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Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 857908, 3 pages
Terminal Hosoya Polynomial of Line Graphs
1Department of Mathematics, Gogte Institute of Technology, Udyambag, Belgaum 590008, India
2Department of Mathematics, Mangalore University, Mangalore 574199, India
3Department of Mathematics, Vishwanathrao Deshpande Rural Institute of Technology, Haliyal 581329, India
Received 31 January 2013; Accepted 2 June 2013
Academic Editor: Wai Chee Shiu
Copyright © 2013 H. S. Ramane et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  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.  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 . That is,
Gutman et al.  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
In analogy of (1), the terminal Hosoya polynomial of a graph was put forward by Narayankar et al.  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 , 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 , . Thus, if , then and (or vice versa), .
Consider two edges and of , where and , .
Therefore . Similarly . Therefore and are pendant vertices of . Thus all edges of are the pendant vertices of , and , and . Therefore
Theorem 2. Let be a connected graph with vertices and the graph obtained by removing pendant vertices of . If (i) or (ii) has no edge such that one of the components of is , and , then
Case 1. It is obvious that (8) holds for .
Case 2. Let , where , and one neighbor of is pendant, . If and if there is no edge in such that one of the component of is , , then the vertices of the set become pendant in . Therefore Substituting this in (6) we get
Corollary 3. Let be a connected graph with vertices and the graph obtained by removing pendant vertices of . If all pendant edges of are mutually independent, then
Proof. It follows from Theorem 2(ii).
Let be the graph with vertices , , , then is the graph obtained from by adding new vertices , , , and joining to by an edge, . The graph is called the corona of .
Theorem 4. Let be a connected graph, then
Proof. If the graph has number of vertices, then has pendant edges, and all are mutually independent. Removing pendant vertices of we get the graph . Therefore from Corollary 3,
Theorem 5. Let be a connected graph and , , where and ; then
Proof. We prove this by induction on .
From Theorem 4, Let Therefore
The subdivision graph is obtained from by inserting a new vertex of degree 2 on each edge of . The graph is obtained from by inserting new vertices of degree 2 on each edge of . Thus .
Theorem 6. Let be a connected graph with pendant vertices , , , . Then for ,
Proof. The pendant vertices of and pendant vertices of are the same. Further all pendant edges of are mutually independent.
Let and be the pendant vertices of ; then and are pendant vertices of . Let and be the subdivision vertices of , where is adjacent to and is adjacent to in . Let be the graph obtained by removing all pendant vertices of . Therefore and are pendant vertices of and . Now, since the pendant edges of are mutually independent, from Corollary 3,
Corollary 7. Let be a connected graph with pendant vertices , , , . If for all , where , then, for ,
Proof. It follows from Theorem 6.
K. P. Narayankar thanks University Grants Commission (UGC), Government of India, for support through MRP no. 39-36/2010 (SR).
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