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 signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

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A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

This chapter represents an overview of research related to a notion of the “rank of a graph" and the dual concept known as the “nullity of a graph," from the point of view of associating a fixed collection of symmetric or Hermitian matrices to a given graph. This topic and related questions have enjoyed a fairly large history within discrete mathematics, and have become very popular among linear algebraists recently, partly based on its connection to certain inverse eigenvalue problems, but also because of the many interesting applications (e.g., to communication complexity in computer science and to control of quantum systems in mathematical physics) and implications to several facets of graph theory. This chapter is divided into eight parts, beginning in Section 1 with what we feel is the standard minimum rank problem concerning symmetric matrices over the real numbers associated with a simple graph. We continue with important variants of the standard minimum rank problem and related parameters, including the concept of minimum rank over other fields (Section 2), the positive semidefinite minimum rank of a graph (Section 3), graph coloring parameters known as zero forcing numbers (Section 4) and the more classical and celebrated parameters due to Y. Colin de Verdière (Section 5). Section 6 contains more advanced topics relevant to the previous five sections, and Section 7 discusses two well-known conjectures related to minimum rank. Whereas the first seven sections concern primarily symmetric matrices and the diagonal of the matrix is free, in Section 8 we discuss minimum rank problems with no symmetry assumption but the diagonal constrained. NB: The topics discussed in this chapter are in an active research area and the facts presented here represent the state of knowledge as of the writing of this chapter in 2012. Disciplines Algebra | Applied Mathematics | Discrete Mathematics and Combinatorics Comments This is an Accepted Manuscript of the book chapter Fallat, Shaun M., and Leslie Hogben. "Minimum Rank, Maximum Nullity, and Zero Forcing Number of Graphs." In Handbook of Linear Algebra, Second Edition, edited by Leslie Hogben. Boca Raton, Florida : CRC Press/Taylor & Francis Group, 2014: 46-1 to 46-36. Posted with permission. This book chapter is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/math_pubs/212 Minimum Rank, Maximum Nullity, and Zero Forcing Number of Graphs Shaun M. Fallat∗ Leslie Hogben† This is the content of Chapter 46 of Handbook of Linear Algebra 2nd ed. copyright 2014 by CRC Press. Reformatted but not updated 5/22/19. This chapter represents an overview of research related to a notion of the “rank of a graph” and the dual concept known as the “nullity of a graph,” from the point of view of associating a fixed collection of symmetric or Hermitian matrices to a given graph. This topic and related questions have enjoyed a fairly large history within discrete mathematics, and have become very popular among linear algebraists recently, partly based on its connection to certain inverse eigenvalue problems, but also because of the many interesting applications (e.g., to communication complexity in computer science and to control of quantum systems in mathematical physics) and implications to several facets of graph theory. This chapter is divided into eight parts, beginning in Section 1 with what we feel is the standard minimum rank problem concerning symmetric matrices over the real numbers associated with a simple graph. We continue with important variants of the standard minimum rank problem and related parameters, including the concept of minimum rank over other fields (Section 2), the positive semidefinite minimum rank of a graph (Section 3), graph coloring parameters known as zero forcing numbers (Section 4) and the more classical and celebrated parameters due to Y. Colin de Verdière (Section 5). Section 6 contains more advanced topics relevant to the previous five sections, and Section 7 discusses two well-known conjectures related to minimum rank. Whereas the first seven sections concern primarily symmetric matrices and the diagonal of the matrix is free, in Section 8 we discuss minimum rank problems with no symmetry assumption but the diagonal constrained. NB: The topics discussed in this chapter are in an active research area and the facts presented here represent the state of knowledge as of the writing of this chapter in 2012. 1 Standard Minimum Rank of Graphs For a given simple graph G, we associate, in a natural way, a collection of matrices whose entries reflect adjacency in G. We are interested in quantities such as the smallest possible rank and the largest possible nullity over all matrices in this collection. Of particular ∗Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada, shaun.fallat@uregina.ca. †Department of Mathematics, Iowa State University, Ames, IA 50011, USA, and American Institute of Mathematics, 600 E. Brokaw Road, San Jose, CA 95112, USA, hogben@aimath.org.

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1218&context=math_pubs

Minimum Rank, Maximum Nullity, and Zero Forcing Number of Graphs

/pdf/on-commutative-matrices-qi61clizb7.pdf

On Commutative Matrices

The minimum rank problem for a (simple) graph $G$ is to determine the smallest possible rank over all real symmetric matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. This paper surveys the many developments on the (standard) minimum rank problem and its variants since the survey paper \cite{FH}. In particular, positive semidefinite minimum rank, zero forcing parameters, and minimum rank problems for patterns are discussed.

Variants on the minimum rank problem: A survey II

In 1979, Ferguson characterized the periodic Jacobi matrices with given eigenvalues and showed how to use the Lanzcos Algorithm to construct each such matrix. This article provides general characterizations and constructions for the complex analogue of periodic Jacobi matrices. As a consequence of the main procedure, we prove that the multiplicity of an eigenvalue of a periodic Jacobi matrix is at most 2.

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The inverse eigenvalue problem for Hermitian matrices whose graphs are cycles

On the spectra of some graphs like weighted rooted trees

The covering number of the elements of a matroid and generalized matrix functions

https://repositorio-aberto.up.pt/bitstream/10216/110839/2/253702.pdf

Rosário Fernandes

Papers

The inverse eigenvalue problem for Hermitian matrices whose graphs are cycles

On the spectra of some graphs like weighted rooted trees

The covering number of the elements of a matroid and generalized matrix functions

Minimal matrices in the Bruhat order for symmetric (0,1)-matrices

On the Bruhat order of labeled graphs