The famous shortest path tree lemma states that, for any node s in a graph G = (V,E), there is a subgraph on O(n) edges that preserves all distances between node pairs in the set {s}×V . A very basic question in distance sketching research, with applications to other problems in the field, is to categorize when else graphs admit sparse subgraphs that preserve distances between a set P of p node...

Norm preserver maps of Jordan product on the algebra Mn of n×n complex matrices are studied, with respect to various norms. A description of such surjective maps with respect to the Frobenius norm is obtained: Up to a suitable scaling and unitary similarity, they are given by one of the four standard maps (identity, transposition, complex conjugation, and conjugate transposition) on Mn, except ...

Wemake improvements to the current upper bounds on pairwise distance preservers and additive spanners. A distance preserver is a sparse subgraph that exactly preserves all distances in a given pair set P . We show that every undirected unweighted graph has a distance preserver on O(max{n|P |, n|P |}) edges, and we conjecture that O(n|P |+ n) is possible. An additive subset spanner is a sparse s...

In this paper we generalize (finite) block diagonal matrices to infinite dimensions and then by using row stochastic (as a special case), define the relation < bdr on c0, which is said majorization. We also obtain some important properties of Pbdr, set all bounded linear operators T : c0 ? c0; preserve Further, it obtained necessary conditions for operator be preserver majorization bdr. Also...