4,987 research outputs found
The center of an infinite graph
In this note we extend the notion of the center of a graph to infinite graphs. Thus, a vertex is in the center of the infinite graph G if it is in the center of an increasing family of finite subgraphs covering G. We give different characterizations of when a vertex is in the center of an infinite graph and we prove that any infinite graph with at least two ends has a center
On the deformation chirality of real cubic fourfolds
According to our previous results, the conjugacy class of the involution
induced by the complex conjugation in the homology of a real non-singular cubic
fourfold determines the fourfold up to projective equivalence and deformation.
Here, we show how to eliminate the projective equivalence and to obtain a pure
deformation classification, that is how to respond to the chirality question:
which cubics are not deformation equivalent to their image under a mirror
reflection. We provide an arithmetical criterion of chirality, in terms of the
eigen-sublattices of the complex conjugation involution in homology, and show
how this criterion can be effectively applied taking as examples -cubics
(that is those for which the real locus has the richest topology) and
-cubics (the next case with respect to complexity of the real locus). It
happens that there is one chiral class of -cubics and three chiral classes
of -cubics, contrary to two achiral classes of -cubics and three
achiral classes of -cubics.Comment: 25 pages, 8 figure
Random incidence matrices: moments of the spectral density
We study numerically and analytically the spectrum of incidence matrices of
random labeled graphs on N vertices : any pair of vertices is connected by an
edge with probability p. We give two algorithms to compute the moments of the
eigenvalue distribution as explicit polynomials in N and p. For large N and
fixed p the spectrum contains a large eigenvalue at Np and a semi-circle of
"small" eigenvalues. For large N and fixed average connectivity pN (dilute or
sparse random matrices limit), we show that the spectrum always contains a
discrete component. An anomaly in the spectrum near eigenvalue 0 for
connectivity close to e=2.72... is observed. We develop recursion relations to
compute the moments as explicit polynomials in pN. Their growth is slow enough
so that they determine the spectrum. The extension of our methods to the
Laplacian matrix is given in Appendix.
Keywords: random graphs, random matrices, sparse matrices, incidence matrices
spectrum, momentsComment: 39 pages, 9 figures, Latex2e, [v2: ref. added, Sect. 4 modified
On the bordification of outer space
We give a simple construction of an equivariant deformation retract of Outer
space which is homeomorphic to the Bestvina-Feighn bordification. This results
in a much easier proof that the bordification is (2n-5)-connected at infinity,
and hence that is a virtual duality group.Comment: Accepted version, to appear in the Journal of the London MS. Section
7, giving the homeomorphism to the Bestvina-Feighn bordification, has been
substantially revise
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