## Abstract

Many physico-chemical parameters of molecules are determined by or are dependent upon their HOMO–LUMO gaps, as has become of special interest for conjugated-carbon nano-structures obtained from graphene and its congeners. Here, we deduce an elegant yet simple upper-bound estimate to the HOMO–LUMO gaps for such molecular *π*-networks corresponding to connected subgraphs of graphene. The result elucidates the general situations (involving larger fragments) for which the HOMO–LUMO gap is small.

## 1. Introduction

Conjugated-carbon nano-structures have played a central part in chemistry for well over a century, with intense interest (and three Nobel prizes) during the last couple of decades for work on conjugated (possibly metallically conducting) polymers [1,2], novel fullerene cages [3] and single-layer graphene [4,5]. But also comparable experimental interest has developed concerning carbon nanotubes [6,7], graphenic strips [8,9] and different modest decorations to these species, while interest in traditional benzenoids has been maintained, and much extended [10–12]; further, for graphene, as well as large benzenoids and buckytubes, much interest has developed regarding decorations [13–18]. In the last few decades, ‘chemical graph theory’ has developed, with a significant fraction of the work directed [19–22] towards conjugated *π*-networks in the context of the tight-binding Hückel model, although only a rather modest effort was initially directed [23–28] to the HOMO–LUMO gap. In the last decade or so attempts to counterbalance this have been made [29–41]. Of course, throughout the more traditional chemical literature (say from the 1950s on), there was an extensive effort to deal with the HOMO–LUMO gap mostly in an explicit consideration of individual molecules case by case. Notably, some of the early work (e.g. [42–44]) was essentially ‘chemical graph theory’, albeit not then so described.

Here, the aim is to develop further general results, for the HOMO–LUMO gap, for the whole class of benzenoids (i.e. finite graphene fragments) and buckytubes, bringing our understanding beyond that of individual case-by-case numerical values. We use conventional notation with *G* a simple connected (molecular) graph with vertex set *V* (*G*)={*v*_{1},*v*_{2},…,*v*_{n}} and edge set *E*(*G*). The *adjacency matrix* of *G*, denoted by *A*(*G*)=(*a*_{ij}), is an *n*×*n* matrix such that *a*_{ij}=1 if *v*_{i} and *v*_{j} are adjacent in *G* and 0 otherwise. The collection of eigenvalues of *A*(*G*), denoted by *S*(*G*), is called the *spectrum* of *G*. As *A*(*G*) is real symmetric, its spectrum is real and can be described as a multi-set *S*(*G*)={λ_{1},λ_{2},…,λ_{n}}, where λ_{1}≥λ_{2}≥⋯≥λ_{n}. The eigenvalues near the middle play an important role in the Hückel model of *π*-electron systems. Thus, the number #_{0} of 0-eigenvalues is widely emphasized (e.g. starting with Longuet–Higgins [44], if not earlier). A chemically important invariant is the *HOMO–LUMO gap*, which when the number of electrons matches the number of vertices is given as
*H*=⌊(*n*+1)/2⌋ and *L*=⌈(*n*+1)/2⌉. HOMO and LUMO, respectively, denote the highest occupied molecular orbital and the lowest unoccupied molecular orbital. Coulson & Rushbrooke [42] noted that several special things occur for graphs which are bipartite, in the sense that their vertices can be partitioned into two sets (starred and unstarred) such that every edge is between a starred and unstarred vertex. A simple related structural characteristic is *k* starred (resp. unstarred) vertices of *G* (with *k*=1,2,3 for our conjugated-carbon species; figure 1). We let

In this paper, a simple upper bound to the HOMO–LUMO gaps of connected subgraphs of graphene is obtained, as given by:

### Theorem 1.1

*Let G be a connected n-vertex subgraph of graphene. If #***≠#*°*, then* Δ(*G*)=0. *Otherwise, the HOMO–LUMO gap of G satisfies:
*

Further, we refine this for the highly interesting cases (benzenoids, coranoids, multi-coranoids, annulenes, biphenyl, polyphenylenes, etc.) where there are no degree-1 carbons.

### Theorem 1.2

*Let G be a graphene subgraph with no degree-1 sites. If* *, then* Δ(*G*)=0. *Otherwise, the HOMO–LUMO gap of G satisfies:
*

An even more special type of sub-structure is that of a *benzenoid* graph defined as a connected subgraph of the honeycomb net enclosed in a Jordan curve on the net. These have no cut edges or cut vertices and no degree-1 sites, though besides the benzenoids there are others having no degree-1 sites: biphenyl, annulenes, coranoids, multi-coranoids, polyphenylenes, etc.—with our theorem 1.2 encompassing these. Benzenoids allow a nice bound in terms of the number #_{∂} of sites on the boundary of the benzenoid:

### Theorem 1.3

*Let G be a benzenoid graph. Then #*_{∂}=2#_{2}−6, *and the HOMO–LUMO gap of G satisfies
*

This and theorem 1.2 nicely elucidate sufficient conditions for a small HOMO–LUMO gap—that if a graphene fragment lacking primary carbons has ever fewer boundary sites of degree-2 than interior sites, then the gap necessarily closes towards 0. That is, as a molecule resembles ever more closely graphene, it approaches the (zero) band-gap of graphene—and, moreover, we have a suggested functional form for the manner of the approach. Most of the proofs are found in §2, and an illustrative application to a few particular benzenoid nano-structures is found in §3.

The case of fragments in carbon nanotubes is somewhat more complicated. A seemingly central point in the proofs of our graphene theorems is the tri-partitioning of each of the starred and unstarred sets of sites (though this tri-partitioning does not explicitly appear in our above three theorem statements). For the carbon nanotubes, this tri-partitioning does not generally apply (in two-thirds of the cases) and so introduces extra complexities. Thus, we delay the theorems for the carbon nanotubes until §4.

## 2. HOMO–LUMO gaps of connected subgraphs of graphene

Ultimately, we focus on connected subgraphs of graphene. To start, let *G* be a connected bipartite graph with starred and unstarred sets denoted *V*_{*}(*G*)={*v*_{1},*v*_{2},…,*v*_{s}} and *V*_{°}(*G*)={*v*_{s+1},*v*_{s+2},…,*v*_{n}}, respectively. The *biadjacency matrix* of *G* is the *s*×(*n*−*s*) matrix *B*(*G*)=(*b*_{ij}) with *b*_{ij}=1, if *v*_{i}∈*V*_{*}(*G*) and *v*_{j}∈*V*_{°}(*G*) are adjacent, and otherwise *b*_{ij}=0. Then, it is well known that
*M*, *M*^{T} and *M*^{†}, respectively, denote the transpose and conjugate transpose of *M*.

Further, we often abbreviate *V* (*G*), *E*(*G*), *V*_{*}(*G*), *V*_{°}(*G*), *A*(*G*) and *B*(*G*) to *V* , *E*, *V*_{*}, *V*_{°}, *A* and *B*, respectively. A useful fundamental result is the Coulson–Rushbrooke [42] ‘pairing theorem’:

### Theorem 2.1 ([42])

*If G is a bipartite graph, then its spectrum is symmetrical with respect to 0, i.e. if* λ *is an eigenvalue of G, then* −λ *is also an eigenvalue of G.*

Now, clearly, the eigenvalues of the square
*A*(*G*). Further, the eigenspectrum of *A*^{2} consists of eigenvalues of *B*_{*} and *B*_{°}, and the positive eigenvalues of *B*_{*} and*B*_{°} are the same. Hence we conclude that the following.

### Proposition 2.2

*Let G be a connected graph, and let* *and* *be least eigenvalues of B*_{*} *and B*_{°}. *Then*

Thence to give bounds for the HOMO–LUMO gap Δ(*G*), it suffices to find bounds for

### Theorem 2.3 (Rayleigh–Ritz quotient)

*Let M*∈*C*^{n×n} *be a Hermitian matrix. Then, the least eigenvalue* *of M is
*

For *k*=1,2,3, recall that *k* in *G*. When more than one graph is involved, we use *B*_{*}.

### Theorem 2.4

*Let G be a finite connected subgraph of graphene with B*_{*} *defined as above. Then
*

### Proof.

For a matrix *M*, let [*M*]_{ij} denote its *ij*th element. Now *a*_{ik}*a*_{kj}=1 only when *a*_{ik}=*a*_{kj}=1, that is, both *v*_{i}*v*_{k} and *v*_{k}*v*_{j} are edges of *G*, which is to say *v*_{i}*v*_{k}*v*_{j} is a two-walk from *v*_{i} to *v*_{j} in *G*. As *G* has no cycles of length less than 6, the number of such two-walks between two distinct vertices *v*_{i} and *v*_{j} is non-zero if and only if *v*_{i} and *v*_{j} are at distance 2 in *G*, whence [*A*^{2}]_{ij}=1. Thus
*d*_{vi}(*G*) is the degree of *v*_{i} (i.e. the number of edges incident to *v*_{i}) in *G*.

For convenience, we construct two new graphs *G*_{*} and *G*_{°}. Let *G*_{*} (resp. *G*_{°}) be the graph with vertex set *V*_{*} (resp. *V*_{°}) and two vertices *v*_{i} and *v*_{j} are adjacent if and only if the distance between *v*_{i} and *v*_{j} in *G* is 2 (figure 2). Then the adjacency matrix of *G*_{*} (resp.*G*_{°}) is the same as *B*_{*} (resp. *B*_{°}) except for the diagonal elements.

As illustrated in figure 2, in the graphene net, if we connected all pairs of starred vertices (resp. unstarred vertices) which are at distance 2, then we obtain a triangular net which contains *G*_{*} (resp. *G*_{°}) as a subgraph. Thus it follows that both *G*_{*} and *G*_{°} are tripartite via the standard tri-partition of the triangular lattice net, where vertices of one type (*α*,*β* or *γ*) have only vertices of the other two types as neighbours. Hence, the vertex set *V* (*G*_{*})=*V*_{*} (resp. *V* (*G*_{°})=*V*_{°}) partitions into three parts *B*_{*} can be written as a block matrix
*D*_{α}, *D*_{β} and *D*_{γ} are diagonal matrices of vertex degrees, and, for *x*,*y*∈{*α*,*β*,*γ*}, *A*_{xy} is the matrix such that [*A*_{xy}]_{ij}=1 if *G*_{*} and otherwise 0.

Let *ε*^{2} and 1 be the cube roots of 1, with *ε* and *ε*^{2} conjugates to each other. Suppose that *α*|, |*β*| and |*γ*| vertices, respectively. Let 1_{ξ} for *ξ*∈{*α*,*β*,*γ*} denote the column vector of |*ξ*| 1s, and let *ξ*∈{*α*,*β*,*γ*}, one finds

The ‘off-diagonal’ elements come in pairs *ξζ*∈{*αβ*,*βγ*,*αγ*}, which are equal to one another and also equal to the number *G*_{*}.

Moreover, one member of such a pair is multiplied by *ε*, while the other is multiplied by *ε*^{2}. Noticing that *ε*+*ε*^{2}=−1, we have
*G* is bipartite with bipartition sets *V*_{*} and *V*_{°}, it follows that
*v*∈*V*_{°} has three neighbours in *V*_{*}, and any two of them are connected by an edge in *G*_{*}. Thus every degree-3 unstarred vertex corresponds to three edges of *G*_{*}. Similarly, every degree-2 unstarred vertex corresponds to one edge of *G*_{*}. As every edge of *G*_{*} is so obtained, we have

We are now ready for the proof of our main theorem.

### Proof of theorem 1.1

The first part of this theorem for the circumstance that #*≠#° has long been realized [28,30] to be an immediate consequence of the pairing theorem. This result has been included here to emphasize the complementary nature of the remaining part of our theorem where #*=*n*/2=#°. If #*=*n*/2=#°, then theorem 2.4 yields
*V*_{*} and *V*_{°} are interchangeable in theorem 2.4, we also have

For example, for the HOMO–LUMO gap of the graph in figure 1, we have #*=#°=23, and

In [39], the result of Mohar implies that the upper bound for the HOMO–LUMO gap of a subcubic planar bipartite graph is 2. For connected subgraphs of graphene, our result in theorem 1.1 improves the upper bound of 2 to

### Corollary 2.5

*Let G be a connected subgraph of graphene. Then*
*with equality if and only if G is the ethylene graph or the benzene graph*.

### Proof.

By theorem 1.1, it is trivial that Δ(*G*)≤2. If the equality holds, then clearly *G* is an even path or an even cycle. Now the eigenvalues of the *n*-vertex path are *n*-vertex cycle are *k*=0,1,2,…,*n*−1 [45]. Thence, it is verified that Δ(*G*)=2 if and only if *G* is the ethylene graph or *G* is the benzene graph. ▪

Next, we work towards our second announced theorem:

### Lemma 2.6

*Let G be a graphene subgraph with no degree-1 sites. Then*,

### Proof.

By hypothesis *G* is bipartite, it follows that

Thence, we have:

Next, we address particular interesting benzenoids.

## 3. Application to benzenoids

### (a) Benzenoids

Again a *benzenoid* graph is defined as a connected subgraph of the honeycomb net enclosed in a Jordan curve on the net. And we let #_{∂} denote the number of sites on the boundary of the benzenoid. A proof of theorem 1.3 (that

### Proof of theorem 1.3

As all the degree-2 vertices of *G* lie on the boundary of *G*, we know that *G*)=0. Also as one goes around the periphery counterclockwise, there is a 60° left turn at each degree-2 site. There are 60° right turns at each boundary degree-3 site. But the net turn is 6×60°=360°. Thus, if we denote by #_{∂3} the number of degree-3 vertices on the boundary of *G*, then #_{2}−#_{∂3}=6 and #_{∂}=#_{2}+#_{∂3}=2#_{2}−6. Hence

It turns out that the bound of theorem 1.3 is quite effective for benzenoids (confining Δ(*G*) to a narrow range, near 0) when a benzenoid has a sizable interior compared with its boundary.

### (b) Cata-condensed benzenoids

A *cata-condensed benzenoid* (or *outerplanar benzenoid*) is a benzenoid such that all the vertices lie on its perimeter. For example, see figure 3.

For a cata-condensed benzenoid with *N* hexagons, note that #_{∂}=*n*=4*N*+2, so that by theorem 1.3 for a cata-condensed benzenoid with *N* hexagons

It is easily seen that the sequence *G*)→0 as

### (c) Circum-coronene series of benzenoids

Consider large polycycles composed via successive circumscriptions by benzene rings, starting from the benzene molecule (denoted as PC[1]). We add benzene rings around this to reach coronene (PC[2]), around which we can add further benzene rings, to obtain circum-coronene (PC[3]), and so forth (as in figure 4).

For PC[*N*], it may be readily verified that *n*=6*N*^{2} and #_{∂}=12*N*−6. Hence by theorem 1.3, we have

### (d) A ring-type benzenoid

We consider a ring-type benzenoid *R*[*N*]. *R*[2], *R*[3] and *R*[4] are illustrated in figure 5.

For *R*[*N*], it is not hard to verify that *n*=18*N*^{2}−18*N*+6 and #_{∂}=24*N*−18. Hence by theorem 1.3, we have

In particular, the large benzenoid from the Müllen group [11] *R*[4] gives *R*[*N*] are Clar-sextet resonant but still have a gap Δ→0 as

### (e) Parallelogram-like benzenoids

The *parallelogram-like benzenoid* with *N* and *M* hexagons along the intersections of the parallelogram, denoted by *P*[*M*,*N*], is shown in figure 6.

For *P*[*M*,*N*], one may observe that *n*=2(*MN*+*M*+*N*) and #_{∂}=4*M*+4*N*−2. Hence by theorem 1.3, we have
*M*≤*N* with *r*≡*M*/*N*, then go to the large *N* limit, this devolves to

## 4. HOMO–LUMO gaps of connected subgraphs of carbon nanotubes

In this section, we consider the HOMO–LUMO gaps of finite connected subgraphs of carbon nanotubes. Carbon nanotube [46–54], particularly single-wall nanotubes, can be imagined to be formed by rolling a single-layer graphite (i.e. graphene) sheet into a seamless cylinder and, depending on the mode of rolling up, it can be of armchair, zigzag or chiral format. The wrapping of the graphene sheet can be represented by a pair of indices (*h*,*k*). The non-negative integers, *h* and *k*, are [46] the number of unit vectors along two hexagon centre–centre directions on the graphene sheet such that on performing this transformation one ends up in the starting hexagon. For example, figure 7 shows a (4,2) nanotube. If *k*=0, the nanotube is zigzag (*h*,0); if *h*=*k*, it is armchair (*h*,*h*); and if *h*>*k*>0 but none is zero, the nanotube is chiral (*h*,*k*).

### Theorem 4.1

*Let G be a finite connected n-vertex subgraph of an* (*h,k*)-*nanotube. If #**≠#°, *then* Δ(*G*)=0. *Otherwise, if h*−*k*=0 (mod3), *then*
*while if h*−*k*≠0 (mod3), *then*
*where* *and*

### Proof.

Let *G* be a connected subgraph of an (*h*,*k*)-nanotube. Then, it is obvious that *G* is bipartite. Using the same arguments and symbols given in §2, to determine the bound for the HOMO–LUMO gap of *G*, it suffices to obtain bounds for the least eigenvalues of *B*_{*} and *B*_{°}. To this end, we again use the Rayleigh quotient technique, trying to find trial vectors for *B*_{*} and *B*_{°}. First consider an upper bound for the least eigenvalue of *B*_{*}. Only the case that |*V*_{*}|=|*V*_{°}|=*n*/2 need be considered, for otherwise Δ(*G*)=0. In this case, then, *B*_{*} is a square matrix of order *n*/2 such that
*G*_{*} with vertex set *V*_{*} such that *v*_{i},*v*_{j}∈*V*_{*} are adjacent in *G*_{*} if and only if the distance between *v*_{i} and *v*_{j} is 2 in *G* (figure 8). Then the adjacency matrix of *G*_{*} could be obtained from *B*_{*} by replacing all the diagonal elements of *B*_{*} by 0.

For the trial vector required in the Rayleigh–Ritz quotient, we assign a complex number to each vertex in *V*_{*}. However, compared with the case in §2 where each vertex is assigned the value 1, or *ε*, or *ε*^{2}, the assignment of values in this case is more complicated, involving an assignment of coordinates to each vertex. Choose an arbitrary starred vertex as the origin (with *x*=*y*=0), then establish the affine coordinates as given in figure 9. For the vertex with coordinate (*x*,*y*), we assign the value *x*,*y*) and (*x*+*k*,*y*+*h*) have the same value, while intervening vertices accommodate somewhat closely to values such as used in the subgraphenic case of §2.

As every component of the trial vector *φ*=(*φ*_{1},*φ*_{2},…,*φ*_{n/2})^{t} is a unit magnitude complex number, it follows that
*B*_{*}]_{ii}=*d*_{vi}(*G*), thus it gives
*i*≠*j*, noticing that [*B*_{*}]_{ij}=1 if and only if *v*_{i} and *v*_{j} are adjacent in *G*_{*} (otherwise [*B*_{*}]_{ij}=0), it follows that
*θ*−*θ*′:

### Case 1

*v*_{i}*v*_{j} is a horizontal edge. This means that the coordinates of *v*_{i} and *v*_{j} are (*x*,*y*) and (*x*+1,*y*), respectively. Then *θ*′−*θ*=(4*π*/3)+*δ* or *θ*′−*θ*=(−2*π*/3)+*δ*, so that

### Case 2

*v*_{i}*v*_{j} is a vertical edge. This means that the coordinates of *v*_{i} and *v*_{j} are (*x*,*y*) and (*x*,*y*+1), respectively. Then *θ*′−*θ*=(2*π*/3)+*δ* or *θ*′−*θ*=(−4*π*/3)+*δ*, so that

### Case 3

*v*_{i}*v*_{j} is a diagonal edge. This means that the coordinates of *v*_{i} and *v*_{j} are (*x*,*y*+1) and (*x*+1,*y*), respectively. Then *θ*′−*θ*=2*π*/3 or *θ*′−*θ*=−4*π*/3, so that

As explained in §2, each degree-3 unstarred vertex corresponds to three (different types of) edges of *G*_{*}, and each degree-2 unstarred vertex corresponds to one edge of *G*_{*}. Conversely, each edge of *G*_{*} is so obtained. For each degree-3 unstarred vertex, the three edges corresponding to this vertex contribute

Substituting equations (4.6), (4.7) and (4.8) into equation (4.5), we get
*δ*, proposition 2.2, and equations (4.10) and (4.11), theorem 4.1 is proved. ▪

For example, for the graph *G* shown in figure 7, it is not hard to verify that
*G*)≤1.84.

## 5. Concluding remarks

Especially because of the emergence (e.g. [4,5,7–12,55–58]) of interest in large conjugated-carbon nano-structures, our results should prove to be of interest. The relevance of the counts *G*)=0, if *G*)→0 as the undefected graphene or buckytube limit is approached. The relevance of the numbers *d*=1,2,3) seems to be becoming manifest, as part of a general systematics for conjugated-carbon nano-structures.

## Data accessibility

This paper comprises theory only and contains no data.

## Author' contributions

D.J.K., Y.Y. and D.Y. performed the research and wrote the paper. All authors gave final approval for publication.

## Conflict of interests

We have no competing interests.

## Funding

The authors acknowledge the support of the Welch Foundation of Houston, Texas (through grant no. BD-0894). Y.Y acknowledges the support of the National Natural Science Foundation of China (through grant no. 11201404), China Postdoctoral Science Foundation (through grant nos. 2012M521318 and 2013T60662) and Special Funds for Postdoctoral Innovative Projects of Shandong Province (through grant no. 201203056).

## Acknowledgements

The authors thank the anonymous referees for their comments on this manuscript.

- Received March 14, 2015.
- Accepted June 9, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.