Notes on the connectivity of Cayley coset digraphs

Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.Comment: 15 page

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