Edge Expansion and Spectral Gap of Nonnegative Matrices

The classic graphical Cheeger inequalities state that if M is an n x n symmetric doubly stochastic matrix, then 1-lambda(2)(M)/2 <= phi(M) <= root 2.(1-lambda(2)(M)) where phi(M) = min(S subset of[n],vertical bar S vertical bar <= n/2) (1/vertical bar S vertical bar Sigma(i is an element of S, j is not an element of S) M-i,M-j) the edge expansion of M, and lambda(2) (M) is the second largest eigenvalue of M. We study the relationship between phi(A) and the spectral gap 1 - Re lambda(2) (A) for any doubly stochastic matrix A (not necessarily symmetric), where lambda(2) (A) is a nontrivial eigenvalue of A with maximum real part. Fiedler showed that the upper bound on phi(A) is unaffected, i.e., phi(A) <= root 2. (1-Re lambda(2)(A)). With regards to the lower bound on phi(A), there are known constructions with phi(A) is an element of Theta (1-Re lambda(2)(A)/log n), indicating that at least a mild dependence on n is necessary to lower bound phi(A). In our first result, we provide an exponentially better construction of n x n doubly stochastic matrices A(n), for which phi(A(n)) <= 1-Re lambda(2)(An)/root n. In fact, all nontrivial eigenvalues of our matrices are 0, even though the matrices are highly nonexpanding. We further show that this bound is in the correct range (up to the exponent of n), by showing that for any doubly stochastic matrix A, phi(A) >= 1-Re lambda(2)(A)/35.n. As a consequence, unlike the symmetric case, there is a (necessary) loss of a factor of n(alpha) for 1/2 <= alpha <= 1 in lower bounding phi by the spectral gap in the nonsymmetric setting. Our second result extends these bounds to general matrices R with nonnegative entries, to obtain a two-sided gapped refinement of the Perron-Frobenius theorem. Recall from the Perron-Frobenius theorem that for such R, there is a nonnegative eigenvalue r such that all eigenvalues of R lie within the closed disk of radius r about 0. Further, if R is irreducible, which means phi(R) > 0 (for suitably defined phi), then r is positive and all other eigenvalues lie within the open disk, so (with eigenvalues sorted by real part), Re lambda(2) (R) < r. An extension of Fiedler's result provides an upper bound and our result provides the corresponding lower bound on phi(R) in terms of r - Re lambda(2) (R), obtaining a two-sided quantitative version of the Perron-Frobenius theorem.