This monograph is based on the ten lectures I gave at Iowa State Uni-
versity in Ames during the week of July 12-16, 2010. The purpose of the
lectures was to show the fascinating and mutually beneficial relationship be-
tween matrices and graphs (the nonzero pattern of a matrix): (i) knowledge
about one of the graphs that can be associated with a matrix is used to il-
luminate matrix properties and to get better information about the matrix,
and (ii) linear algebraic properties of one of the matrices associated with
a graph is used to get useful combinatorial information about the graph.
The lectures were not intended to be comprehensive on any of the topics
treated; they could not have been within the time framework imposed by
ten one-hour lectures. Nor were the lectures intended to cover all instances
in which the interplay between matrices and graphs has turned out to be
useful; again an impossibility within the time framework. 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. It was, of course, influenced by the author’s personal interests and
In this monograph, I have stayed within the context of the lectures, and
have avoided writing a more comprehensive book. In most cases I have not
given original references for results if they are readily available in one or
more books referenced. Just as we did for the lectures, we assume that the
reader is familiar with many of the basic concepts and facts of matrix theory
and graph theory. We define some standard terms but many are presumed
known and can be found in most elementary and advanced books.
I am indebted to Leslie Hogben and Bryan Shader for organizing this
CBMS Regional Conference and for suggesting me as principal lecturer.
They did a superb job, from recruiting a diverse group of participants to
arranging a stimulating and fun daily schedule with afternoon and evening
activities. I would also like to express my gratitude to the participants for
their attention, stimulating questions, and camaraderie. Finally, I want to
thank the Department of Mathematics of Iowa State University for hosting
the conference and the National Science Foundation under grant number
DMS 0938261 for financially supporting it.