Inequalities for monotonic arrangements of eigenvalues

Monotonic Arrangements of Undirected Graphs

The problem of finding a path in a network each edge of which is at least as long as the previous edge has attracted some attention in recent years. For example, problem 10 of the 2008 American Invitational Mathematics Exam asked a variation of this problem, namely to find the number of paths of maximal length in the 4× 4 rectangular grid of dots such that the edge length strictly increases fro...

New inequalities for subspace arrangements

For each positive integer n ≥ 4, we give an inequality satisfied by rank functions of arrangements of n subspaces. When n = 4 we recover Ingleton’s inequality; for higher n the inequalities are all new. These inequalities can be thought of as a hierarchy of necessary conditions for a (poly)matroid to be realizable. Some related open questions about the “cone of realizable polymatroids” are also...

On Some Covariance Inequalities for Monotonic and Non-monotonic Functions

Chebyshev’s integral inequality, also known as the covariance inequality, is an important problem in economics, finance, and decision making. In this paper we derive some covariance inequalities for monotonic and non-monotonic functions. The results developed in our paper could be useful in many applications in economics, finance, and decision making.

Isoperimetric Inequalities and Eigenvalues

Isoperimetric Inequalities and Eigenvalues

An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the i-th largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the i-th largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically tight. A recent result by Quenell relating the diameter, the second eigenvalue, and the girth of a...