ABSTRACT
Combinatorial Matrix Theory 38 Combinatorial Matrix Theory Richard A. Brualdi . . . . . . . . . . . . . . . . . . . . . . . 38-1
39 Matrices and Graphs Willem H. Haemers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39-1
40 Digraphs and Matrices Jeffrey L. Stuart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40-1
41 Bipartite Graphs and Matrices Bryan L. Shader . . . . . . . . . . . . . . . . . . . . . . . . 41-1
42 Sign Pattern Matrices Frank J. Hall and Zhongshan Li . . . . . . . . . . . . . . . . . . 42-1
Topics in Combinatorial Matrix Theory 43 Permanents Ian M. Wanless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43-1
44 D-Optimal Matrices Michael G. Neubauer and William Watkins . . . . . . . . . 44-1
45 Tournaments T. S. Michael . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45-1
46 Minimum Rank, Maximum Nullity, and Zero Forcing Number of Graphs Shaun M. Fallat and Leslie Hogben . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46-1
47 Spectral Graph Theory Steve Butler and Fan Chung . . . . . . . . . . . . . . . . . . . . . 47-1
48 Algebraic Connectivity Steve Kirkland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-1
49 Matrix Completion Problems Luz M. DeAlba, Leslie Hogben, and Amy Wangsness Wehe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49-1
38 Combinatorial Matrix Theory Richard A. Brualdi . . . . . . . . . . . . . . . . . . . . . . . 38-1
39 Matrices and Graphs Willem H. Haemers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39-1
40 Digraphs and Matrices Jeffrey L. Stuart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40-1
41 Bipartite Graphs and Matrices Bryan L. Shader . . . . . . . . . . . . . . . . . . . . . . . . 41-1
42 Sign Pattern Matrices Frank J. Hall and Zhongshan Li . . . . . . . . . . . . . . . . . . 42-1
Richard A. Brualdi University of Wisconsin-Madison
38.1 Combinatorial Structure and Invariants . . . . . . . . . . 38-1 38.2 Square Matrices and Strong Combinatorial
The combinatorial structure of a matrix generally refers to the locations of the nonzero entries of a matrix, or it might be used to refer to the locations of the zero entries. To study and take advantage of the combinatorial structure of a matrix, graphs are used as models. Associated with a matrix are several graphs that represent the combinatorial structure of a matrix in various ways. The type of graph (undirected graph, bipartite graph, digraph) used depends on the kind of matrices (symmetric, rectangular, square) being studied [BR91], [Bru92], [BS04]. Conversely, associated with a graph, bipartite graph, or digraph are matrices that allow one to consider it as an algebraic object. These matrices — their algebraic properties — can often be used to obtain combinatorial information about a graph that is not otherwise obtainable. These are two of three general aspects of combinatorial matrix theory. A third aspect concerns intrinsic combinatorial properties of matrices viewed simply as an array of numbers.
