173 research outputs found
Isospectral deformations of the Dirac operator
We give more details about an integrable system in which the Dirac operator
D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a
Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) =
d(t) + b(t) + d(t)^* defines a new exterior derivative d(t) and a new Dirac
operator C(t) = d(t) + d(t)^* and Laplacian M(t) = d(t) d(t)^* + d(t)* d(t) and
so a new distance on G or a new metric on M.Comment: 32 pages, 8 figure
On the Dimension and Euler characteristic of random graphs
The inductive dimension dim(G) of a finite undirected graph G=(V,E) is a
rational number defined inductively as 1 plus the arithmetic mean of the
dimensions of the unit spheres dim(S(x)) at vertices x primed by the
requirement that the empty graph has dimension -1. We look at the distribution
of the random variable "dim" on the Erdos-Renyi probability space G(n,p), where
each of the n(n-1)/2 edges appears independently with probability p. We show
here that the average dimension E[dim] is a computable polynomial of degree
n(n-1)/2 in p. The explicit formulas allow experimentally to explore limiting
laws for the dimension of large graphs. We also study the expectation E[X] of
the Euler characteristic X, considered as a random variable on G(n,p). We look
experimentally at the statistics of curvature K(v) and local dimension dim(v) =
1+dim(S(v)) which satisfy the Gauss-Bonnet formula X(G) = sum K(v) and by
definition dim(G) = sum dim(v)/|V|. We also look at the signature functions
f(p)=E[dim], g(p)=E[X] and matrix values functions A(p) = Cov[{dim(v),dim(w)],
B(p) = Cov[K(v),K(w)] on the probability space G(p) of all subgraphs of a host
graph G=(V,E) with the same vertex set V, where each edge is turned on with
probability p. If G is the complete graph or a union of cyclic graphs with have
explicit formulas for the signature polynomials f and g.Comment: 18 pages, 14 figures, 4 table
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