638,998 research outputs found
Condition numbers and scale free graphs
In this work we study the condition number of the least square matrix
corresponding to scale free networks. We compute a theoretical lower bound of
the condition number which proves that they are ill conditioned. Also, we
analyze several matrices from networks generated with the linear preferential
attachment model showing that it is very difficult to compute the power law
exponent by the least square method due to the severe lost of accuracy expected
from the corresponding condition numbers.Comment: Submitted to EP
Some families of density matrices for which separability is easily tested
We reconsider density matrices of graphs as defined in [quant-ph/0406165].
The density matrix of a graph is the combinatorial laplacian of the graph
normalized to have unit trace. We describe a simple combinatorial condition
(the "degree condition") to test separability of density matrices of graphs.
The condition is directly related to the PPT-criterion. We prove that the
degree condition is necessary for separability and we conjecture that it is
also sufficient. We prove special cases of the conjecture involving nearest
point graphs and perfect matchings. We observe that the degree condition
appears to have value beyond density matrices of graphs. In fact, we point out
that circulant density matrices and other matrices constructed from groups
always satisfy the condition and indeed are separable with respect to any
split. The paper isolates a number of problems and delineates further
generalizations.Comment: 14 pages, 4 figure
Pseudo orbit expansion for the resonance condition on quantum graphs and the resonance asymptotics
In this note we explain the method how to find the resonance condition on
quantum graphs, which is called pseudo orbit expansion. In three examples with
standard coupling we show in detail how to obtain the resonance condition. We
focus on non-Weyl graphs, i.e. the graphs which have fewer resonances than
expected. For these graphs we explain benefits of the method of "deleting
edges" for simplifying the graph.Comment: 11 pages, 8 figure
On highly regular strongly regular graphs
In this paper we unify several existing regularity conditions for graphs,
including strong regularity, -isoregularity, and the -vertex condition.
We develop an algebraic composition/decomposition theory of regularity
conditions. Using our theoretical results we show that a family of non rank 3
graphs known to satisfy the -vertex condition fulfills an even stronger
condition, -regularity (the notion is defined in the text). Derived from
this family we obtain a new infinite family of non rank strongly regular
graphs satisfying the -vertex condition. This strengthens and generalizes
previous results by Reichard.Comment: 29 page
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