Calculus without limits # Arboricity of spheres

I explain a bit more why for graphs containing a d dimensional simplex, there are Barycentric refinements for which the arboricity is at least c(d), where c(d) can be obtained from the Perron-Frobenius eigenvector of the Barycentric refinement operator. This number c(d) is very interesting for me because it is the half the mean of the universal Barycentric limit measure in dimension d. In dimension d=1, it is c(1)=1, in dimension 2 it is c(2)=3. In 3 dimensions d=3, it is c(3)=13/2 which means that any sufficiently refined graph of dimension d has arboricity at least 7. In the animation put into the youtube video, I took a rather random 3-dimensional graph and took its second Barycentric refinement, then rotated it. The 3 dimensional simplex has become already a rather large 3-dimensional ball. Because it grows like (d+1)!, the lower dimensional parts (there is a 2 dimensional triangle at the bottom which twice refined produces a 2 dimensional disk). I also mention that I believe it should be relatively easy to get this down to 8 and that bringing it down to 7 (what I believe it to be the case because for manifolds it appears that the Barycentric bound is the actual arboricity), would probably require a harder analysis. I always like to get back to the Barycentric limit story because of its universality (Barycentric central limit theorem), because the tree numbers and forest numbers and tree forest ratio converge in the limit to universal numbers which have a potential theoretical flavor. Here are again the pictures of the potential and integrated density of states in dimension 1 and dimension 2. The expectation in d=1 is 2=2c(1), in dimension d=2 it is 6 =2c(2) =2*3 (“three trees suffice!”).

A little bit out of context, here are the universal Barycentric limiting measures in the case of the connection Laplacian which is the matrix L(x,y) = 1 if x and y intersect and L(x,y)=0 else. The matrix L has the same size than the Hodge Laplacian $(d+d^*)^2$. What is interesting about the potential of the measure here is that it can become negative. In the Kirchhoff Laplacian case it looked to be non-negative (this is not proven). But the amazing thing about the connection Laplacian is that it is invertible.