2,285 research outputs found
Beyond graph energy: norms of graphs and matrices
In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia
The algorithm by Ferson et al. is surprisingly fast: An NP-hard optimization problem solvable in almost linear time with high probability
We start with the algorithm of Ferson et al. (\emph{Reliable computing} {\bf
11}(3), p.~207--233, 2005), designed for solving a certain NP-hard problem
motivated by robust statistics.
First, we propose an efficient implementation of the algorithm and improve
its complexity bound to , where is the
clique number in a certain intersection graph. Then we treat input data as
random variables (as it is usual in statistics) and introduce a natural
probabilistic data generating model. On average, we get and . This results in
average computing time for arbitrarily
small, which may be considered as ``surprisingly good'' average time complexity
for solving an NP-hard problem. Moreover, we prove the following tail bound on
the distribution of computation time: ``hard'' instances, forcing the algorithm
to compute in time , occur rarely, with probability tending to
zero faster than exponentially with
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