Research Article | Open Access
The Global Cyclicity Index of Benzenoid Chains
The resistance distance between vertices and of a connected (molecular) graph is computed as the effective resistance between nodes and in the corresponding network constructed from by replacing each edge of with a unit resistor. The conductance excess between any and of is the difference between and the reciprocal of the distance between and . The global cyclicity index of is defined as the sum of conductance excesses between all pairs of adjacent vertices. In this paper, by computing resistance distances between pairs of adjacent vertices in linear polyacenes, an explicit formula for the global cyclicity index of a benzenoid chain is obtained in terms of its number of hexagons.
As the number of possible chemical compounds is so big, their huge part will never be experimentally tested. For this reason, there is a need for mathematical modeling and analysis of certain classes of compounds. To this end, many topological indices are defined and applied in the modeling of chemical and pharmacological properties of molecules. In the present work, we will take a new molecular cyclicity measure into consideration.
There are different possible measures of “cyclicity” of a molecular graph . One simple such traditional fundamental measure is the cyclomatic number (also called the first Betti number, the nullity, or the cycle rank) which is defined for a connected graph with vertices and edges as Motivated from electrical network theory, Klein and Ivanciuc proposed a new cyclicity measure. This new cyclicity measure is established on the basis of the novel concept of resistance distance [1–7]. As an intrinsic graph metric, the resistance distance between vertices and of a connected (molecular) graph is computed as the effective resistance between nodes and in the corresponding network constructed from by replacing each edge of with a unit resistor. Comparing to the traditional (shortest path) distance between and , it is well known that equals the length of the shortest path between and if there is a unique single path between and , while if there is more than one path, then is strictly less than . Thence, the conductance excess indicates in some manner the presence of cyclicity in the portion of the graph interconnecting and , where is known as the effective conductance between and . To measure the cyclicity of a graph , Klein and Ivanciuc  proposed the global cyclicity index as where means that and are adjacent and the sum is over all edges of . Since for , can also be written as
As a new measure of cyclicity of graphs, the global cyclicity index has less degeneracy than the standard cyclomatic number and has some intuitively appealing features. Since the idea of cyclicity is related to measures of connectivity or complexity  and characterization of “cyclicity” is an aspect of key importance in the study of molecular graphs [10, 11], it is worth studying the global cyclicity index from both mathematical and chemical points of view.
A benzenoid system is a 1-connected collection of congruent regular hexagons arranged in the plane in such a way that any two hexagons having a common point intersect in a whole edge. The vertices lying on the border of the unbounded face of a benzenoid system are called external; other vertices, if present, are called internal. A benzenoid system without internal vertices is called catacondensed. If no hexagon in a catacondensed benzenoid is adjacent to three other hexagons, we say that the benzenoid is a benzenoid chain. If a benzenoid chain has no turn hexagons, then it is called a linear polyacene.
In , Klein and Ivanciuc established a number of theorems for the global cyclicity index of graphs (even not connected). In , one of the present authors obtained bounds for the global cyclicity index of fullerene graphs. In , one of the present authors also obtained some further results on the global cyclicity number, including the strictly monotone increasing property, some lower and upper bounds, and some Nordhaus-Gaddum-type results, a relationship between and the cyclomatic number . In this paper, by computing resistance distances between pairs of adjacent vertices in linear polyacenes, an explicit formula for the global cyclicity index of benzenoid chains is obtained in terms of the number of hexagons.
2.1. Resistance Distances between Adjacent Vertices in Linear Polyacenes
In this subsection, we will compute resistance distances between pairs of adjacent vertices in linear polyacenes. Denote by the linear polyacene with hexagons. For convenience, we label the vertices of as depicted in Figure 1.
To compute resistance distances between pairs of adjacent vertices in , we need to employ the classical result of computing resistance distances in terms of spanning trees and spanning bi-trees. A spanning tree (resp., forest) of a connected graph is a subgraph that contains all the vertices and is a tree (resp., forest). A spanning bi-tree of a connected graph is defined as a spanning forest of the graph with exactly two components. A spanning bi-tree is said to separate vertices and if the vertices and are in distinct components of the bi-tree. For a connected graph and for any two vertices , we denote by and the number of spanning trees of and the number of spanning bi-trees of separating and , respectively. Then resistance distances can be computed as given in the following Lemma.
Lemma 1 (see ). Let be a connected graph. Then the resistance distance between any two vertices and in can be computed as
Now suppose that and are adjacent in , and let be an edge connecting them. If is a spanning tree that contains , then by the deletion of from we could obtain a spanning bi-tree separating and . Conversely, if is a spanning bi-tree separating and , then by adding an new edge connecting and to we could obtain a spanning tree of . So, the number of spanning trees containing is equal to the number of spanning bi-trees separating and . While, on the other hand, the number of spanning trees of that contain is equal to the number of spanning trees of the graph , where denotes the graph obtained from by contracting , that is, deleting and then identifying two end vertices of . Thus, as a consequence of Lemma 1, we have the following result.
Lemma 2. Let be a connected graph with . Then
In the enumeration of spanning trees of graphs, there is a famous recursion formula, known as the deletion-contraction recurrence. As stated in the following lemma, the recursion formula is applicable to multiple graphs (i.e., graphs with multiple edges).
Lemma 3. Let be a multiple graph, and let be an edge of . Then where denotes the graph obtained from by the deletion of .
Lemma 4. Let be a connected graph with being an edge of . Then
For computing resistance distances between adjacent vertices in , the number of spanning trees of plays an essential rule.
Now we are arriving at Theorem 6.
Theorem 6. Resistance distances between pairs of adjacent vertices of a linear polyacene could be computed as follows.
For , for ,
Proof. We first compute resistance distances between adjacent degree two and degree three vertices. For , by the structure of , we could see that
By Lemma 4, we have
Notice that the number of spanning trees of the graph is equal to the product of the number of spanning trees of and the number of spanning trees of (here the number of spanning trees of is regarded as 1, and the formula in Lemma 5 is also applicable to ); that is,
Substituting (8) into the above equality, we could obtain (9). It should be mentioned that to obtain (9), one should notice that is the reciprocal of .
Now, we compute resistance distances between . Denote by the graph obtained from by contracting the edge ; that is, . Noticing that the number of spanning trees of is equal to the number of spanning trees of , by Lemma 3, simple calculation leads to If we regard the number of spanning trees of as 1, then (15) also holds for . Let be the graph obtained from by contracting the edge . Then it is not hard to observe that Hence, by Lemma 2, for , Substituting (8) and (15) into the above equality, we could obtain (10).
2.2. The Global Cyclicity Index of Benzenoid Chains
Lemma 7 (see ). All benzenoid chains with the same number of hexagons have the same global cyclicity index.
Now we are ready to give the main result of the paper.
Theorem 8. Let be a benzenoid chain with hexagons. Then
Proof. By Lemma 7, we only need to compute the global cyclicity index of . On one hand, by (9) in Theorem 6, for , we have On the other hand, by (10), for , we have By the definition of the global cyclicity index, it follows that Then (18) could be obtained by substituting (19) and (20) into (21).
As numerical results, global cyclicity indices of benzenoid chains from to are listed in Table 1.
The authors acknowledge the support of the National Natural Science Foundation of China under Grant no. 11371307. Y. Yang acknowledges the support of the National Natural Science Foundation of China under Grant no. 11201404, China Postdoctoral Science Foundation under Grant nos. 2012M521318 and 2013T60662, Special Funds for Postdoctoral Innovative Projects of Shandong Province under Grant no. 201203056, and Shandong Province Higher Educational Science and Technology Program through Grant J12LI05. Y. Wang and Y. Li acknowledge the support of the Natural Science Foundation of Shandong Province under Grant no. ZR2011AM005.
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