Graph energy and Hadamard's inequality

We use Hadamard's determinantal inequality and its generalization to prove some upper bounds on the energy of a graph in terms of degrees, average 2-degrees and number of common neighbors of its vertices. Also, we prove an inequality relating the energy of a graph and one arbitrary subgraph of it.

A Determinantal Inequality Involving Partial Traces

Let A be a density matrix in . Audenaert [J. Math. Phys. 48(2007) 083507] proved an inequality for Schatten p-norms:where Tr1 and Tr2 stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality

On a determinantal inequality arising from diffusion tensor imaging

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality [Formula: see text] where [Formula: see text] are [Formula: see text] positive semidefinite matrices. We complement his result by proving [Formula: see text] Our proofs feature the fruitful interplay between determinantal inequalities and majorization relations. Some related questions are mentioned.