The Global Cyclicity Index of Benzenoid Chains

Abstract

The resistance distance Ω_𝐺(𝑖,𝑗) between vertices 𝑖 and 𝑗 of a connected (molecular) graph 𝐺 is computed as the effective resistance between nodes 𝑖 and 𝑗 in the corresponding network constructed from 𝐺 by replacing each edge of 𝐺 with a unit resistor. The conductance excess between any 𝑖 and 𝑗 of 𝐺 is the difference between 1/Ω_𝐺(𝑖,𝑗) and the reciprocal of the distance between 𝑖 and 𝑗. The global cyclicity index of 𝐺 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.

1. Introduction

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 𝐺=(𝑉(𝐺),𝐸(𝐺)). One simple such traditional fundamental measure is the cyclomatic number 𝜇(𝐺) (also called the first Betti number, the nullity, or the cycle rank) which is defined for a connected graph 𝐺 with 𝑛 vertices and 𝑚 edges as 𝜇(𝐺)=𝑚−𝑛+1. (1) 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  Ω_𝐺(𝑖,𝑗) between vertices 𝑖 and 𝑗 of a connected (molecular) graph 𝐺 is computed as the effective resistance between nodes 𝑖 and 𝑗 in the corresponding network constructed from 𝐺 by replacing each edge of 𝐺 with a unit resistor. Comparing to the traditional (shortest path) distance 𝑑_𝐺(𝑖,𝑗) between 𝑖 and 𝑗, it is well known that Ω_𝐺(𝑖,𝑗) equals the length 𝑑_𝐺(𝑖,𝑗) of the shortest path between 𝑖 and 𝑗 if there is a unique single path between 𝑖 and 𝑗, while if there is more than one path, then Ω_𝐺(𝑖,𝑗) is strictly less than 𝑑_𝐺(𝑖,𝑗). Thence, the conductance excess  𝜎_𝐺(𝑖,𝑗)−1/𝑑_𝐺(𝑖,𝑗) indicates in some manner the presence of cyclicity in the portion of the graph interconnecting 𝑖 and 𝑗, where 𝜎_𝐺(𝑖,𝑗)=1/Ω_𝐺(𝑖,𝑗) is known as the effective conductance between 𝑖 and 𝑗. To measure the cyclicity of a graph 𝐺, Klein and Ivanciuc [8] proposed the global cyclicity index  𝐶(𝐺) as 𝐶(𝐺)=_𝑖∼𝑗𝜎_𝐺−1𝑖,𝑗𝑑_𝐺,𝑖,𝑗 (2) where 𝑖∼𝑗 means that 𝑖 and  𝑗 are adjacent and the sum is over all edges of 𝐺. Since 𝑑_𝐺(𝑖,𝑗)=1 for 𝑖∼𝑗, 𝐶(𝐺) can also be written as 𝐶(𝐺)=_𝑖∼𝑗𝜎_𝐺𝑖,𝑗−1=_𝑖∼𝑗𝜎_𝐺𝑖,𝑗−|𝐸(𝐺)|. (3)

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 [9] 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 [8], Klein and Ivanciuc established a number of theorems for the global cyclicity index of graphs (even not connected). In [12], one of the present authors obtained bounds for the global cyclicity index of fullerene graphs. In [13], 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 𝐶(𝐺) and the cyclomatic number 𝜇(𝐺). 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. Results

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 𝐿_𝑛 the linear polyacene with 𝑛 hexagons. For convenience, we label the vertices of 𝐿_𝑛 as depicted in Figure 1.

(a)

(b)

Figure 1 

A benzenoid chain with 𝑛 hexagons (a) and the linear polyacene 𝐿𝑛 (b).

To compute resistance distances between pairs of adjacent vertices in 𝐿_𝑛, 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 𝐺 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 𝑖 and 𝑗 if the vertices 𝑖 and 𝑗 are in distinct components of the bi-tree. For a connected graph 𝐺 and for any two vertices 𝑖,𝑗∈𝑉(𝐺), we denote by 𝑡(𝐺) and 𝑡(𝐺;𝑖,𝑗) the number of spanning trees of 𝐺 and the number of spanning bi-trees of 𝐺 separating 𝑖 and 𝑗, respectively. Then resistance distances can be computed as given in the following Lemma.

Lemma 1 (see [14]). Let 𝐺 be a connected graph. Then the resistance distance between any two vertices 𝑖 and 𝑗 in 𝐺 can be computed as Ω_𝐺=𝑡𝑖,𝑗𝐺;𝑖,𝑗𝑡(𝐺). (4)

Now suppose that 𝑖 and 𝑗 are adjacent in 𝐺, and let 𝑒 be an edge connecting them. If 𝑇 is a spanning tree that contains 𝑒, then by the deletion of 𝑒 from 𝑇 we could obtain a spanning bi-tree 𝐹 separating 𝑖 and 𝑗. Conversely, if 𝐹 is a spanning bi-tree separating 𝑖 and 𝑗, then by adding an new edge 𝑒 connecting 𝑖 and 𝑗 to 𝐹 we could obtain a spanning tree of 𝐺. So, the number of spanning trees containing 𝑒 is equal to the number of spanning bi-trees separating 𝑖 and 𝑗. While, on the other hand, the number of spanning trees of 𝐺 that contain 𝑒 is equal to the number of spanning trees of the graph 𝐺/𝑒, where 𝐺/𝑒 denotes the graph obtained from 𝐺 by contracting 𝑒, that is, deleting 𝑒 and then identifying two end vertices of 𝑒. Thus, as a consequence of Lemma 1, we have the following result.

Lemma 2. Let 𝐺 be a connected graph with 𝑒=𝑖𝑗∈𝐸(𝐺). Then Ω_𝐺=𝑡()𝑖,𝑗𝐺/𝑒𝑡(𝐺). (5)

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 𝐺 be a multiple graph, and let 𝑒 be an edge of 𝐺. Then 𝑡(𝐺)()𝐺=𝑡𝐺−𝑒+𝑡𝑒, (6) where 𝐺−𝑒 denotes the graph obtained from 𝐺 by the deletion of 𝑒.

Combining Lemmas 2 and 3, we readily have Lemma 4.

Lemma 4. Let 𝐺 be a connected graph with 𝑒=𝑖𝑗 being an edge of 𝐺. Then Ω_𝐺=𝑡(𝐺)()𝑖,𝑗−𝑡𝐺−𝑒𝑡(𝐺). (7)

For computing resistance distances between adjacent vertices in 𝐿_𝑛, the number of spanning trees of 𝐿_𝑛 plays an essential rule.

Lemma 5 (see [15–17]). Consider 𝑡𝐿_𝑛=√283+2√2^𝑛+1−3−2√2^𝑛+1. (8)

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 1≤𝑘≤𝑛, Ω_𝐿𝑛𝑎_𝑘,𝑏_𝑘=Ω_𝐿𝑛𝑏_𝑘,𝑎_𝑘+1=Ω_𝐿𝑛𝑎^′_𝑘,𝑏^′_𝑘=Ω_𝐿𝑛𝑏^′_𝑘,𝑎^′_𝑘+1=√1−28+√283−2√2^2𝑘3+2√2^𝑛+1−√1+28−√283+2√2^2𝑘3−2√2^𝑛+1×3+2√2^𝑛+1−3−2√2^𝑛+1−1, (9) for 1≤𝑘≤𝑛+1, Ω_𝐿𝑛𝑎_𝑘,𝑎^′_𝑘=√221+3−2√2^2𝑘−13+2√2^𝑛+1+1+3+2√2^2𝑘−13−2√2^𝑛+1×3+2√2^𝑛+1−3−2√2^𝑛+1−1. (10)

Proof. We first compute resistance distances between adjacent degree two and degree three vertices. For 1≤𝑘≤𝑛, by the structure of 𝐿_𝑛, we could see that Ω_𝐿𝑛𝑎_𝑘,𝑏_𝑘=Ω_𝐿𝑛𝑏_𝑘,𝑎_𝑘+1=Ω_𝐿𝑛𝑎^′_𝑘,𝑏^′_𝑘=Ω_𝐿𝑛𝑏^′_𝑘,𝑎^′_𝑘+1. (11) By Lemma 4, we have Ω_𝐿𝑛𝑎_𝑘,𝑏_𝑘=𝑡𝐿_𝑛−𝑡𝐿_𝑛−𝑎_𝑘𝑏_𝑘𝑡𝐿_𝑛. (12) Notice that the number of spanning trees of the graph 𝐿_𝑛−𝑎_𝑘𝑏_𝑘 is equal to the product of the number of spanning trees of 𝐿_𝑘−1 and the number of spanning trees of 𝐿_𝑛−𝑘 (here the number of spanning trees of 𝐿₀ is regarded as 1, and the formula in Lemma 5 is also applicable to 𝐿₀); that is, 𝑡𝐿_𝑛−𝑎_𝑘𝑏_𝑘=𝑡𝐿_𝑘−1𝑡𝐿_𝑛−𝑘. (13) Hence, Ω_𝐿𝑛𝑎_𝑘,𝑏_𝑘=𝑡𝐿_𝑛−𝑡𝐿_𝑘−1𝑡𝐿_𝑛−𝑘𝑡𝐿_𝑛. (14) Substituting (8) into the above equality, we could obtain (9). It should be mentioned that to obtain (9), one should notice that 3+2√2 is the reciprocal of 3−2√2.
Now, we compute resistance distances between {𝑎_𝑘,𝑎^′_𝑘} (1≤𝑘≤𝑛+1). Denote by 𝑆_𝑛 the graph obtained from 𝐿_𝑛 by contracting the edge 𝑎₁𝑏₁; that is, 𝑆_𝑛=𝐿_𝑛/𝑎₁𝑏₁. Noticing that the number of spanning trees of 𝐿_𝑛−𝑎₁𝑏₁ is equal to the number of spanning trees of 𝐿_𝑛−1, by Lemma 3, simple calculation leads to 𝑡𝑆_𝑛=𝑡𝐿_𝑛−𝑡𝐿_𝑛−𝑎₁𝑏₁=𝑡𝐿_𝑛−𝑡𝐿_𝑛−1=√24√2+1^2𝑛+1+√2−1^2𝑛+1. (15) If we regard the number of spanning trees of 𝑆₀ as 1, then (15) also holds for 𝑛=0. Let 𝐿_𝑛/𝑎_𝑘𝑎^′_𝑘 be the graph obtained from 𝐿_𝑛 by contracting the edge 𝑎_𝑘𝑎^′_𝑘. Then it is not hard to observe that 𝑡𝐿_𝑛𝑎_𝑘𝑎^′_𝑘=𝑡𝑆_𝑘−1𝑡𝑆_{𝑛−𝑘+1}. (16) Hence, by Lemma 2, for 1≤𝑘≤𝑛+1, Ω_𝐿𝑛𝑎_𝑘,𝑎^′_𝑘=𝑡𝐿_𝑛/𝑎_𝑘𝑎^′_𝑘𝑡𝐿_𝑛=𝑡𝑆_𝑘−1𝑡𝑆_{𝑛−𝑘+1}𝑡𝐿_𝑛. (17) Substituting (8) and (15) into the above equality, we could obtain (10).

2.2. The Global Cyclicity Index of Benzenoid Chains

In [8], Klein and Ivanciuc obtained the following result in Lemma 7.

Lemma 7 (see [8]). 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 𝐻_𝑛 be a benzenoid chain with 𝑛 hexagons. Then 𝐶𝐻_𝑛=4^𝑛_𝑘=11−3−2√2^2𝑘3+2√2^𝑛+1+1+3+2√2^2𝑘3−2√2^𝑛+1×4√2−1+3−2√2^2𝑘3+2√2^𝑛+1−4√2+1−3+2√2^2𝑘3−2√2^𝑛+1−1+^𝑛+1_𝑘=1√2−1−3−2√2^2𝑘−13+2√2^𝑛+1−√2+1+3+2√2^2𝑘−13−2√2^𝑛+1×1−3−2√2^2𝑘−13+2√2^𝑛+1+1+3+2√2^2𝑘−13−2√2^𝑛+1−1. (18)

Proof. By Lemma 7, we only need to compute the global cyclicity index of 𝐿_𝑛. On one hand, by (9) in Theorem 6, for 1≤𝑘≤𝑛, we have 𝜎_𝐿𝑛𝑎_𝑘,𝑏_𝑘−1=𝜎_𝐿𝑛𝑏_𝑘,𝑎_𝑘+1−1=𝜎_𝐿𝑛𝑎^′_𝑘,𝑏^′_𝑘−1=𝜎_𝐿𝑛𝑏^′_𝑘,𝑎^′_𝑘+1=√−13+22^𝑛+1−3−2√2^𝑛+1×√1−28+√283−2√2^2𝑘3+2√2^𝑛+1−√1+28−√283+2√2^2𝑘3−2√2^𝑛+1−1=√−11−3−22^2𝑘3+2√2^𝑛+1+1+3+2√2^2𝑘3−2√2^𝑛+1×4√2−1+3−2√2^2𝑘3+2√2^𝑛+1−4√2+1−3+2√2^2𝑘3−2√2^𝑛+1−1. (19) On the other hand, by (10), for 1≤𝑘≤𝑛+1, we have 𝜎_𝐿𝑛𝑎_𝑘,𝑎^′_𝑘=√−123+2√2^𝑛+1−3−2√2^𝑛+1×1+3−2√2^2𝑘−13+2√2^𝑛+1+1+3+2√2^2𝑘−13−2√2^𝑛+1−1=√−12−1−3−2√2^2𝑘−13+2√2^𝑛+1−√2+1+3+2√2^2𝑘−13−2√2^𝑛+1×1−3−2√2^2𝑘−13+2√2^𝑛+1+1+3+2√2^2𝑘−13−2√2^𝑛+1−1. (20) By the definition of the global cyclicity index, it follows that 𝐶(𝐺)=^𝑛_𝑘=1𝜎_𝐿𝑛𝑎_𝑘,𝑏_𝑘−1+𝜎_𝐿𝑛𝑏_𝑘,𝑎_𝑘+1−1+𝜎_𝐿𝑛𝑎^′_𝑘,𝑏^′_𝑘−1+𝜎_𝐿𝑛𝑏^′_𝑘,𝑎^′_𝑘+1+−1_𝑛+1𝑘=1𝜎_𝐿𝑛𝑎_𝑘,𝑎^′_𝑘−1=4^𝑛_𝑘=1𝜎_𝐿𝑛𝑎_𝑘,𝑏_𝑘−1+^𝑛+1_𝑘=1𝜎_𝐿𝑛𝑎_𝑘,𝑎^′_𝑘−1. (21) Then (18) could be obtained by substituting (19) and (20) into (21).

As numerical results, global cyclicity indices of benzenoid chains from 𝐻₁ to 𝐻₅₀ are listed in Table 1.

Table 1 

Acknowledgments

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.