  We study the combinatorial and algebraic properties of Nonnegative Matrices.
Our results are divided into three different categories.
  1. We show a quantitative generalization of the 100 year-old Perron-Frobenius
theorem, a fundamental theorem which has been used within diverse areas of
mathematics. The Perron-Frobenius theorem states that every irreducible
nonnegative matrix $R$ has a largest positive eigenvalue $r$, and every other
eigenvalue $\lambda$ of $R$ is such that $\text{Re}\lambda<r$ and
$|\lambda|\leq r$. We capture the notion of irreducibility through the widely
studied notion of edge expansion $\phi$ of $R$ which intuitively measures how
well-connected the underlying digraph of $R$ is, and show a quantitative
relation between the spectral gap $\Delta=1-\text{Re}\lambda/r$ (where
$\lambda\not=r$ is the eigenvalue of $R$ with the largest real part) and the
edge expansion $\phi$, providing a more general result than the Cheeger-Buser
inequalities, as
follows.\[\dfrac{1}{15}\cdot\dfrac{\Delta(R)}{n}\leq\phi(R)\leq\sqrt{2\cdot\Delta(R)}.\]
  2. We study constructions of specific nonsymmetric matrices (or nonreversible
Markov Chains) that have small edge expansion but large spectral gap, and
provide a novel construction of a nonreversible chain for
which\[\phi(R)\leq\dfrac{\Delta(R)}{\sqrt{n}},\] and we also present a
candidate construction of matrices for
which\[\phi(R)\leq2\dfrac{\Delta(R)}{n},\] which is the most beautiful
contribution of this thesis.
  3. We connect edge expansion and spectral gap to other combinatorial
properties of nonsymmetric matrices, such as mixing time and capacity, and
provide elementary proofs or unified views of known results and new results
relating the different combinatorial/algebraic properties. Notably, we show the
monotonicity of capacity for nonsymmetric nonnegative matrices.
