Abstract
Let be a connected graph and let be the set of pendent vertices of . The terminal Hosoya polynomial of is defined as , where denotes the distance between the pendent vertices and . In this note paper we obtain closed formulae for the terminal Hosoya polynomial of rooted product graphs and corona product graphs.
1. Introduction
Throughout this paper denotes a connected, simple, and finite graph with vertex set and edge set . The distance between two vertices and is the minimum of the lengths of paths between and .
The Wiener index of a graph with vertex set , defined as the sum of distances between all pairs of vertices of , 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 . 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 .
The Hosoya polynomial has been obtained for trees, composite graphs, benzenoid graphs, tori, zig-zag open-ended nanotubes, certain graph decorations, armchair open-ended nanotubes, zigzag polyhex nanotorus, nanotubes, pentachains, polyphenyl chains, the circumcoronene series, Fibonacci and Lucas cubes, Hanoi graphs, and so forth. See the references in [7].
The degree of a vertex in is the number of edges incident to it and is denoted by . If , then is called a pendent vertex of . The set of pendent vertices of will be denoted by .
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 [8–11] 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 for the case of line graphs [13]. As we can expect, the terminal Wiener index, denoted by , is obtained from 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.
2. 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 be a sequence of rooted graphs . The rooted product graph 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 consists of isomorphic rooted graphs [15]. More formally, assuming that and that the root vertex of is , we define the rooted product graph as the graph whose vertex set is and whose edge set is
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 , and whose molecular graph can be expressed as , where is the cycle graph of order , is a path of order , and is the central vertex of (see Figure 1).
To begin with, we need some additional notation. We define the terminal Hosoya polynomial with respect to as
Theorem 1. Let be a connected graph of order and let be a connected nontrivial graph. Then for any ,
Proof. Let and let be the th-“copy” of in ; that is, is the subgraph of generated by the set , which is isomorphic to . Then,
So, if , then and, as a consequence, . So, we assume that .
Notice that for every pair of pendent vertices of and we have
Also, for , , we have
Hence, splits as the sum of two polynomials , where is obtained from pairs of pendent vertices in the same and is obtained from pairs , where . On the one hand,
and, on the other hand,
Therefore, the result follows.
Given a vertex , we define Notice that we can see as the terminal Wiener index with respect to ; that is, is the first derivative of the polynomial at . 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 and any connected graph of order ,
Now we consider the case of rooted product graphs where the root is a pendent vertex of .
Theorem 3. Let be a connected graph of order and let be a connected nontrivial graph. Then for any ,
Proof. We proceed as in the proof of Theorem 1, with the only difference that in this case is obtained as follows:
Corollary 4. Let be a connected nontrivial graph. Then for any and any connected graph of order , one has
3. 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 the set of vertices of and by the copy of such that is adjacent to for every . Thus, the vertex set of is and, consequently, has order and, if is connected, then is also connected and it has diameter .
Notice that the corona graph is isomorphic to the join graph . 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 . 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.
Corollary 5. For any connected graph of order and any graph having isolated vertices,
Proof. First of all, notice that , where is the trivial graph composed by the vertex . Now, since and , the result immediately follows from Theorem 1.
Finally, Corollary 5 (and also Corollary 2) leads to the following formula for the terminal Wiener index of .
Remark 6. For any connected graph of order and any graph having isolated vertices,
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.