Hindawi Publishing Corporation Journal of Discrete Mathematics Volume 2013, Article ID 857908, 3 pages http://dx.doi.org/10.1155/2013/857908
Research Article Terminal Hosoya Polynomial of Line Graphs H. S. Ramane,1 A. B. Ganagi,1 K. P. Narayankar,2 and S. S. Shirkol3 1
Department of Mathematics, Gogte Institute of Technology, Udyambag, Belgaum 590008, India Department of Mathematics, Mangalore University, Mangalore 574199, India 3 Department of Mathematics, Vishwanathrao Deshpande Rural Institute of Technology, Haliyal 581329, India 2
Correspondence should be addressed to H. S. Ramane;
[email protected] 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 𝐺 is defined as TH(𝐺, 𝜆) = ∑𝑘≥1 𝑑𝑇 (𝐺, 𝑘)𝜆𝑘 , where 𝑑𝑇 (𝐺, 𝑘) is the number of pairs of pendant vertices of 𝐺 that are at distance 𝑘. In this paper we obtain terminal Hosoya polynomial of line graphs.
1. Introduction Let 𝐺 be a connected graph with vertex set 𝑉(𝐺) = {V1 , V2 , . . . , V𝑛 } and edge set 𝐸(𝐺) = {𝑒1 , 𝑒2 , . . . , 𝑒𝑚 }. The degree of a vertex V in 𝐺 is the number of edges incident to it and is denoted by deg𝐺(V). If deg𝐺(V) = 1, then V is called a pendant vertex or terminal vertex. An edge 𝑒 = 𝑢V of a graph 𝐺 is called a pendant edge if deg𝐺(𝑢) = 1 or deg𝐺(V) = 1. Two edges are said to be independent if they are not adjacent to each other. The distance between the vertices V𝑖 and V𝑗 in 𝐺 is equal to the length of a shortest path joining them and is denoted by 𝑑(V𝑖 , V𝑗 | 𝐺). 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 𝐺, the Hosoya polynomial denoted by 𝐻(𝐺, 𝜆) is defined as 𝐻 (𝐺, 𝜆) = ∑ 𝑑 (𝐺, 𝑘) 𝜆𝑘 = 𝑘≥1
∑ 𝜆𝑑(V𝑖 ,V𝑗 |𝐺) , 1≤𝑖
