1 3
Theor Chem Acc (2015) 134:104
DOI 10.1007/s00214-015-1708-5
REGULAR ARTICLE
Topological coordinates for bar polyhex carbon structures
István László
1
Received: 11 June 2015 / Accepted: 25 July 2015 / Published online: 12 August 2015
© Springer-Verlag Berlin Heidelberg 2015
initial position of the atoms. This information is an inevitable requirement for a molecular viewer as well. In many
cases, however, only the neighbouring structure of the
atoms is given. That is, we know the adjacency matrix
A = A(G) = (a uv ) of the structure which is represented
by a graph G(V , E) . Here V is the set of vertices, and E
is the set of edges, the adjacency matrix element a uv = 1
if (u, v) ∈ E and a uv = 0 ; otherwise, we suppose further
that the graph has n = |V | vertices. In this graph, the atoms
correspond to the vertices and the fi rst-neighbour bonds
to the edges. The topology can be described with the help
of the Q(G) = D − A Laplacian matrix as well, where
D = (d vv ) is the diagonal matrix with d vv =
u:(u,v)∈E a uv .
In the topological coordinate method, some eigenvectors of the adjacency matrix (or the Laplacian matrix) are
used to generate the Descartes coordinates of the atoms.
These eigenvectors are the so-called bi-lobal eigenvectors. An eigenvector c k is bi-lobal, if in the graph of the
atomic structure after deleting the vertices i if c k
i = 0 and
the edges (i, j) if the signs of c k
i and c k
j are different, the
resulting graph will have two components. The increasing index k for the eigenvector c k corresponds to decreasing k eigenvector for the adjacency matrix and increasing
eigenvector to the Laplacian matrix. The fi rst systematic
application of this method was presented by Fowler and
Manolopoulos [ 1 , 2 ] for general fullerene isomers C 20 to
C 50 and isolated-pentagon isomers C 60 to C 100 . Similar
method was found by Pisanski and Shaw-Taylor [ 3 – 5 ]. All
of these methods were applicable where the carbon atoms
were on a spherical or near-spherical surface. In mathematics, the problem was stated and proved as embedding
graphs into the Euclidean space R 3 or R 2 [ 5 – 7 ]. Under
embedding a graph G(V , E) into R k , we mean a mapping
τ : V (G) → R k . Let τ i be the n -dimensional vector formed
by taking the i -th coordinate τ (v) i of τ (v) for all v ∈ V .
Abstract Very often, the basic information about a nanostructure is a topological one. Based on this topological
information, we have to determine the Descartes coordinates of the atoms. In the present paper, we review fi rst the
previous results obtained by drawing graphs with the help
of various matrices as the adjacency matrix, the Laplacian
matrix and the Colin de Verdière matrix. We explain why
they are applicable if the atoms are on spherical surfaces.
We have found recently a matrix W which could generate
the Descartes coordinates for fullerenes, nanotubes and
nanotori and also for nanotube junctions and coils as well.
Here will be shown with examples of bar polyhex structures that using the matrix elements of smaller structures,
the W matrix of larger structures can be generated.
Keywords Drawing · Eigenvectors · Embedding ·
Graphs · Molecular structures · Nanostructures
1 Introduction
In order to perform a quantum chemical calculation, usually one of the most important input data is the
This paper is dedicated to Professor P. R. Surján on the occasion
of his 60th birthday.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surján”.
* István László
laszlo@eik.bme.hu
1
Department of Theoretical Physics, Institute of Physics ,
Budapest University of Technology and Economics ,
Budapest 1521 , Hungary
73
Reprinted from the journal
Theor Chem Acc (2015) 134:104
DOI 10.1007/s00214-015-1708-5
REGULAR ARTICLE
Topological coordinates for bar polyhex carbon structures
István László
1
Received: 11 June 2015 / Accepted: 25 July 2015 / Published online: 12 August 2015
© Springer-Verlag Berlin Heidelberg 2015
initial position of the atoms. This information is an inevitable requirement for a molecular viewer as well. In many
cases, however, only the neighbouring structure of the
atoms is given. That is, we know the adjacency matrix
A = A(G) = (a uv ) of the structure which is represented
by a graph G(V , E) . Here V is the set of vertices, and E
is the set of edges, the adjacency matrix element a uv = 1
if (u, v) ∈ E and a uv = 0 ; otherwise, we suppose further
that the graph has n = |V | vertices. In this graph, the atoms
correspond to the vertices and the fi rst-neighbour bonds
to the edges. The topology can be described with the help
of the Q(G) = D − A Laplacian matrix as well, where
D = (d vv ) is the diagonal matrix with d vv =
u:(u,v)∈E a uv .
In the topological coordinate method, some eigenvectors of the adjacency matrix (or the Laplacian matrix) are
used to generate the Descartes coordinates of the atoms.
These eigenvectors are the so-called bi-lobal eigenvectors. An eigenvector c k is bi-lobal, if in the graph of the
atomic structure after deleting the vertices i if c k
i = 0 and
the edges (i, j) if the signs of c k
i and c k
j are different, the
resulting graph will have two components. The increasing index k for the eigenvector c k corresponds to decreasing k eigenvector for the adjacency matrix and increasing
eigenvector to the Laplacian matrix. The fi rst systematic
application of this method was presented by Fowler and
Manolopoulos [ 1 , 2 ] for general fullerene isomers C 20 to
C 50 and isolated-pentagon isomers C 60 to C 100 . Similar
method was found by Pisanski and Shaw-Taylor [ 3 – 5 ]. All
of these methods were applicable where the carbon atoms
were on a spherical or near-spherical surface. In mathematics, the problem was stated and proved as embedding
graphs into the Euclidean space R 3 or R 2 [ 5 – 7 ]. Under
embedding a graph G(V , E) into R k , we mean a mapping
τ : V (G) → R k . Let τ i be the n -dimensional vector formed
by taking the i -th coordinate τ (v) i of τ (v) for all v ∈ V .
Abstract Very often, the basic information about a nanostructure is a topological one. Based on this topological
information, we have to determine the Descartes coordinates of the atoms. In the present paper, we review fi rst the
previous results obtained by drawing graphs with the help
of various matrices as the adjacency matrix, the Laplacian
matrix and the Colin de Verdière matrix. We explain why
they are applicable if the atoms are on spherical surfaces.
We have found recently a matrix W which could generate
the Descartes coordinates for fullerenes, nanotubes and
nanotori and also for nanotube junctions and coils as well.
Here will be shown with examples of bar polyhex structures that using the matrix elements of smaller structures,
the W matrix of larger structures can be generated.
Keywords Drawing · Eigenvectors · Embedding ·
Graphs · Molecular structures · Nanostructures
1 Introduction
In order to perform a quantum chemical calculation, usually one of the most important input data is the
This paper is dedicated to Professor P. R. Surján on the occasion
of his 60th birthday.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surján”.
* István László
laszlo@eik.bme.hu
1
Department of Theoretical Physics, Institute of Physics ,
Budapest University of Technology and Economics ,
Budapest 1521 , Hungary
73
Reprinted from the journal