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Applications of combinatorial matrix theory to Laplacian matrices of graphs

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On the surface, matrix theory and graph theory seem like very different branches of mathematics. However, adjacency, Laplacian, and incidence matrices are commonly used to represent graphs, and many properties of matrices can give us useful information about the structure of graphs. Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs is a compilation of many of the exciting results concerning

Abstract: On the surface, matrix theory and graph theory seem like very different branches of mathematics. However, adjacency, Laplacian, and incidence matrices are commonly used to represent graphs, and many properties of matrices can give us useful information about the structure of graphs. Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs is a compilation of many of the exciting results concerning Laplacian matrices developed since the mid 1970s by well-known mathematicians such as Fallat, Fiedler, Grone, Kirkland, Merris, Mohar, Neumann, Shader, Sunder, and more. The text i.

"Preface On the surface, matrix theory and graph theory are seemingly very different branches of mathematics. However, these two branches of mathematics interact since it is often convenient to represent a graph as a matrix. Adjacency, Laplacian, and incidence matrices are commonly used to represent graphs. In 1973, Fiedler published his first paper on Laplacian matrices of graphs and showed how many properties of the Laplacian matrix, especially the eigenvalues, can give us useful information about the structure of the graph. Since then, many papers have been published on Laplacian matrices. This book is a compilation of many of the exciting results concerning Laplacian matrices that have been developed since the mid 1970's. Papers written by well-known mathematicians such as (alphabetically) Fallat, Fiedler, Grone, Kirkland, Merris, Mohar, Neumann, Shader, Sunder, and several others are consolidated here. Each theorem is referenced to its appropriate paper so that the reader can easily do more in-depth research on any topic of interest. However, the style of presentation in this book is not meant to be that of a journal but rather a reference textbook. Therefore, more examples and more detailed calculations are presented in this book than would be in a journal article. Additionally, most sections are followed by exercises to aid the reader in gaining a deeper understanding of the material. Some exercises are routine calculations that involve applying the theorems presented in the section. Other exercises require a more in-depth analysis of the theorems and require the reader to prove theorems that go beyond what was presented in the section. Many of these exercises are taken from relevant papers and they are referenced accordingly"

Author(s): Molitierno, Jason J
Series: Discrete mathematics and its applications
Publisher: CRC Press
Year: 2012

Language: English
Pages: 423
City: Boca Raton, Fla
Tags: Graph connectivity.;Laplacian matrices.;MATRIXMETHODEN DER GRAPHENTHEORIE + SPEKTRALE GRAPHENTHEORIE;MATRIX METHODS IN GRAPH THEORY + SPECTRAL GRAPH THEORY;MÉTHODES MATRICIELLES EN THÉORIE DES GRAPHES + THÉORIE SPECTRALE DES GRAPHES

Content: Matrix Theory PreliminariesVector Norms, Matrix Norms, and the Spectral Radius of a MatrixLocation of EigenvaluesPerron-Frobenius TheoryM-MatricesDoubly Stochastic MatricesGeneralized InversesGraph Theory PreliminariesIntroduction to GraphsOperations of Graphs and Special Classes of GraphsTreesConnectivity of GraphsDegree Sequences and Maximal GraphsPlanar Graphs and Graphs of Higher GenusIntroduction to Laplacian MatricesMatrix Representations of GraphsThe Matrix Tree TheoremThe Continuous Version of the LaplacianGraph Representations and EnergyLaplacian Matrices and NetworksThe Spectra of Laplacian MatricesThe Spectra of Laplacian Matrices Under Certain Graph OperationsUpper Bounds on the Set of Laplacian EigenvaluesThe Distribution of Eigenvalues Less than One and Greater than OneThe Grone-Merris ConjectureMaximal (Threshold) Graphs and Integer SpectraGraphs with Distinct Integer SpectraThe Algebraic ConnectivityIntroduction to the Algebraic Connectivity of GraphsThe Algebraic Connectivity as a Function of Edge WeightThe Algebraic Connectivity with Regard to Distances and DiametersThe Algebraic Connectivity in Terms of Edge Density and the Isoperimetric NumberThe Algebraic Connectivity of Planar GraphsThe Algebraic Connectivity as a Function Genus k where k is greater than 1The Fiedler Vector and Bottleneck Matrices for TreesThe Characteristic Valuation of VerticesBottleneck Matrices for TreesExcursion: Nonisomorphic Branches in Type I TreesPerturbation Results Applied to Extremizing the Algebraic Connectivity of TreesApplication: Joining Two Trees by an Edge of Infinite WeightThe Characteristic Elements of a TreeThe Spectral Radius of Submatrices of Laplacian Matrices for TreesBottleneck Matrices for GraphsConstructing Bottleneck Matrices for GraphsPerron Components of GraphsMinimizing the Algebraic Connectivity of Graphs with Fixed GirthMaximizing the Algebraic Connectivity of Unicyclic Graphs with Fixed GirthApplication: The Algebraic Connectivity and the Number of Cut VerticesThe Spectral Radius of Submatrices of Laplacian Matrices for GraphsThe Group Inverse of the Laplacian MatrixConstructing the Group Inverse for a Laplacian Matrix of a Weighted TreeThe Zenger Function as a Lower Bound on the Algebraic ConnectivityThe Case of the Zenger Equalling the Algebraic Connectivity in TreesApplication: The Second Derivative of the Algebraic Connectivity as a Function of Edge Weight