4,763 research outputs found
Resolving sets for breaking symmetries of graphs
This paper deals with the maximum value of the difference between the
determining number and the metric dimension of a graph as a function of its
order. Our technique requires to use locating-dominating sets, and perform an
independent study on other functions related to these sets. Thus, we obtain
lower and upper bounds on all these functions by means of very diverse tools.
Among them are some adequate constructions of graphs, a variant of a classical
result in graph domination and a polynomial time algorithm that produces both
distinguishing sets and determining sets. Further, we consider specific
families of graphs where the restrictions of these functions can be computed.
To this end, we utilize two well-known objects in graph theory: -dominating
sets and matchings.Comment: 24 pages, 12 figure
Developments on Spectral Characterizations of Graphs
In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs
Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization
Attached to both Yang-Mills and General Relativity about Minkowski spacetime
are distinguished gauge independent objects known as the on-shell tree
scattering amplitudes. We reinterpret and rigorously construct them as
minimal model brackets. This is based on formulating YM and GR as
differential graded Lie algebras. Their minimal model brackets are then given
by a sum of trivalent (cubic) Feynman tree graphs. The amplitudes are gauge
independent when all internal lines are off-shell, not merely up to
isomorphism, and we include a homological algebra proof of this fact. Using the
homological perturbation lemma, we construct homotopies (propagators) that are
optimal in bringing out the factorization of the residues of the amplitudes.
Using a variant of Hartogs extension for singular varieties, we give a rigorous
account of a recursive characterization of the amplitudes via their residues
independent of their original definition in terms of Feynman graphs (this does
neither involve so-called BCFW shifts nor conditions at infinity under such
shifts). Roughly, the amplitude with legs is the unique section of a sheaf
on a variety of complex momenta whose residues along a finite list of
irreducible codimension one subvarieties (prime divisors) factor into
amplitudes with less than legs. The sheaf is a direct sum of rank one
sheaves labeled by helicity signs. To emphasize that amplitudes are robust
objects, we give a succinct list of properties that suffice for a dgLa so as to
produce the YM and GR amplitudes respectively.Comment: 51 page
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