  We explore relations between various variational problems for graphs like
Euler characteristic chi(G), characteristic length mu(G), mean clustering
nu(G), inductive dimension iota(G), edge density epsilon(G), scale measure
sigma(G), Hilbert action eta(G) and spectral complexity xi(G). A new insight in
this note is that the local cluster coefficient C(x) in a finite simple graph
can be written as a relative characteristic length L(x) of the unit sphere S(x)
within the unit ball B(x) of a vertex. This relation L(x) = 2-C(x) will allow
to study clustering in more general metric spaces like Riemannian manifolds or
fractals. If eta is the average of scalar curvature s(x), a formula mu ~
1+log(epsilon)/log(eta) of Newman, Watts and Strogatz relates mu with the edge
density epsilon and average scalar curvature eta telling that large curvature
correlates with small characteristic length. Experiments show that the
statistical relation mu ~ log(1/nu) holds for random or deterministic
constructed networks, indicating that small clustering is often associated to
large characteristic lengths and lambda=mu/log(nu) can converge in some graph
limits of networks. Mean clustering nu, edge density epsilon and curvature
average eta therefore can relate with characteristic length mu on a statistical
level. We also discovered experimentally that inductive dimension iota and
cluster-length ratio lambda correlate strongly on Erdos-Renyi probability
spaces.
