Hosoya polynomial of some cactus chains

Let be a simple graph. Hosoya polynomial of G is , where d(u, v) denotes the distance between vertices u and v. A cactus graph is a connected graph in which no edge lies in more than one cycle. In this paper we compute the Hosoya polynomial of some cactus chains. As a consequence, Wiener and hyper-Wiener indices of these kind of chains are also obtained.


ABOUT THE AUTHORS
Ali Sadeghieh is an assistant professor of Pure Mathematics (Algebra) in the department of Mathematics, College of Science of Yazd Branch of Islamic Azad University. Saeid Alikhani is an associate professor of Mathematics at Yazd University, Yazd, Iran. He is managing editor of journal entitled "Algebraic structures and their applications (ASTA)" and is a member of editorial board of six international journals and a reviewer of more than 20 international journals.
Nima Ghanbari is a PhD student of Pure Mathematics (Algebraic Graph Theory) at Yazd University. He is interested to some kind of graph colorings and Mathematical Chemistry.
Abdul Jalil M. Khalaf is an assistant professor of Mathematics (Graph Theory) in Faculty of Computer Science and Mathematics, at University of Kufa, Iraq. He is interested to chromatic polynomial and chromaticity of graphs, labeling of graphs and some another graph polynomials.

PUBLIC INTEREST STATEMENT
A simple graph G = (V, E) is a finite nonempty set V of objects called vertices together with a (possibly empty) set E of unordered pairs of distinct vertices of G called edges. In chemical graphs, the vertices of the graph correspond to the atoms of the molecule, and the edges represent the chemical bonds. A graphical invariant is a number related to a graph which is structural invariant, that is to say it is fixed under graph automorphisms. In chemistry and for chemical graphs, these invariant numbers are known as the topological indices. One of the most important topological indices is the Wiener index of a connected graph G is denoted by W(G), is the sum of distances between all pairs of vertices in G. It found numerous applications. The first derivative of the Hosoya polynomial at x = 1 is equal to the Wiener index. In this paper we computed the Hosoya polynomial of some cactus chains that are of importance in chemistry. has many chemical applications (Deutsch & Klavžar, 2013;Estrada, Ivanciuc, Gutman, Gutierrez, & Rodrguez, 1998;Gutman, Klavžar, Petkovsek, & Zigert, 2001;Gutman et al., 2012). Especially, the two well-known topological indices, i.e. Wiener index and hyper-Wiener index, can be directly obtained from the Hosoya polynomial. The Wiener index of a connected graph G is denoted by W(G), is defined as the sum of distances between all pairs of vertices in G (Hosoya, 1971), i.e.
The hyper-Wiener index is denoted by WW(G) and defined as follows: Note that the first derivative of the Hosoya polynomial at x = 1 is equal to the Wiener index: Also we have the following relation: In this paper we consider a class of simple linear polymers called cactus chains. Cactus graphs were first known as Husimi tree, they appeared in the scientific literature sixty years ago in papers by Husimi and Riddell concerned with cluster integrals in the theory of condensation in statistical mechanics (Harary & Uhlenbeck, 1953;Husimi, 1950;Riddell, 1951). We refer the reader to papers (Chellali, 2006;Majstorović, Došlić, & Klobučar, 2012) for some aspects of parameters of cactus graphs. A cactus graph is a connected graph in which no edge lies in more than one cycle. Consequently, each block of a cactus graph is either an edge or a cycle. If all blocks of a cactus G are cycles of the same size i, the cactus is i-uniform. The cactus graphs whose are i-uniform for i = 3, 4, 6 are of importance in chemistry and so we consider them in this paper. A triangular cactus is a graph whose blocks are triangles, i.e. a 3-uniform cactus. A vertex shared by two or more triangles is called a cut-vertex. If each triangle of a triangular cactus G has at most two cut-vertices, and each cutvertex is shared by exactly two triangles, we say that G is a chain triangular cactus. By replacing triangles in these definitions by cycles of length 4 we obtain cacti whose every block is C 4 . We call such cacti square cacti. Note that the internal squares may differ in the way they connect to their neighbors. If their cut-vertices are adjacent, we say that such a square is an ortho-square; if the cut-vertices are not adjacent, we call the square a para-square (Alikhani, Jahari, Mehryar, & Hasni, 2014).
In the next section, we compute the Hosoya polynomial of triangular and square cacti chains. In Section 3, we compute this polynomial for two kind of chain hexagonal cactus. As a consequence, the Wiener and the hyper-Wiener indices of these kind of chains are also obtained.

Hosoya polynomial of triangular and square cactus chains
In this section we compute the Hosoya polynomial of triangular and square cactus chains. First we consider a chain triangular. An example of a chain triangular cactus is shown in Figure 1. We call the number of triangles in G, the length of the chain. Obviously, all chain triangular cacti of the same length are isomorphic. Hence, we denote the chain triangular cactus of length n by T n . Here we compute the Hosoya polynomial of T n .
Finally for k = n, there are four pair of vertices u, v ∈ V(G) with deg(u) = deg(v) = 2, and so the coefficient of x n is 4. Therefore by definition of Hosoya polynomial we have the result. ✷ The following corollary gives the Wiener index and hyper-Wiener index of T n : Proof (i) It follows from Theorem 2.1 and the identity W(G) = (H(G, x)) � | x=1 . (ii) It follows from Theorem 2.1 and the identity ✷ By replacing triangles in the definitions of triangular cactus T n , by cycles of length 4 we obtain cacti whose every block is C 4 . We call such cacti, square cacti. An example of a square cactus chain is shown in Figure 2. We see that the internal squares may differ in the way they connect to their neighbors. If their cut-vertices are adjacent, we say that such a square is an ortho-square; if the cutvertices are not adjacent, we call the square a para-square. We consider a para-chain of length n, 2s(5n − 5s + 1).

Hosoya polynomial of chain hexagonal cactus
In this section we shall compute the Hosoya polynomial of some hexagonal cactus chains. By replacing triangles in the definitions of triangular cactus, by cycles of length 6 we obtain cacti whose every block is C 6 . We call such cacti, hexagonal cacti. An example of a hexagonal cactus chain is shown in Figure 4. We see that the internal hexagonal may differ in the way they connect to their neighbors. If their cut-vertices are adjacent, we say that such a square is an ortho-hexagonal; if the cut-vertices are not adjacent, we call the square a para-hexagonal. We consider a para-chain of length n, which is denoted by L n as shown in Figure 4. The following theorem gives the Hosoya polynomial of L n . In this section, we shall compute the Hosoya polynomial of two kinds of para-chain hexagonal cactus. The following theorem gives the Hosoya polynomial of L n .