Abstract
A phenylene is a conjugated hydrocarbons molecule composed of six- and four-membered rings. The matching energy of a graph is equal to the sum of the absolute values of the zeros of the matching polynomial of , while the Hosoya index is defined as the total number of the independent edge sets of . In this paper, we determine the extremal graph with respect to the matching energy and Hosoya index for all phenylene chains.
1. Introduction
Phenylenes are a class of conjugated hydrocarbons composed of six- and four-membered rings, where the six-membered rings (hexagons) are adjacent only to four-membered rings, and every four-membered ring is adjacent to a pair of hexagons. They are nanostructures that can be precisely designed and manufactured for a wide variety of applications; see [1–3] and the references therein.
A topological index is a numerical quantity derived in an unambiguous manner from the structure graph of a molecule, as a graph structural invariant; that is, it does not depend on the labeling or the pictorial representation of a graph. Various topological indices usually reflect molecular size and shape. One topological index is Hosoya index, which was first introduced by Hosoya [4]. It plays an important role in the so-called inverse structure-property relationship problems. For details of Hosoya index and its applications, the readers are suggested to refer to [5, 6]. A new topological index in chemistry, matching energy, is first introduced by Gutman and Wagner [7] in 2012 to study topological resonance energy of conjugated molecules, which has received a lot of attention from researchers in recent years. For more background and applications about matching energy, see [8–16].
In this paper, our aim is to determine the phenylenes with minimum and maximum matching energy (Hosoya index) among all the phenylenes with hexagons.
In the following we present some definitions and notations.
Let be a graph with the vertex set and the edge set . Let and be an edge and a vertex in , respectively. We denote by the graph obtained from by removing edge and by the graph obtained from by deleting vertex .
By we denote the number of -matchings of a graph . The matching polynomial of a graph with vertices is fined aswhere and for all . This expression induces a quasi-order relation (i.e., reflexive and transitive relation) on the set of all graphs with vertices. If and are two graphs with matching polynomial in the form (1), then the quasi-order is defined by Particularly, if and there exists some such that , then we write .
Gutman and Wagner in [7] first proposed the concept of the matching energy of a graph, denoted by , as
Meanwhile, they gave also another form of definition of matching energy of a graph. That is, where denotes the root of matching polynomial of . By (2) and (3), we easily obtain the fact as follows:
The -counting polynomial was defined by Hosoya [4] asParticularly, set ; then is called Hosoya index of . Furthermore, The -counting polynomial of graphs has the property as follows.
Lemma 1 (see [4]). (a) Let be a graph consisting of two components and . Then
(b) Let be an edge of . Then .
A phenylene chain containing hexagons, denoted by , is a phenylene with the properties that (a) no vertex is incident with two hexagons or squares and (b) no hexagon is adjacent to more than two squares. We denote by the set of all phenylene chains with hexagons. Let . If the subgraph induced by the vertices with degree 3 is the union of disjoint copies of a square, then is called a linear phenylene chain and denoted by (see Figure 1). If the subgraph of induced by the vertices with degree 3 is isomorphic to the graph having squares (see Figure 1), then is called a zigzag phenylene chain and is denoted by (see Figure 1). It is easy to see that and . Finally, by the definition of a phenylene chain, any element in can be obtained from an appropriately chosen graph by attaching to it a new graph , where is obtained from an edge of a square attaching an edge of a hexagon; see Figure 2.
2. Main Results
Theorem 2. Let be the set of all phenylene chains with hexagons. For any , then where the equalities on the left side hold only if , and the equalities on the right side hold only if .
By (2) and (5), we know that Theorem 2 holding only needs to prove the following result.
Theorem 3. For any and for each ,where the equalities on the left side hold only if and the equalities on the right side hold only if .
Let and be two polynomials of . We say if for all . If and there exists some such that , then we say . By (6), it is easy to obtain the following result which is equivalent to Theorem 3.
Theorem 4. For any ,(I)if , then ,(II)if , then .
In the following we will use the notation for , when it would lead to no confusion.
Proof. Checking Figure 2, by Lemma 1, we obtained thatBy (10) and (11), we have(i) and ;(ii) and . Particularly, if , then(i′),(ii′),(iii′)+.We prove Theorem 4(I) by mathematical induction.
First we consider . In this case, .
By (9), we have By (i′)–(iii′), we have .
Suppose that Theorem 4(I) is right for all phenylene chains with few hexagons. Let be a phenylene chain with hexagons, which is obtained from by attaching to it a new (see Figure 2). We show that if , then . By (9) we obtain that
By the inductive hypotheses we have , , and . Since , either or , and hence at least one of the three inequalities is strict. Therefore, we get that .
In the following we prove Theorem 4(II) by induction.
By the proof of Theorem 4(I), we know that .
Similarly, suppose that Theorem 4(II) is right for all phenylene chains with few hexagons. Let be a phenylene chain with hexagons, which is obtained from by attaching to it a new (see Figure 2). We show that if , then . By (9) we have By the inductive hypotheses we have , , and . Since , either or , and hence at least one of the three inequalities is strict. Therefore, we get that .
The proof is complete.
By the definition of Hosoya index and Theorem 4, we can obtain the following result.
Theorem 5. Let be the set of all phenylene chains with hexagons. For any , then where the equalities on the left side hold only if and the equalities on the right side hold only if .
Competing Interests
The author declares that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work is financially supported by the National Natural Science Foundation of China (11371180 and 11561056), the Project of QHMU (2015XJZ12), and the Qinghai Province Natural Science Foundation (2016).