The Terminal Hosoya Polynomial of Some Families of Composite Graphs

is the first mathematical invariant reflecting the topological structure of a molecular graph. This topological index has been extensively studied; for instance, a comprehensive survey on the direct calculation, applications, and the relation of the Wiener index of trees with other parameters of graphs can be found in [1]. Moreover, a list of 120 references of the main works on the Wiener index of graphs can be found in the referred survey. The Hosoya polynomial of a graph was introduced in Hosoya’s seminal paper [2] in 1988 and received a lot of attention afterwards.The polynomial was later independently introduced and considered by Sagan et al. [3] under the name Wiener polynomial of a graph. Both names are still used for the polynomial but the term Hosoya polynomial is nowadays used by the majority of researchers. The main advantage of the Hosoya polynomial is that it contains a wealth of information about distance based graph invariants. For instance, knowing the Hosoya polynomial of a graph, it is straightforward to determine the Wiener index of a graph as the first derivative of the polynomial at the point t = 1. Cash [4] noticed that the hyper-Wiener index can be obtained from the Hosoya polynomial in a similar simple manner. A comprehensive survey on the main properties of the Hosoya polynomial, which includes a wide list of bibliographic references, can be found in Gutman et al. [5]. Also, Estrada et al. [6] studied several chemical applications of the Hosoya polynomial. TheHosoya polynomial of a connected graphG is defined as


Introduction
Throughout this paper  = (, ) denotes a connected, simple, and finite graph with vertex set  = () and edge set  = ().The distance   (, V) between two vertices  and V is the minimum of the lengths of paths between  and V.
The Wiener index () of a graph  with vertex set {V 1 , V 2 , . . ., V  }, defined as the sum of distances between all pairs of vertices of ,  () := 1 2 is the first mathematical invariant reflecting the topological structure of a molecular graph.This topological index has been extensively studied; for instance, a comprehensive survey on the direct calculation, applications, and the relation of the Wiener index of trees with other parameters of graphs can be found in [1].Moreover, a list of 120 references of the main works on the Wiener index of graphs can be found in the referred survey.
The Hosoya polynomial of a graph was introduced in Hosoya's seminal paper [2] in 1988 and received a lot of attention afterwards.The polynomial was later independently introduced and considered by Sagan et al. [3] under the name Wiener polynomial of a graph.Both names are still used for the polynomial but the term Hosoya polynomial is nowadays used by the majority of researchers.The main advantage of the Hosoya polynomial is that it contains a wealth of information about distance based graph invariants.For instance, knowing the Hosoya polynomial of a graph, it is straightforward to determine the Wiener index of a graph as the first derivative of the polynomial at the point  = 1.Cash [4] noticed that the hyper-Wiener index can be obtained from the Hosoya polynomial in a similar simple manner.A comprehensive survey on the main properties of the Hosoya polynomial, which includes a wide list of bibliographic references, can be found in Gutman et al. [5].Also, Estrada et al. [6] studied several chemical applications of the Hosoya polynomial.
The Hosoya polynomial of a connected graph  is defined as As we pointed out above, the Wiener index of a graph  is determined as the first derivative of the polynomial (, ) at  = 1.
The degree of a vertex V in  is the number of edges incident to it and is denoted by deg  (V).If deg  (V) = 1, then V is called a pendent vertex of .The set of pendent vertices of  will be denoted by Ω().
Gutman et al. proposed the concept of terminal Wiener index, which is defined as the sum of distances between all pairs of pendent vertices of  [8].For detailed information on the terminal Wiener index the readers may refer to the recent papers [8][9][10][11] and the references cited therein.
By analogy to the definition of Hosoya polynomial, the terminal Hosoya polynomial was defined by Narayankar et al. [12] as and recently Ramane et al. studied TH(, ) for the case of line graphs [13].As we can expect, the terminal Wiener index, denoted by TW(), is obtained from TH(, ) as Among the most known products of graphs, namely, Cartesian product, lexicographic product, strong product, rooted product, and corona product, the two latest ones are the only cases in which we can obtain connected graphs having pendent vertices.So, in this paper we obtain formulae for the terminal Hosoya polynomial of both types of graphs, rooted product graphs and corona product graphs.
The remaining definitions will be given the first time that the concept is found in the text.

The Terminal Hosoya Polynomial of Rooted Product Graphs
A rooted graph is a graph in which one vertex is labeled in a special way so as to distinguish it from the other vertices.The special vertex is called the root of the graph.Let  be a labeled graph on  vertices.Let F be a sequence of  rooted graphs   1 ,   2 , . . .,    .The rooted product graph (F) is the graph obtained by identifying the root of    with the th vertex of  [14].In this paper we consider a particular case of rooted product graphs where F consists of  isomorphic rooted graphs [15].More formally, assuming that () = { 1 , . . .,   } and that the root vertex of   is V, we define the rooted product graph ∘ V   as the graph whose vertex set is (∘ V   ) = () × (  ) and whose edge set is Figure 1: The graph C 6 ∘ V P 3 , where C 6 is the cycle graph of order 6, P 3 is a path of order 3, and V is the central vertex of P 3 .This chemical graph that corresponds to the Cyclohexane (C6H12) is a naturally occurring chemical that is also produced synthetically and used as a solvent in numerous industries.
As an example of a chemical compound whose graph is obtained as a rooted product graph we consider the Cycloalkanes with a single ring, whose chemical formula is C  H 2 , and whose molecular graph can be expressed as C  ∘ V P 3 , where C  is the cycle graph of order , P 3 is a path of order 3, and V is the central vertex of P 3 (see Figure 1).
Given a vertex V ∈ (  ), we define Notice that we can see   (V) as the terminal Wiener index with respect to V; that is,   (V) is the first derivative of the polynomial TH V (  , ) at  = 1.As a direct consequence of Theorem 1 we obtain the following result on the terminal Wiener index.
Corollary 2. Let   be a connected nontrivial graph.Then for any V ∈ (  ) − Ω(  ) and any connected graph  of order  ≥ 2, Now we consider the case of rooted product graphs ∘ V   where the root V is a pendent vertex of   .Theorem 3. Let  be a connected graph of order  ≥ 2 and let   be a connected nontrivial graph.Then for any V ∈ Ω(  ), Proof.We proceed as in the proof of Theorem 1, with the only difference that in this case () is obtained as follows: (17)

The Terminal Hosoya Polynomial of Corona Graphs
Let  and   be two graphs of order  and   , respectively.The corona product  ⊙   is defined as the graph obtained from  and   by taking one copy of  and  copies of   and joining by an edge each vertex from the th-copy of   with the th-vertex of  [16].We will denote by  = {V 1 , V 2 , . . ., V  } the set of vertices of  and by    = (  ,   ) the copy of   such that V  is adjacent to V for every V ∈   .Thus, the vertex set of  ⊙   is  ∪ (∪   =1   ) and, consequently,  ⊙   has order (  + 1) and, if  is connected, then  ⊙   is also connected and it has diameter ( ⊙   ) = () + 2.
Notice that the corona graph  1 ⊙   is isomorphic to the join graph  1 +  .Now, if  is a tree and   =   is an empty graph (An empty graph on  nodes consists of  isolated nodes with no edges.Such graphs are sometimes also called edgeless graphs or null graphs), then the corona graph  ⊙   is a tree and the set of leaves (pendent vertices) of  ⊙   is ∪  =1   .In particular, if  =  is a path, then  ⊙   is a caterpillar graph and the path , that is formed by the nonleaves of  ⊙   , is known as the backbone of the caterpillar.
The formula for the Hosoya polynomial of corona graphs was previously obtained in [17].In this section we obtain the formula for the terminal Hosoya polynomial of corona graphs  ⊙   .Of course,  ⊙   has pendent vertices if and only if   has isolated vertices.Proof.First of all, notice that  ⊙   = ∘ V (⟨V⟩ +   ), where ⟨V⟩ is the trivial graph composed by the vertex V. Now, since TH(⟨V⟩ +   , ) = (  2 )  2 and TH V (⟨V⟩ +   , ) = , the result immediately follows from Theorem 1.