutman et al. [9] put forward another topological index called terminal Wiener index defined as the sum of the distances between all pairs of pendant vertices of？？ . Thus if is the set of pendant vertices of , then For recent work on the terminal Wiener index, see [10–14]. In analogy of (1), the terminal Hosoya polynomial of a graph was put forward by Narayankar et al. [15] and is defined as follows: if , , , are the pendant vertices of , then where is the number of pairs of pendant vertices of the graph that are at distance . It is easy to check that In [15], the terminal Hosoya polynomial of thorn graphs is obtained. In the present paper we obtain the terminal Hosoya polynomial of line graphs. If the graph has no pendant vertex or has only one pendant vertex, then we write , for . If we write , where is the diameter of , then for all graphs of order and for , a complete graph on two vertices. 2. Terminal Hosoya Polynomial of Line Graphs The line graph of , denoted by , is the graph whose vertices are the edges of , and two vertices of are adjacent if and only if the corresponding edges are adjacent in . Theorem 1. Let be a connected graph with vertices, let , where , and one neighbor of is pendant, . Then Proof. Let be the set of pendant edges of and the subset of , where, for each , the edge is incident to the vertex