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The Wiener index and the Szeged index of benzenoid systems in linear time
 J. Chem. Inf. Comput. Sci
, 1997
"... A linear time algorithm is presented which, for a given benzenoid system G, computes the Wiener index of G. The algorithm is based on an isometric embedding of G into the Cartesian product of three trees, combined with the notion of the Wiener index of vertexweighted graphs. An analogous approach y ..."
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A linear time algorithm is presented which, for a given benzenoid system G, computes the Wiener index of G. The algorithm is based on an isometric embedding of G into the Cartesian product of three trees, combined with the notion of the Wiener index of vertexweighted graphs. An analogous approach yields also a linear algorithm for computing the Szeged index of benzenoid systems. 1.
Chemical Graph Theory of Fibonacenes
"... Fibonacenes (zigzag unbranched catacondensed benzenoid hydrocarbons) are a class of polycyclic conjugated systems whose molecular graphs possess remarkable properties, often related with the Fibonacci numbers. This article is a review of the chemical graph theory of fibonacenes, with emphasis on th ..."
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Fibonacenes (zigzag unbranched catacondensed benzenoid hydrocarbons) are a class of polycyclic conjugated systems whose molecular graphs possess remarkable properties, often related with the Fibonacci numbers. This article is a review of the chemical graph theory of fibonacenes, with emphasis on their Kekulestructurerelated and Clarstructurerelated properties.
Distances in Benzenoid Systems: Further Developments
, 1998
"... We present some new results on distances in benzenoids. An algorithm is proposed which, for a given benzenoid system G bounded by a simple circuit Z with n vertices, computes the Wiener index of G in O(n) number of operations. We also show that benzenoid systems have a convenient dismantling scheme, ..."
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We present some new results on distances in benzenoids. An algorithm is proposed which, for a given benzenoid system G bounded by a simple circuit Z with n vertices, computes the Wiener index of G in O(n) number of operations. We also show that benzenoid systems have a convenient dismantling scheme, which can be derived by applying the breadthfirst search to their dual graphs. Our last result deals with the clustering problem of sets of atoms of benzenoids systems. We demonstrate how the
Area distribution and scaling function for punctured polygons, preprint available at http://arxiv.org/abs/math/0701633
"... Punctured polygons are polygons with internal holes which are also polygons. The external and internal polygons are of the same type, and they are mutually as well as selfavoiding. We rigorously analyse the effect of a finite number of punctures on the limiting area distribution in a uniform ensemb ..."
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Cited by 3 (1 self)
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Punctured polygons are polygons with internal holes which are also polygons. The external and internal polygons are of the same type, and they are mutually as well as selfavoiding. We rigorously analyse the effect of a finite number of punctures on the limiting area distribution in a uniform ensemble, where punctured polygons with equal perimeter have the same probability of occurrence. The results rely on an assumption on the limiting area distribution for unpunctured polygons. Our analysis leads to conjectures about the possible scaling behaviour of the models. We also analyse exact enumeration data. For staircase polygons with punctures of fixed size, we find exact generating functions for the first few areamoments. For staircase polygons with punctures of arbitrary size, a careful numerical analysis yields very accurate estimates for the areamoments. Interestingly, we find that the leading correction term for each areamoment is proportional to the corresponding areamoment with one less puncture. We finally analyse corresponding quantities for punctured selfavoiding polygons and find agreement with the exact formulas to at least 3–4 significant digits. 1
Polyhedral Combinatorics of Benzenoid Problems
 Lect. Notes Comput. Sci
, 1998
"... Many chemical properties of benzenoid hydrocarbons can be understood in terms of the maximum number of mutually resonant hexagons, or Clar number, of the molecules. Hansen and Zheng (1994) formulated this problem as an integer program and conjectured, based on computational results, that solving the ..."
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Many chemical properties of benzenoid hydrocarbons can be understood in terms of the maximum number of mutually resonant hexagons, or Clar number, of the molecules. Hansen and Zheng (1994) formulated this problem as an integer program and conjectured, based on computational results, that solving the linear programming relaxation always yields integral solutions. We prove this conjecture showing that the constraint matrices of these problems are unimodular. This establishes the integrality of the relaxation polyhedra since the linear programs are in standard form. However, the matrices are not, in general, totally unimodular as is often the case with other combinatorial optimization problems that give rise to integral polyhedra. Similar results are proved for the Fries number, another optimization problem for benzenoids. In a previous paper, Hansen and Zheng (1992) showed that a certain minimum weight cut cover problem defined for benzenoids yields an upper bound for the Clar number and...
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
Wiener Number of Hexagonal Bitrapeziums and Trapeziums
 Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur
, 1997
"... The Wiener number of a connected graph is equal to the sum of distances between all pairs of its vertices. A graph formed by a row of n hexagonal cells is called an nhexagonal chain. A graph consisting of m shexagonal chains, where s runs from n to n \Gamma m+1 , forming the shape of a trapezium ..."
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The Wiener number of a connected graph is equal to the sum of distances between all pairs of its vertices. A graph formed by a row of n hexagonal cells is called an nhexagonal chain. A graph consisting of m shexagonal chains, where s runs from n to n \Gamma m+1 , forming the shape of a trapezium is called an n \Theta m hexagonal trapezium. A graph obtained by merging the base nhexagonal chains of two n \Theta m hexagonal trapeziums, forming a convex 6sided polygram, is called an n \Theta m hexagonal bitrapezium. In this paper, we obtain the Wiener numbers of the n \Theta m hexagonal trapezium and of the n \Theta m hexagonal bitrapezium. Key words and phrases: Wiener number, distance in graphs, hexagonal systems AMS 1991 Subject Classifications: 05C12, 05C90. 1 Introduction An important invariant of connected graphs is called the Wiener number (or Wiener index) W . This number is equal to the sum of distances between all pairs of vertices of the respective graph. The quantity W...
Situ Chemical Oxidation of Creosote/Coal Tar Residuals: Experimental and Numerical Investigation
, 2004
"... I herby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final version, as accepted by my examiners. I understand that my thesis may be made electronically available to the public ..."
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I herby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final version, as accepted by my examiners. I understand that my thesis may be made electronically available to the public
The Lattice Dimension of Benzenoid Systems
, 2006
"... A labeling of vertices of a benzenoid system B is proposed that reflects the graph distance in B and is significantly shorter that the labeling obtained from a hypercube embedding of B. The new labeling corresponds to an embedding of B into the integer lattice Z and is shown to be optimal for all ..."
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A labeling of vertices of a benzenoid system B is proposed that reflects the graph distance in B and is significantly shorter that the labeling obtained from a hypercube embedding of B. The new labeling corresponds to an embedding of B into the integer lattice Z and is shown to be optimal for all practical purposes. A coordinatization algorithm is presented and it is demonstrated that it can be easily carried out by hand.
A Constructive Enumeration of Fusenes and Benzenoids
"... In this paper, a fast and complete method to constructively enumerate fusenes and benzenoids is given. It is fast enough to construct several million non isomorphic structures per second. The central idea is to represent fusenes as labelled inner duals and generate them in a two step approach using ..."
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In this paper, a fast and complete method to constructively enumerate fusenes and benzenoids is given. It is fast enough to construct several million non isomorphic structures per second. The central idea is to represent fusenes as labelled inner duals and generate them in a two step approach using the canonical construction path method and the homomorphism principle.