Terminal Hosoya Polynomial of Line Graphs

Abstract

The terminal Hosoya polynomial of a graph 𝐺 is defined as TH(𝐺,𝜆)=∑_𝑘≥1𝑑_𝑇(𝐺,𝑘)𝜆^𝑘, where 𝑑_𝑇(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,…,𝑣_𝑛} and edge set 𝐸(𝐺)={𝑒₁,𝑒₂,…,𝑒_𝑚}. The degree of a vertex 𝑣 in 𝐺 is the number of edges incident to it and is denoted by deg_𝐺(𝑣). If deg_𝐺(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 deg_𝐺(𝑢)=1 or deg_𝐺(𝑣)=1. Two edges are said to be independent if they are not adjacent to each other. The distance between the vertices 𝑣_𝑖 and 𝑣_𝑗 in 𝐺 is equal to the length of a shortest path joining them and is denoted by 𝑑(𝑣_𝑖,𝑣_𝑗∣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)=_𝑘≥1𝑑(𝐺,𝑘)𝜆^𝑘=_{1≤𝑖<𝑗≤𝑛}𝜆^{𝑑(𝑣𝑖,𝑣𝑗∣𝐺)}, (1) where 𝑑(𝐺,𝑘) is the number of pairs of vertices of 𝐺 that are at distance 𝑘 and 𝜆 is the parameter.

Estrada et al. [7] studied the chemical applications of Hosoya polynomial. The interesting property of 𝐻(𝐺,𝜆) is that its first derivative at 𝜆=1 is equal to the well-known Wiener index 𝑊(𝐺) of 𝐺, the sum of the distances between all pairs of vertices of 𝐺 [8]. That is, 𝑊(𝐺)=_{1≤𝑖<𝑗≤𝑛}𝑑𝑣_𝑖,𝑣_𝑗=𝑑∣𝐺(𝐻(𝑑𝜆𝐺,𝜆))|_𝜆=1. (2)

Gutman et al. [9] put forward another topological index called terminal Wiener index TW(𝐺) defined as the sum of the distances between all pairs of pendant vertices of  𝐺. Thus if 𝑉_𝑇(𝐺)={𝑣₁,𝑣₂,…,𝑣_𝑘} is the set of pendant vertices of 𝐺, then TW(𝐺)=_{{}𝑢,𝑣⊆𝑉𝑇(𝐺)}𝑑(𝑢,𝑣∣𝐺)=_{1≤𝑖<𝑗≤𝑘}𝑑𝑣_𝑖,𝑣_𝑗∣𝐺. (3)

For recent work on the terminal Wiener index, see [10–14].

In analogy of (1), the terminal Hosoya polynomial TH(𝐺,𝜆) of a graph 𝐺 was put forward by Narayankar et al. [15] and is defined as follows: if 𝑣₁, 𝑣₂, …, 𝑣_𝑘 are the pendant vertices of 𝐺, then TH(𝐺,𝜆)=_𝑘≥1𝑑_𝑇(𝐺,𝑘)𝜆^𝑘=_{1≤𝑖<𝑗≤𝑘}𝜆^{𝑑(𝑣𝑖,𝑣𝑗∣𝐺)}, (4) where 𝑑_𝑇(𝐺,𝑘) is the number of pairs of pendant vertices of the graph 𝐺 that are at distance 𝑘.

It is easy to check that (𝐺)=𝑑TW((𝑑𝜆TH𝐺,𝜆))|_𝜆=1. (5)

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 𝐺 has no pendant vertex or has only one pendant vertex, then we write TH(𝐺,𝜆)=0, for 𝑛≠2.

If we write TH(𝐺,𝜆)=𝑎₁𝜆+𝑎₂𝜆²+⋯+𝑎_diam(𝐺)𝜆^diam(𝐺), where diam(𝐺) is the diameter of 𝐺, then 𝑎₁=0 for all graphs of order 𝑛≥3 and 𝑎₁=1 for 𝐺=𝐾₂, a complete graph on two vertices.

2. Terminal Hosoya Polynomial of Line Graphs

The line graph of 𝐺, denoted by 𝐿(𝐺), is the graph whose vertices are the edges of 𝐺, and two vertices of 𝐿(𝐺) are adjacent if and only if the corresponding edges are adjacent in 𝐺.

Theorem 1. Let 𝐺 be a connected graph with 𝑛≥4 vertices, let 𝐷₂(𝐺)={𝑣₁,𝑣₂,…,𝑣_𝑞}, where deg_𝐺(𝑣_𝑖)=2, and one neighbor of 𝑣_𝑖 is pendant, 𝑖=1,2,…,𝑞. Then TH(𝐿(𝐺),𝜆)=𝜆_{1≤𝑖<𝑗≤𝑞}𝜆^{𝑑(𝑣𝑖,𝑣𝑗∣𝐺)}. (6)

Proof. Let 𝐸_𝑘={𝑒₁,𝑒₂,…,𝑒_𝑘} be the set of pendant edges of 𝐺 and 𝐸_𝑞={𝑒₁,𝑒₂,…,𝑒_𝑞} the subset of 𝐸_𝑘, where, for each 𝑒_𝑖∈𝐸_𝑞, the edge 𝑒_𝑖 is incident to the vertex 𝑣_𝑖∈𝐷₂(𝐺), 𝑖=1,2,…,𝑞. Thus, if 𝑒_𝑖=𝑢𝑣∈𝐸_𝑞, then deg_𝐺(𝑢)=1 and deg_𝐺(𝑣)=2 (or vice versa), 𝑖=1,2,…,𝑞.
Consider two edges 𝑒_𝑖=𝑢𝑣_𝑖 and 𝑒_𝑗=𝑣_𝑗𝑤 of 𝐸_𝑞, where deg_𝐺(𝑢)=deg_𝐺(𝑤)=1 and deg_𝐺(𝑣_𝑖)=deg_𝐺(𝑣_𝑗)=2, 𝑖,𝑗=1,2,…,𝑞.
Therefore deg_𝐿(𝐺)(𝑒_𝑖)=deg_𝐺(𝑢)+deg_𝐺(𝑣_𝑖)−2=1. Similarly deg_𝐿(𝐺)(𝑒_𝑗)=1. Therefore 𝑒_𝑖 and 𝑒_𝑗 are pendant vertices of 𝐿(𝐺). Thus all edges of 𝐸_𝑞 are the pendant vertices of 𝐿(𝐺), and 𝑑(𝑒_𝑖,𝑒_𝑗∣𝐿(𝐺))=1+𝑑(𝑣_𝑖,𝑣_𝑗∣𝐺), 𝑖≠𝑗 and 𝑖,𝑗=1,2,…,𝑞. Therefore TH(𝐿(𝐺),𝜆)=_{1≤𝑖<𝑗≤𝑞}𝜆^{𝑑(𝑒𝑖,𝑒𝑗∣𝐿(𝐺))}=_{1≤𝑖<𝑗≤𝑞}𝜆^{1+𝑑(𝑣𝑖,𝑣𝑗∣𝐺)}=𝜆_{1≤𝑖<𝑗≤𝑞}𝜆^{𝑑(𝑣𝑖,𝑣𝑗∣𝐺)}. (7)

Theorem 2. Let 𝐺 be a connected graph with 𝑛≥4 vertices and 𝐺^′ the graph obtained by removing pendant vertices of 𝐺. If (i) 𝐺=𝐾_1,𝑛−1 or (ii) 𝐺 has no edge 𝑒 such that one of the components of 𝐺−𝑒 is 𝐾_1,𝑠, 𝑠≥2 and 𝐺≠𝐾_1,𝑛−1, then TH(𝐿(𝐺),𝜆)=𝜆TH𝐺^′,𝜆. (8)

Proof.
Case 1. It is obvious that (8) holds for 𝐺=𝐾_1,𝑛−1.
Case 2. Let 𝐷₂(𝐺)={𝑣₁,𝑣₂,…,𝑣_𝑞}, where deg_𝐺(𝑣_𝑖)=2, and one neighbor of 𝑣_𝑖 is pendant, 𝑖=1,2,…,𝑞. If 𝐺≠𝐾_1,𝑛−1 and if there is no edge 𝑒 in 𝐺 such that one of the component of 𝐺−𝑒 is 𝐾_1,𝑠, 𝑠≥2, then the vertices of the set 𝐷₂(𝐺) become pendant in 𝐺^′. Therefore 𝑑𝑣_𝑖,𝑣_𝑗∣𝐺=𝑑𝑣_𝑖,𝑣_𝑗∣𝐺^′,where𝑣_𝑖,𝑣_𝑗∈𝐷₂(𝐺). (9) Substituting this in (6) we get TH(𝐿(𝐺),𝜆)=𝜆_{1≤𝑖<𝑗≤𝑞}𝜆^{𝑑(𝑣𝑖,𝑣𝑗∣𝐺′)}=𝜆TH𝐺^′,𝜆. (10)

Corollary 3. Let 𝐺 be a connected graph with 𝑛≥4 vertices and 𝐺^′ the graph obtained by removing pendant vertices of 𝐺. If all pendant edges of 𝐺 are mutually independent, then TH(𝐿(𝐺),𝜆)=𝜆TH𝐺^′,𝜆. (11)

Proof. It follows from Theorem 2(ii).

Let 𝐺 be the graph with vertices 𝑣₁, 𝑣₂, …, 𝑣_𝑛 then 𝐺⁺ is the graph obtained from 𝐺 by adding 𝑛 new vertices 𝑣^′₁, 𝑣^′₂, …, 𝑣^′_𝑛 and joining 𝑣^′_𝑖 to 𝑣_𝑖 by an edge, 𝑖=1,2,…,𝑛. The graph 𝐺⁺ is called the corona of 𝐺 [16].

Theorem 4. Let 𝐺 be a connected graph, then TH𝐿𝐺⁺,𝜆=𝜆TH(𝐺,𝜆). (12)

Proof. If the graph 𝐺 has 𝑛 number of vertices, then 𝐺⁺ has 𝑛 pendant edges, and all are mutually independent. Removing pendant vertices of 𝐺⁺ we get the graph 𝐺. Therefore from Corollary 3, TH𝐿𝐺⁺,𝜆=𝜆TH(𝐺,𝜆). (13)

Theorem 5. Let 𝐺 be a connected graph and 𝐹_𝑡=𝐿(𝐹⁺_𝑡−1), 𝑡=1,2,…, where 𝐹₀=𝐺 and 𝐹₁=𝐿(𝐺⁺); then TH𝐹_𝑡,𝜆=𝜆^𝑡TH(𝐺,𝜆). (14)

Proof. We prove this by induction on 𝑡.
From Theorem 4, TH𝐹₁,𝜆=TH𝐿𝐺⁺,𝜆=𝜆TH(𝐺,𝜆). (15) Let TH𝐹_𝑡−1,𝜆=𝜆^𝑡−1TH(𝐺,𝜆). (16) Therefore TH𝐹_𝑡,𝜆=TH𝐿𝐹⁺_𝑡−1𝐹,𝜆=𝜆TH_𝑡−1,𝜆=𝜆𝜆^𝑡−1TH(𝐺,𝜆)=𝜆^𝑡TH(𝐺,𝜆). (17)

The subdivision graph 𝑆(𝐺) is obtained from 𝐺 by inserting a new vertex of degree 2 on each edge of 𝐺. The graph 𝑆_𝑙(𝐺) is obtained from 𝐺 by inserting 𝑙 new vertices of degree 2 on each edge of 𝐺. Thus 𝑆₁(𝐺)=𝑆(𝐺).

Theorem 6. Let 𝐺 be a connected graph with 𝑘 pendant vertices 𝑣₁, 𝑣₂, …, 𝑣_𝑘. Then for 𝑙≥1, TH𝐿𝑆_𝑙(𝐺=1),𝜆𝜆_{1≤𝑖<𝑗≤𝑘}𝜆^{(𝑙+1)𝑑(𝑣𝑖,𝑣𝑗∣𝐺)}. (18)

Proof. The pendant vertices of 𝐺 and pendant vertices of 𝑆_𝑙(𝐺) are the same. Further all pendant edges of 𝑆_𝑙(𝐺) are mutually independent.
Let 𝑢 and 𝑣 be the pendant vertices of 𝐺; then 𝑢 and 𝑣 are pendant vertices of 𝑆_𝑙(𝐺). Let 𝑢^′ and 𝑣^′ be the subdivision vertices of 𝑆_𝑙(𝐺), where 𝑢^′ is adjacent to 𝑢 and 𝑣^′ is adjacent to 𝑣 in 𝑆_𝑙(𝐺). Let (𝑆_𝑙(𝐺))^′ be the graph obtained by removing all pendant vertices of 𝑆_𝑙(𝐺). Therefore 𝑢^′ and 𝑣^′ are pendant vertices of (𝑆_𝑙(𝐺))^′ and 𝑑(𝑢^′,𝑣^′∣(𝑆_𝑙(𝐺))^′)=(𝑙+1)𝑑(𝑢,𝑣∣𝐺)−2. Now, since the pendant edges of 𝑆_𝑙(𝐺) are mutually independent, from Corollary 3, TH𝐿𝑆_𝑙(𝐺),𝜆=𝜆TH𝑆_𝑙(𝐺)^′,𝜆=𝜆_{𝑢′,𝑣′⊆𝑉𝑇𝑆𝑙(𝐺)′}𝜆^{𝑑(𝑢′,𝑣′∣(𝑆𝑙(𝐺))′)}=𝜆_{{𝑢,𝑣}⊆𝑉𝑇(𝐺)}𝜆^{(𝑙+1)𝑑(𝑢,𝑣∣𝐺)−2}=1𝜆_{{𝑢,𝑣}⊆𝑉𝑇(𝐺)}𝜆^{(𝑙+1)𝑑(𝑢,𝑣∣𝐺)}=1𝜆_{1≤𝑖<𝑗≤𝑘}𝜆^{(𝑙+1)𝑑(𝑣𝑖,𝑣𝑗∣𝐺)}. (19)

Corollary 7. Let 𝐺 be a connected graph with 𝑘 pendant vertices 𝑣₁, 𝑣₂, …, 𝑣_𝑘. If 𝑑(𝑣_𝑖,𝑣_𝑗∣𝐺)=𝑑 for all 𝑖,𝑗=1,2,…,𝑘, where 𝑖≠𝑗, then, for 𝑙≥1, TH𝐿𝑆_𝑙(𝐺)=𝑘(),𝜆𝑘−12𝜆^{(𝑙+1)𝑑−1}. (20)

Proof. It follows from Theorem 6.

Acknowledgment

K. P. Narayankar thanks University Grants Commission (UGC), Government of India, for support through MRP no. 39-36/2010 (SR).