37 research outputs found
Inequalities for C-S seminorms and Lieb functions
AbstractLet Mn be the space of n Γ n complex matrices. A seminorm β Β· β on Mn is said to be a C-S seminorm if βA*Aβ = βAA*β for all A β Mn and βAββ€βBβ whenever A, B, and B-A are positive semidefinite. If β Β· β is any nontrivial C-S seminorm on Mn, we show that ββ£Aββ£ is a unitarily invariant norm on Mn, which permits many known inequalities for unitarily invariant norms to be generalized to the setting of C-S seminorms. We prove a new inequality for C-S seminorms that includes as special cases inequalities of Bhatia et al., for unitarily invariant norms. Finally, we observe that every C-S seminorm belongs to the larger class of Lieb functions, and we prove some new inequalities for this larger class
Graphs with many independent vertex cuts
The cycles are the only -connected graphs in which any two nonadjacent
vertices form a vertex cut. We generalize this fact by proving that for every
integer there exists a unique graph satisfying the following
conditions: (1) is -connected; (2) the independence number of is
greater than (3) any independent set of cardinality is a vertex cut of
The edge version of this result does not hold. We also consider the
problem when replacing independent sets by the periphery