**CBMS Regional Conference Series in Mathematics**

Volume: 115;
2011;
96 pp;
Softcover

MSC: Primary 05;
Secondary 15

Print ISBN: 978-0-8218-5315-3

Product Code: CBMS/115

List Price: $36.00

Individual Price: $28.80

**Electronic ISBN: 978-1-4704-1573-0
Product Code: CBMS/115.E**

List Price: $36.00

Individual Member Price: $28.80

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#### Supplemental Materials

# The Mutually Beneficial Relationship of Graphs and Matrices

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*Richard A. Brualdi*

A co-publication of the AMS and CBMS

Graphs and matrices enjoy a fascinating and mutually beneficial
relationship. This interplay has benefited both graph theory and linear
algebra. In one direction, knowledge about one of the graphs that can
be associated with a matrix can be used to illuminate matrix properties
and to get better information about the matrix. Examples include the
use of digraphs to obtain strong results on diagonal dominance and
eigenvalue inclusion regions and the use of the Rado-Hall theorem to
deduce properties of special classes of matrices. Going the other way,
linear algebraic properties of one of the matrices associated with a
graph can be used to obtain useful combinatorial information about the
graph. The adjacency matrix and the Laplacian matrix are two well-known
matrices associated to a graph, and their eigenvalues encode important
information about the graph. Another important linear algebraic
invariant associated with a graph is the Colin de Verdière number,
which, for instance, characterizes certain topological properties of the
graph.

This book is not a comprehensive study of graphs and matrices. The
particular content of the lectures was chosen for its accessibility,
beauty, and current relevance, and for the possibility of enticing the
audience to want to learn more.

A co-publication of the AMS and CBMS.

#### Readership

Graduate students and research mathematicians interested in graph theory.

#### Reviews & Endorsements

This delightful short book ... could be used as a supplemental course book in an upper level undergraduate course or first year graduate course in graph theory. ... its contents are beautiful, current, relevant, accessible to undergraduates, and have the potential to entice the audience to want to learn more.

-- MAA Reviews

#### Table of Contents

# Table of Contents

## The Mutually Beneficial Relationship of Graphs and Matrices

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface ix10 free
- Some fundamentals 112 free
- Eigenvalues of graphs 1324
- Rado-Hall theorem and applications 2536
- Colin de Verdière number 3344
- Classes of matrices of zeros and ones 3950
- Matrix sign patterns 4960
- Eigenvalue inclusion and diagonal products 5970
- Tournaments 6778
- Two matrix polytopes 7788
- Digraphs and eigenvalues of (0,1)-matrices 8798
- Index 95106 free
- Back Cover Back Cover1110