Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix �� which is in a prescribed way defined for any graph. This theory is called �� -theory. We outline a spectral theory of graphs based on the signless Laplacians �� and compare it with other spectral theories, in particular with those based on the adjacency matrix �� and the Laplacian �� . The �� -theory can be composed using various connections to other theories: equivalency with �� -theory and �� -theory for regular graphs, or with �� -theory for bipartite graphs, general analogies with �� -theory and analogies with �� -theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.

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Towards a spectral theory of graphs based on the signless Laplacian, I

In order to store a graph or digraph in a computer, we need something other than the diagram or the formal definition. This something is the adjacency matrix, a matrix of O’s and l’s. The l’s correspond to the arcs of the digraph. Certain matrix operations will be seen to correspond to digraph concepts.

Graphs and Matrices

Combinatorial Matrix Theory

This chapter on chemical graph theory forms part of the natural science and processes section of the handbook

Chemical Graph Theory

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum It is known that η(G) ≤ n − 2 if G is a simple graph on n vertices and G is notisomorphic to nK1 In this paper, we characterize the extremal graphs attaining the upper bound n − 2 and the second upper bound n − 3 The maximum nullity of simple graphs with n vertices and e edges, M(n, e), is also discussed We obtain an upper bound of M(n, e), and characterize n and e for which the upper bound is achieved

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On the nullity of graphs

On the nullity of unicyclic graphs

Combinatorics and Matrix Theory have a symbiotic, or mutually beneficial, relationship. This relationship is discussed in my paper The symbiotic relationship of combinatorics and matrix theoryl where I attempted to justify this description. One could say that a more detailed justification was given in my book with H. J. Ryser entitled Combinatorial Matrix Theon? where an attempt was made to give a broad picture of the use of combinatorial ideas in matrix theory and the use of matrix theory in proving theorems which, at least on the surface, are combinatorial in nature. In the book by Liu and Lai, this picture is enlarged and expanded to include recent developments and contributions of Chinese mathematicians, many of which have not been readily available to those of us who are unfamiliar with Chinese journals. Necessarily, there is some overlap with the book Combinatorial Matrix Theory. Some of the additional topics include: spectra of graphs, eulerian graph problems, Shannon capacity, generalized inverses of Boolean matrices, matrix rearrangements, and matrix completions. A topic to which many Chinese mathematicians have made substantial contributions is the combinatorial analysis of powers of nonnegative matrices, and a large chapter is devoted to this topic. This book should be a valuable resource for mathematicians working in the area of combinatorial matrix theory. Richard A. Brualdi University of Wisconsin - Madison 1 Linear Alg. Applies., vols. 162-4, 1992, 65-105 2Camhridge University Press, 1991.

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Matrices in Combinatorics and Graph Theory

Let G be a simple graph of order n ,a nd letμ1 ≥ μ2 ≥ · ·· ≥ μn =0b e the Laplacian spectrum of G. The Laplacian-energy-like invariant of G, LEL for short, is defined as LEL(G )= � n−1 i=1 √ μi. In this paper, we survey results on LEL, with

Bolian Liu

Papers

On the nullity of graphs

On the nullity of unicyclic graphs

Matrices in Combinatorics and Graph Theory

A Survey on the Laplacian-Energy-Like Invariant ∗

The signless Laplacian spread