The Global Cyclicity Index of Benzenoid Chains

The resistance distanceΩ 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.


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  = ((), ()).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  () =  −  + 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][2][3][4][5][6][7].As an intrinsic graph metric, the resistance distance Ω  (, ) 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 Ω  (, ) 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 Ω  (, ) is strictly less than   (, ).
Thence, the conductance excess   (, ) − 1/  (, ) indicates in some manner the presence of cyclicity in the portion of the graph interconnecting  and , where   (, ) = 1/Ω  (, ) is known as the effective conductance between  and .To measure the cyclicity of a graph , Klein and Ivanciuc [8] proposed the global cyclicity index () as where  ∼  means that  and  are adjacent and the sum is over all edges of .Since   (, ) = 1 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 [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 () 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.

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 com puting 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 [14]).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 𝐺 be a connected graph with 𝑒 = 𝑖𝑗 ∈ 𝐸(𝐺).
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 .
For computing resistance distances between adjacent vertices in   , the number of spanning trees of   plays an essential rule.

Theorem 6. Resistance distances between pairs of adjacent vertices of a linear polyacene could be computed as follows.
For 1 ≤  ≤ , Proof.We first compute resistance distances between adjacent degree two and degree three vertices.For 1 ≤  ≤ , 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  −1 and the number of spanning trees of  − (here the number of spanning trees of  0 is regarded as 1, and the formula in Lemma 5 is also applicable to  0 ); that is, Hence, 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 {  ,    } (1 ≤  ≤  + 1).Denote by   the graph obtained from   by contracting the edge  1  1 ; that is,   =   / 1  1 .Noticing that the number of spanning trees of   −  1  1 is equal to the number of spanning trees of  −1 , by Lemma 3, simple calculation leads to If we regard the number of spanning trees of  0 as 1, then (15) also holds for  = 0. Let   /     be the graph obtained from   by contracting the edge      .Then it is not hard to observe that Hence, by Lemma 2, for 1 ≤  ≤  + 1, Substituting ( 8) and (15) into the above equality, we could obtain (10).

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.(20)