Maximizing the number of independent subsets over trees with bounded degree
01 May 2008-Journal of Graph Theory (Wiley Subscription Services, Inc., A Wiley Company)-Vol. 58, Iss: 1, pp 49-68
TL;DR: In this article, the authors give a characterization of the trees with given maximum degree which maximize the number of independent subsets, and show that these trees also minimize the independent edge subsets.
Abstract: The number of independent vertex subsets is a graph parameter that is, apart from its purely mathematical importance, of interest in mathematical chemistry. In particular, the problem of maximizing or minimizing the number of independent vertex subsets within a given class of graphs has already been investigated by many authors. In view of the applications of this graph parameter, trees of restricted degree are of particular interest. In the current article, we give a characterization of the trees with given maximum degree which maximize the number of independent subsets, and show that these trees also minimize the number of independent edge subsets. The structure of these trees is quite interesting and unexpected: it can be described by means of a novel digital system—in the case of maximum degree 3, we obtain a binary system using the digits 1 and 4. The proof mainly depends on an exchange lemma for branches of a tree. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 49–68, 2008
Dedicated to Prof. Robert Tichy on the occasion of his 50th birthday.
This article was written while C. Heuberger was a visitor at the Center of Experimental Mathematics at the University of Stellenbosch. He thanks the center for its hospitality.
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TL;DR: The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure as discussed by the authors.
Abstract: The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.
121 citations
Cites background from "Maximizing the number of independen..."
...A related result that is specifically geared towards trees with fixed maximum degree can be found in [32]....
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...Theorem 6 ([32]) Given the number n of vertices and the maximum degree d ≥ 3, the tree that minimizes the Hosoya index and maximizes the Merrifield-Simmons index has the following shape: ....
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...The result for the minimum Hosoya index and maximum Merrifield-Simmons index appears to be much less intuitive: a partial solution to this problem was found by Lv and Yu [68] for trees with large maximum degree (at least one third of the number of vertices), the general problem was settled by Heuberger and one of the authors of this survey [32]:...
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...Quite often, these formulas together with elementary methods such as induction or the trivial fact that deleting edges results in a decrease of the Hosoya index and an increase of the Merrifield-Simmons index are sufficient, if applied in the right way (see [1, 3, 4, 7, 8, 9, 10, 11, 12, 22, 32, 51, 52, 53, 57, 59, 61, 63, 65, 67, 68, 82, 84, 85, 86, 87, 89, 90, 92, 93, 98, 99, 100, 101, 103, 105, 109, 110, 111, 112, 115, 116, 117, 119, 120] for various examples)....
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TL;DR: In this paper, it was shown that among the trees with fixed maximum degree Δ, the broom B istg n, Δ, consisting of a star S istg Δ+1 and a path of length n−Δ−1 attached to an arbitrary pendent vertex of the star, is the unique tree which minimizes even spectral moments and the Estrada index.
Abstract: Let G be a simple graph with n vertices and let λ1, λ2, . . . , λ
n
be the eigenvalues of its adjacency matrix. The Estrada index of G is a recently introduced molecular structure descriptor, defined as $${EE (G) = \sum_{i = 1}^n e^{\lambda_i}}$$
, proposed as a measure of branching in alkanes. In order to support this proposal, we prove that among the trees with fixed maximum degree Δ, the broom B
n,Δ, consisting of a star S
Δ+1 and a path of length n−Δ−1 attached to an arbitrary pendent vertex of the star, is the unique tree which minimizes even spectral moments and the Estrada index, and then show the relation EE(S
n
) = EE(B
n,n−1) > EE(B
n,n−2) > . . . > EE(B
n,3) > EE(B
n,2) = EE(P
n
). We also determine the trees with minimum Estrada index among the trees with perfect matching and maximum degree Δ. On the other hand, we strengthen a conjecture of Gutman et al. [Z. Naturforsch. 62a (2007), 495] that the Volkmann trees have maximal Estrada index among the trees with fixed maximum degree Δ, by conjecturing that the Volkmann trees also have maximal even spectral moments of any order. As a first step in this direction, we characterize the starlike trees which maximize even spectral moments and the Estrada index.
57 citations
TL;DR: In this paper, it was shown that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al.
Abstract: The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph's eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. Trees minimizing the energy under various additional conditions have been determined in the past, e.g., trees with a given diameter or trees with a perfect matching. However, it is quite a natural problem to minimize the energy of trees with bounded maximum degree--clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimal energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.
38 citations
TL;DR: In this article, it was shown that the trees with given maximum degree that minimize the energy are the same trees that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus proving a conjecture due to Fischermann et al.
Abstract: The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph’s eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. It is quite a natural problem to minimize the energy of trees with bounded maximum degree—clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimum energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.
37 citations
TL;DR: How far the negative correlation between distances and subtrees go if the authors look for (and characterize) the extremal values of F"T (w)/F"T(u), F" t(w/F"t(w)", and F(T)/F'T(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and w is a leaf in T is tested.
29 citations
Cites background from "Maximizing the number of independen..."
...It is worthwhile to investigate how far the dual behaviour goes as the papers [26] and [28] generated considerable interest in different disciplines [4, 6, 10, 11, 12, 13, 14, 17, 20, 21, 23, 24, 31, 34, 38]....
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References
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TL;DR: In this article, the authors present a classification scheme for Monocyclic systems based on the Huckel Spectrum and the Cayley Generation Functions. But they do not discuss the role of Kekule structures in chemistry.
Abstract: INTRODUCTION. ELEMENTS OF GRAPH THEORY. The Definition of a Graph. Isomorphic Graphs and Graph Automorphism. Walks, Trails, Paths, Distances and Valencies in Graphs. Subgraphs. Regular Graphs. Trees. Planar Graphs. The Story of the Koenigsberg Bridge Problem and Eulerian Graphs. Hamiltonian Graphs. Line Graphs. Vertex Coloring of a Graph. CHEMICAL GRAPHS. The Concept of a Chemical Graph. Molecular Topology. Huckel Graphs. Polyhexes and Benzenoid Graphs. Weighted Graphs. GRAPH-THEORETICAL MATRICES. The Adjacency Matrix. The Distance Matrix. THE CHARACTERISTIC POLYNOMIAL OF A GRAPH. The Definition of the Characteristic Polynomial. The Method of Sachs for Computing the Characteristic Polynomial. The Characteristic Polynomials of Some Classes of Simple Graphs. The Le Verrier-Faddeev-Frame Method for Computing the Characteristic Polynomial. TOPOLOGICAL ASPECTS OF HUECKEL THEORY. Elements of Huckel Theory. Isomorphism of Huckel Theory and Graph Spectral Theory. The Huckel Spectrum. Charge Densities and Bond Orders in Conjugated Systems. The Two-Color Problem in Huckel Theory. Eigenvalues of Linear Polyenes. Eigenvalues of Annulenes. Eigenvalues of Moebius Annulenes. A Classification Scheme for Monocyclic Systems. Total p-Electron Energy. TOPOLOGICAL RESONANCE ENERGY. Huckel Resonance Energy. Dewar Resonance Energy. The Concept of Topological Resonance Energy. Computation of the Acyclic Polynomial. Applications of the TRE Model. ENUMERATION OF KEKULE VALENCE STRUCTURES. The Role of Kekule Valence Structures in Chemistry. The Identification of Kekule Systems. Methods for the Enumeration of Kekule Structures. The Concept of Parity of Kekule Structures. THE CONJUGATED-CIRCUIT MODEL. The Concept of Conjugated Circuits. The p-Resonance Energy Expression. Selection of the Parameters. Computational Procedure. Applications of the Conjugated-Circuit Model. Parity of Conjugated Circuits. TOPOLOGICAL INDICES. Definitions of Topological Indices. The Three-Dimensional Wiener Number. ISOMER ENUMERATION. The Cayley Generation Functions. The Henze-Blair Approach. The Polya Enumeration Method. The Enumeration Method Based on the N-Tuple Code.
1,473 citations
"Maximizing the number of independen..." refers background in this paper
...Meanwhile, the number of independent subsets of a graph is called the Merrifield-Simmons index in mathematical chemistry, and there is already a substantial amount of literature on chemical applications as well as on graph-theoretical properties of this index (see [3, 19] and the references therein)....
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Book•
01 Jan 1986
TL;DR: In this paper, the authors define the topology of a graph as follows: 1.1 Topology in Chemistry, 2.2 Geometry, Symmetry, Topology, Graph Automorphisms, and Graph Topology.
Abstract: A Chemistry and Topology.- 1 Topological Aspects in Chemistry.- 1.1 Topology in Chemistry.- 1.2 Abstraction in Science and How Far One Can Go.- 2 Molecular Topology.- 2.1 What is Molecular Topology?.- 2.2 Geometry, Symmetry, Topology.- 2.3 Definition of Molecular Topology.- B Chemistry and Graph Theory.- 3 Chemical Graphs.- 4 Fundamentals of Graph Theory.- 4.1 The Definition of a Graph.- 4.1.1 Relations.- 4.1.2 The First Definition of a Graph.- 4.1.3 The Second Definition of a Graph.- 4.1.4 Vertices and Edges.- 4.1.5 Isomorphic Graphs and Graph Automorphisms.- 4.1.6 Special Graphs.- 4.2 Subgraphs.- 4.2.1 Sachs Graphs.- 4.2.2 Matchings.- 4.3 Graph Spectral Theory.- 4.3.1 The Adjacency Matrix.- 4.3.2 The Spectrum of a Graph.- 4.3.3 The Sachs Theorem.- 4.3.4 The ?-Polynomial.- 4.4 Graph Operations.- 5 Graph Theory and Molecular Orbitals.- 6 Special Molecular Graphs.- 6.1 Acyclic Molecules.- 6.1.1 Trees.- 6.1.2 The Path and the Star.- 6.1.3 The Characteristic Polynomial of Trees.- 6.1.4 Trees with Greatest Number of Matchings.- 6.1.5 The Spectrum of the Path.- 6.2 The Cycle.- 6.3 Alternant Molecules.- 6.3.1 Bipartite Graphs.- 6.3.2 The Pairing Theorem.- 6.3.3 Some Consequences of the Pairing Theorem.- 6.4 Benzenoid Molecules.- 6.4.1 Benzenoid Graphs.- 6.4.2 The Characteristic Polynomial of Benzenoid Graphs.- 6.5 Hydrocarbons and Molecules with Heteroatoms.- 6.5.1 On the Question of the Molecular Graph.- 6.5.2 The Characteristic Polynomial of Weighted Graphs.- 6.5.3 Some Regularities in the Electronic Structure of Heteroconjugated Molecules.- C Chemistry and Group Theory.- 7 Fundamentals of Group Theory.- 7.1 The Symmetry Group of an Equilateral Triangle.- 7.2 Order, Classes and Representations of a Group.- 7.3 Reducible and Irreducible Representations.- 7.4 Characters and Reduction of a Reducible Representation.- 7.5 Subgroups and Sidegroups - Products of Groups.- 7.6 Abelian Groups.- 7.7 Abstract Groups and Group Isomorphism.- 8 Symmetry Groups.- 8.1 Notation of Symmetry Elements and Representations.- 8.2 Some Symmetry Groups.- 8.2.1 Rotation Groups.- 8.2.2 Groups with More than One n-Fold Axis, n > 2.- 8.2.3 Groups of Collinear Molecules.- 8.3 Transformation Properties and Direct Products of Irreducible Representations.- 8.3.1 Transformation Properties.- 8.3.2 Rules Concerning the Direct Product of Irreducible Representations.- 8.4 Some Applications of Symmetry Groups.- 8.4.1 Electric Dipole Moment.- 8.4.2 Polarizability.- 8.4.3 Motions of Atomic Nuclei: Translations, Rotations and Vibrations.- 8.4.4 Transition Probabilities for the Absorption of Light.- 8.4.5 Transition Probabilities in Raman Spectra.- 8.4.6 Group Theory and Quantum Chemistry.- 8.4.7 Orbital and State Correlations.- 9 Automorphism Groups.- 9.1 Automorphism of a Graph.- 9.2 The Automorphism Group A(G1).- 9.3 Cycle Structure of Permutations.- 9.4 Isomorphism of Graphs and of Automorphism Groups 112..- 9.5 Notation of some Permutation Groups.- 9.6 Direct Product and Wreath Product.- 9.7 The Representation of Automorphism Groups as Group Products.- 10 Some Interrelations between Symmetry and Automorphism Groups.- 10.1 The Idea of Rigid Molecules.- 10.2 Local Symmetries.- 10.3 Non-Rigid Molecules.- 10.4 What Determines the Respective Orders of the Symmetry and the Automorphism Group of a Given Molecule?.- D Special Topics.- 11 Topological Indices.- 11.1 Indices Based on the Distance Matrix.- 11.1.1 The Wiener Number and Related Quantities.- 11.1.2 Applications of the Wiener Number.- 11.2 Hosoya's Topological Index.- 11.2.1 Definition and Chemical Applications of Hosoya's Index.- 11.2.2 Mathematical Properties of Hosoya's Index.- 11.2.3 Example: Hosoya's Index of the Path and the Cycle.- 11.2.4 Some Inequalities for Hosoya's Index.- 12 Thermodynamic Stability of Conjugated Molecules.- 12.1 Total ?-Electron Energy and Thermodynamic Stability of Conjugated Molecules.- 12.2 Total ?-Electron Energy and Molecular Topology.- 12.3 The Energy of a Graph.- 12.4 The Coulson Integral Formula.- 12.5 Some Further Applications of the Coulson Integral Formula.- 12.6 Bounds for Total ?-Electron Energy.- 12.7 More on the McClelland Formula.- 12.8 Conclusion: Factors Determining the Total ?-Electron Energy.- 12.9 Use of Total ?-Electron Energy in Chemistry.- 13 Topological Effect on Molecular Orbitals.- 13.1 Topologically Related Isomers.- 13.2 Interlacing Rule.- 13.3 PE Spectra of Topomers.- 13.4 TEMO and a-Electron Systems.- 13.5 TEMO and Symmetry.- Appendices.- Appendix 1 Matrices.- Appendix 2 Determinants.- Appendix 3 Eigenvalues and Eigenvectors.- Appendix 4 Polynomials.- Appendix 5 Characters of Irreducible Representations of Symmetry Groups.- Appendix 6 The Symbols Used.- Literature.- References.
1,283 citations
"Maximizing the number of independen..." refers background in this paper
...for instance [3, 10]....
[...]
...Meanwhile, the number of independent subsets of a graph is called the Merrifield-Simmons index in mathematical chemistry, and there is already a substantial amount of literature on chemical applications as well as on graph-theoretical properties of this index (see [3, 19] and the references therein)....
[...]
...The formulæ for z0 and z1 are easy to prove and can be found in [3, 10] again, and the identity for τ(T ) follows at once....
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Book•
01 Jan 1989
TL;DR: In this paper, the authors present a topological description of molecular structure bond topology graph topology duplex spaces topology of chemical reactions and a graph theory connectivity classification of topological spaces -separation axioms combinatorics functions and continuity.
Abstract: Part 1 Finite topology: topological spaces finite topologies and lattices finite topologies and graph theory connectivity classification of topological spaces - separation axioms combinatorics functions and continuity. Part 2 Finite topology and chemistry: topological description of molecular structure bond topology graph topology duplex spaces topology of chemical reactions.
218 citations
"Maximizing the number of independen..." refers background in this paper
...Independently, Merrifield and Simmons [13] introduced the number of independent vertex subsets (which they call the σ-index ) to the chemical literature in 1989, showing connections between the σ-index of a molecular graph and physicochemical properties such as boiling points....
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Journal Article•
177 citations
TL;DR: This paper characterize the trees which minimize the Wiener index among all trees of given order and maximum degree and the treesWhich maximize theWiener indexamong all treesof given order that have only vertices of two different degrees.
157 citations
"Maximizing the number of independen..." refers background in this paper
...For other graph parameters, namely the Wiener index (sum of all distances between pairs of vertices) and the number of subtrees, the extremal trees of given maximum degree are already known (see [2, 7, 16, 17])—basically, the solution is given by the complete d-ary trees....
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