  Graph Laplacians as well as related spectral inequalities and (co-)homology
provide a foray into discrete analogues of Riemannian manifolds, providing a
rich interplay between combinatorics, geometry and theoretical physics. We
apply some of the latest techniques in data science such as supervised and
unsupervised machine-learning and topological data analysis to the Wolfram
database of some 8000 finite graphs in light of studying these correspondences.
Encouragingly, we find that neural classifiers, regressors and networks can
perform, with high efficiently and accuracy, a multitude of tasks ranging from
recognizing graph Ricci-flatness, to predicting the spectral gap, to detecting
the presence of Hamiltonian cycles, etc.
