ABSTRACT
Historically, matrix theory and combinatorics have enjoyed a powerful, mutually beneficial relationship. Examples include:
240Perron–Frobenius theory describes the relationship between the combinatorial arrangement of the entries of a nonnegative matrix and the properties of its eigenvalues and eigenvectors (see [53, Chapter 8]).
The theory of vibrations (e.g., of a system of masses connected by strings) provides many inverse problems (e.g., can the stiffness of the springs be prescribed to achieve a system with a given set of fundamental vibrations?) whose resolution intimately depends upon the families of matrices with a common graph (see [46, Chapter 7]).
