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Graph Polynomials

This book covers both theoretical and practical results for graph polynomials. Graph polynomials have been developed for measuring combinatorial graph invariants and for characterizing graphs.

Author : Yongtang Shi

Release : 2016-11-25

Publisher : CRC Press

ISBN : 1498755917

File Size : 54.4 MB

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This book covers both theoretical and practical results for graph polynomials. Graph polynomials have been developed for measuring combinatorial graph invariants and for characterizing graphs. Various problems in pure and applied graph theory or discrete mathematics can be treated and solved efficiently by using graph polynomials. Graph polynomials have been proven useful areas such as discrete mathematics, engineering, information sciences, mathematical chemistry and related disciplines.

Chromatic Polynomials and Chromaticity of Graphs

We end this chapter by considering some graph polynomials that are related to
the chromatic polynomial in the study of chromaticity of graphs. The first of these
is the o-polynomial introduced by Korfhage (1978). This can be written in terms of

Author : F. M. Dong

Release : 2005

Publisher : World Scientific

ISBN : 9812569464

File Size : 46.74 MB

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This is the first book to comprehensively cover chromatic polynomialsof graphs. It includes most of the known results and unsolved problemsin the area of chromatic polynomials. Dividing the book into threemain parts, the authors take readers from the rudiments of chromaticpolynomials to more complex topics: the chromatic equivalence classesof graphs and the zeros and inequalities of chromatic polynomials.

Encyclopaedia of Mathematics, Supplement III

The matching polynomial itself is one of a general class of graph polynomials,
called F-poly normals (see [2]). Two graphs are called co-matching if and only if
they have the same matching polynomial. A graph is called matching unique if
and ...

Author : Michiel Hazewinkel

Release : 2001-12-31

Publisher : Springer Science & Business Media

ISBN : 9781402001987

File Size : 62.3 MB

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This is the third supplementary volume to Kluwer's highly acclaimed twelve-volume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This supplementary volume enhances the existing twelve volumes, and together, these thirteen volumes represent the most authoritative, comprehensive and up-to-date Encyclopaedia of Mathematics available.

Graphs on Surfaces

In this chapter, after a brief review of some classical graph polynomials as well as
deletion and contraction reductions, we turn to polynomials of embedded graphs,
beginning with the topological transition polynomial of [30]. The topological ...

Author : Joanna A. Ellis-Monaghan

Release : 2013-06-28

Publisher : Springer Science & Business Media

ISBN : 1461469716

File Size : 36.30 MB

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Graphs on Surfaces: Dualities, Polynomials, and Knots offers an accessible and comprehensive treatment of recent developments on generalized duals of graphs on surfaces, and their applications. The authors illustrate the interdependency between duality, medial graphs and knots; how this interdependency is reflected in algebraic invariants of graphs and knots; and how it can be exploited to solve problems in graph and knot theory. Taking a constructive approach, the authors emphasize how generalized duals and related ideas arise by localizing classical constructions, such as geometric duals and Tait graphs, and then removing artificial restrictions in these constructions to obtain full extensions of them to embedded graphs. The authors demonstrate the benefits of these generalizations to embedded graphs in chapters describing their applications to graph polynomials and knots. Graphs on Surfaces: Dualities, Polynomials, and Knots also provides a self-contained introduction to graphs on surfaces, generalized duals, topological graph polynomials, and knot polynomials that is accessible both to graph theorists and to knot theorists. Directed at those with some familiarity with basic graph theory and knot theory, this book is appropriate for graduate students and researchers in either area. Because the area is advancing so rapidly, the authors give a comprehensive overview of the topic and include a robust bibliography, aiming to provide the reader with the necessary foundations to stay abreast of the field. The reader will come away from the text convinced of advantages of considering these higher genus analogues of constructions of plane and abstract graphs, and with a good understanding of how they arise.

Model Theoretic Methods in Finite Combinatorics

Graph invariants and graph polynomials. A graph invariant is a function from the
class of (finite) graphs G into some domain D such that isomorphic graphs have
the same image. Usually such invariants are uniformly defined in some formalism

Author : Martin Grohe

Release : 2011-11-28

Publisher : American Mathematical Soc.

ISBN : 0821849433

File Size : 41.22 MB

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This volume contains the proceedings of the AMS-ASL Special Session on Model Theoretic Methods in Finite Combinatorics, held January 5-8, 2009, in Washington, DC. Over the last 20 years, various new connections between model theory and finite combinatorics emerged. The best known of these are in the area of 0-1 laws, but in recent years other very promising interactions between model theory and combinatorics have been developed in areas such as extremal combinatorics and graph limits, graph polynomials, homomorphism functions and related counting functions, and discrete algorithms, touching the boundaries of computer science and statistical physics. This volume highlights some of the main results, techniques, and research directions of the area. Topics covered in this volume include recent developments on 0-1 laws and their variations, counting functions defined by homomorphisms and graph polynomials and their relation to logic, recurrences and spectra, the logical complexity of graphs, algorithmic meta theorems based on logic, universal and homogeneous structures, and logical aspects of Ramsey theory.

Graph-Theoretic Concepts in Computer Science

A Most General Edge Elimination Polynomial⋆ Ilia Averbouch⋆⋆, Benny
Godlin⋆⋆, and Johann A. Makowsky Faculty of ... We look for graph polynomials
which satisfy recurrence re- lations on three kinds of edge elimination: edge
deletion, ...

Author : Hajo Broersma

Release : 2008-12-18

Publisher : Springer Science & Business Media

ISBN : 3540922474

File Size : 85.99 MB

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This book constitutes the thoroughly refereed post-conference proceedings of the 34th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2008, held in Durham, UK, in June/July 2008. The 30 revised full papers presented together with 3 invited paper were carefully reviewed and selected from 76 submissions. The papers feature original results on all aspects of graph-theoretic concepts in Computer Science, e.g. structural graph theory, sequential, parallel, and distributed graph and network algorithms and their complexity, graph grammars and graph rewriting systems, graph-based modeling, graph-drawing and layout, diagram methods, and support of these concepts by suitable implementations.

Graph Based Representations in Pattern Recognition

Graph Polynomials, Principal Pivoting, and Maximum Independent Sets⋆
Roland Glantz and Marcello Pelillo Dipartimento di Informatica Universit`a Ca'
Foscari di Venezia Via Torino 155, 30172 Venezia Mestre, Italy { } Abstract.

Author : Edwin Hancock

Release : 2003-06-18

Publisher : Springer Science & Business Media

ISBN : 354040452X

File Size : 33.42 MB

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This volume contains the papers presented at the Fourth IAPR Workshop on Graph Based Representations in Pattern Recognition. The workshop was held at the King’s Manor in York, England between 30 June and 2nd July 2003. The previous workshops in the series were held in Lyon, France (1997), Haindorf, Austria (1999), and Ischia, Italy (2001). The city of York provided an interesting venue for the meeting. It has been said that the history of York is the history of England. There have been both Roman and Viking episodes. For instance, Constantine was proclaimed emperor in York. The city has also been a major seat of ecclesiastical power and was also involved in the development of the railways in the nineteenth century. Much of York’s history is evidenced by its buildings, and the King’s Manor is one of the most important and attractive of these. Originally part of the Abbey, after the dissolution of the monasteries by Henry VIII, the building became a center of government for the Tudors and the Stuarts (who stayed here regularly on their journeys between London and Edinburgh), serving as the headquarters of the Council of the North until it was disbanded in 1561. The building became part of the University of York at its foundation in 1963. The papers in the workshop span the topics of representation, segmentation, graph-matching, graph edit-distance, matrix and spectral methods, and gra- clustering.

Logical Approaches to Computational Barriers

From a Zoo to a Zoology: Descriptive Complexity for Graph Polynomials J.A.
Makowsky Department of Computer Science ... We outline a general theory of
graph polynomials which covers all the examples we found in the vast literature,

Author : Arnold Beckmann

Release : 2006-06-26

Publisher : Springer Science & Business Media

ISBN : 3540354662

File Size : 57.50 MB

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This book constitutes the refereed proceedings of the Second International Conference on Computability in Europe, CiE 2006, held in Swansea, UK, in June/July 2006. The 31 revised full papers presented together with 30 invited papers were carefully reviewed and selected from about 80 submissions. Among them are papers corresponding to 8 plenary talks and papers of 6 special sessions entitled proofs and computation, computable analysis, challenges in complexity, foundations of programming, mathematical models of computers and hypercomputers, and G??del centenary: G??del's legacy for computability.

Combinatorics and Graphs

The independence polynomial of G is the polynomial I(G, x) = ∑ nk=0 skxk,
where sk is the number of independent sets of G of size k and s0 = 1. It was
proved that all roots of the independence polynomial of a claw-free graph are

Author : Richard A. Brualdi

Release : 2010

Publisher : American Mathematical Soc.

ISBN : 0821848658

File Size : 21.66 MB

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This volume contains a collection of papers presented at the international conference IPM 20--Combinatorics 2009, which was held at the Institute for Research in Fundamental Sciences in Tehran, Iran, May 15-21, 2009. The conference celebrated IPM's 20th anniversary and was dedicated to Reza Khosrovshahi, one of the founders of IPM and the director of its School of Mathematics from 1996 to 2007, on the occasion of his 70th birthday. The conference attracted an international group of distinguished researchers from many different parts of combinatorics and graph theory, including permutations, designs, graph minors, graph coloring, graph eigenvalues, distance regular graphs and association schemes, hypergraphs, and arrangements.

Orthogonal Polynomials

Note that then the vi ( 2 ) satisfy the following three - term recurrence relation (
hence they are orthogonal polynomials ) : zvi ... Conversely , it is easily seen that
from a distanceregular graph , a P - polynomial association scheme is
canonically ...

Author : Paul Nevai

Release : 1989-12-31

Publisher : Springer Science & Business Media

ISBN : 9780792305699

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This volume contains the Proceedings of the NATO Advanced Study Institute on "Orthogonal Polynomials and Their Applications" held at The Ohio State University in Columbus, Ohio, U.S.A. between May 22,1989 and June 3,1989. The Advanced Study Institute primarily concentrated on those aspects of the theory and practice of orthogonal polynomials which surfaced in the past decade when the theory of orthogonal polynomials started to experience an unparalleled growth. This progress started with Richard Askey's Regional Confer ence Lectures on "Orthogonal Polynomials and Special Functions" in 1975, and subsequent discoveries led to a substantial revaluation of one's perceptions as to the nature of orthogonal polynomials and their applicability. The recent popularity of orthogonal polynomials is only partially due to Louis de Branges's solution of the Bieberbach conjecture which uses an inequality of Askey and Gasper on Jacobi polynomials. The main reason lies in their wide applicability in areas such as Pade approximations, continued fractions, Tauberian theorems, numerical analysis, probability theory, mathematical statistics, scattering theory, nuclear physics, solid state physics, digital signal processing, electrical engineering, theoretical chemistry and so forth. This was emphasized and convincingly demonstrated during the presentations by both the principal speakers and the invited special lecturers. The main subjects of our Advanced Study Institute included complex orthogonal polynomials, signal processing, the recursion method, combinatorial interpretations of orthogonal polynomials, computational problems, potential theory, Pade approximations, Julia sets, special functions, quantum groups, weighted approximations, orthogonal polynomials associated with root systems, matrix orthogonal polynomials, operator theory and group representations.

Periodic Nanostructures

It is well-known that a graph can be described by: a connection table, a sequence
of numbers, a matrix, a polynomial or a derived number (called a topological
index). In Quantum Chemistry, the early H ̋uckel theory calculated the levels of ...

Author : Mircea V. Diudea

Release : 2007-08-23

Publisher : Springer Science & Business Media

ISBN : 1402060203

File Size : 69.81 MB

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These tiny structures could offer architectural designs for the cities of the future. The authors explore the foam-like carbon structures, which relate to ‘schwarzites’ and which are infinite periodic minimal surfaces of negative curvature. They show that the periodicity of close repeat units of such structures is evident not only in these formations but also in all of the carbon allotropes. The text provides literature and data on the field of nanostructure periodicity and the authors’ own results on nanostructure building and energy calculations.

Language and Automata Theory and Applications

A Graph Polynomial Approach to Primitivity⋆ Francine Blanchet-Sadri1, Michelle
Bodnar2, Nathan Fox3, and Joe ... from the power of other graph polynomials that
appear in the literature such as the matching polynomial, the Tutte polynomial, ...

Author : Adrian-Horia Dediu

Release : 2013-03-15

Publisher : Springer

ISBN : 3642370640

File Size : 50.67 MB

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This book constitutes the refereed proceedings of the 7th International Conference on Language and Automata Theory and Applications, LATA 2013, held in Bilbao, Spain in April 2013. The 45 revised full papers presented together with 5 invited talks were carefully reviewed and selected from 97 initial submissions. The volume features contributions from both classical theory fields and application areas (bioinformatics, systems biology, language technology, artificial intelligence, etc.). Among the topics covered are algebraic language theory; algorithms for semi-structured data mining; algorithms on automata and words; automata and logic; automata for system analysis and program verification; automata, concurrency and Petri nets; automatic structures; cellular automata; combinatorics on words; computability; computational complexity; computational linguistics; data and image compression; decidability questions on words and languages; descriptional complexity; DNA and other models of bio-inspired computing; document engineering; foundations of finite state technology; foundations of XML; fuzzy and rough languages; grammars (Chomsky hierarchy, contextual, multidimensional, unification, categorial, etc.); grammars and automata architectures; grammatical inference and algorithmic learning; graphs and graph transformation; language varieties and semigroups; language-based cryptography; language-theoretic foundations of artificial intelligence and artificial life; parallel and regulated rewriting; parsing; pattern recognition; patterns and codes; power series; quantum, chemical and optical computing; semantics; string and combinatorial issues in computational biology and bioinformatics; string processing algorithms; symbolic dynamics; symbolic neural networks; term rewriting; transducers; trees, tree languages and tree automata; weighted automata.

Mathematical Tools for Physicists

t=1 (4.51) The real connection with the theory of graphs comes from the two terms
UI and F, which are graph polynomials, known as the first and second Symanzik
polynomials (sometimes called Kirchhoff–Symanzik polynomials). We will now ...

Author : Michael Grinfeld

Release : 2014-11-05

Publisher : John Wiley & Sons

ISBN : 3527684271

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The new edition is significantly updated and expanded. This unique collection of review articles, ranging from fundamental concepts up to latest applications, contains individual contributions written by renowned experts in the relevant fields. Much attention is paid to ensuring fast access to the information, with each carefully reviewed article featuring cross-referencing, references to the most relevant publications in the field, and suggestions for further reading, both introductory as well as more specialized. While the chapters on group theory, integral transforms, Monte Carlo methods, numerical analysis, perturbation theory, and special functions are thoroughly rewritten, completely new content includes sections on commutative algebra, computational algebraic topology, differential geometry, dynamical systems, functional analysis, graph and network theory, PDEs of mathematical physics, probability theory, stochastic differential equations, and variational methods.

Graph Theory, Coding Theory and Block Designs

The actual polynomials are, apart from trivial modifications, the Krawtchouk and
Eberlein polynomials respectively for the two schemes. ) Note that the P-
polynomial condition depends on the ordering of the graphs I'i, while the Q-
polynomial ...

Author : P. J. Cameron

Release : 1975-09-18

Publisher : Cambridge University Press

ISBN : 0521207428

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These are notes deriving from lecture courses on the theory of t-designs and graph theory given by the authors in 1973 at Westfield College, London.

Modern Graph Theory

So far we have encountered several polynomials associated with a graph,
including the chromatic polynomial, the characteristic polynomial and the minimal
polynomial. Our aim in this chapter is to study a polynomial that gives us much
more ...

Author : Bela Bollobas

Release : 2013-12-01

Publisher : Springer Science & Business Media

ISBN : 1461206197

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An in-depth account of graph theory, written for serious students of mathematics and computer science. It reflects the current state of the subject and emphasises connections with other branches of pure mathematics. Recognising that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavour of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory, the book presents a detailed account of newer topics, including Szemerédis Regularity Lemma and its use, Shelahs extension of the Hales-Jewett Theorem, the precise nature of the phase transition in a random graph process, the connection between electrical networks and random walks on graphs, and the Tutte polynomial and its cousins in knot theory. Moreover, the book contains over 600 well thought-out exercises: although some are straightforward, most are substantial, and some will stretch even the most able reader.

Graph Theory

North-Holland Publishing Company LINE GRAPHS AND THEIR CHROMATIC
POLYNOMIALS RUTH A. BARI George ... a connected (p,q)-graph with q > O ,
and let L(G) be the line graph of G . Denote the chromatic polynomial of G by P(G,
X), ...

Author : B. Bollobás

Release : 1982-01-01

Publisher : Elsevier

ISBN : 9780080871738

File Size : 25.63 MB

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The Cambridge Graph Theory Conference, held at Trinity College from 11 to 13 March 1981, brought together top ranking workers from diverse areas of the subject. The papers presented were by invitation only. This volume contains most of the contniutions, suitably refereed and revised. For many years now, graph theory has been developing at a great pace and in many directions. In order to emphasize the variety of questions and to preserve the freshness of research, the theme of the meeting was not restricted. Consequently, the papers in this volume deal with many aspects of graph theory, including colouring, connectivity, cycles, Ramsey theory, random graphs, flows, simplicial decompositions and directed graphs. A number of other papers are concerned with related areas, including hypergraphs, designs, algorithms, games and social models. This wealth of topics should enhance the attractiveness of the volume.

Graph-Based Representations in Pattern Recognition

Characteristic Polynomial Analysis on Matrix Representations of Graphs Peng
Ren, Richard C. Wilson, and Edwin R. ... They have a similar form and are
established from a transformed graph, i.e. the oriented line "aah of the original

Author : Andrea Torsello

Release : 2009-07-09

Publisher : Springer Science & Business Media

ISBN : 3642021247

File Size : 87.73 MB

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This book constitutes the refereed proceedings of the 7th IAPR-TC-15 International Workshop on Graph-Based Representations in Pattern Recognition, GbRPR 2009, held in Venice, Italy in May 2009. The 37 revised full papers presented were carefully reviewed and selected from 47 submissions. The papers are organized in topical sections on graph-based representation and recognition, graph matching, graph clustering and classification, pyramids, combinatorial maps, and homologies, as well as graph-based segmentation.

Graph Theory and Applications

Let P(G, Å) be the chromatic polynomial of the graph G. A graph G is uniquely m-
colorable if and only if p(G, m) = m!. Two graphs are said to be chromatically
equivalent if their chromatic polynomials are equal. In [3], Chao and Whitehead ...

Author : J. Akiyama

Release : 1988-01-01

Publisher : Elsevier

ISBN : 9780080867786

File Size : 42.56 MB

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Graph Theory and Applications

Graph Spectra for Complex Networks

9 Polynomials with real coefficients The characteristic polynomial of real matrices
possesses real coefficients. This chapter aims to summarize general results on
the location and determination of the zeros of polynomials with mainly real ...

Author : Piet van Mieghem

Release : 2010-12-02

Publisher : Cambridge University Press

ISBN : 1139492276

File Size : 88.1 MB

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Analyzing the behavior of complex networks is an important element in the design of new man-made structures such as communication systems and biologically engineered molecules. Because any complex network can be represented by a graph, and therefore in turn by a matrix, graph theory has become a powerful tool in the investigation of network performance. This self-contained 2010 book provides a concise introduction to the theory of graph spectra and its applications to the study of complex networks. Covering a range of types of graphs and topics important to the analysis of complex systems, this guide provides the mathematical foundation needed to understand and apply spectral insight to real-world systems. In particular, the general properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks. An ideal resource for researchers and students in communications networking as well as in physics and mathematics.