A topic of current activity in graph theory is combinatorial curvature.  One would like to define notions of curvature that depend only on the combinatorial structure and not on any imposed metric structure.  My approach to this problem is to exploit the ideas of the first topic above.  A graph carries around with it, its geometric spectrum - this only depends on the combinatorial structure.  To an element of the geometric spectrum corresponds a local embedding of a vertex and its neighbours into Euclidean space as an invariant framework.  This defines a Gauss map from which one can deduce an edge curvature, but also vertex curvature through angle deficiency formulae (an analogue of Gauss curvature).  This work will appear as a chapter of a book on Discrete Curvature edited by Laurent Najman and Pascal Romon. 

By studying the smooth analogue of the equations which determine the geometric spectrum, one sees that the spectral parameter corresponds to mean curvature (Differential Geometry and its Applications 2014).  Constant values of this parameter on a graph lead to the constuction of a new graph invariant which I call the gamma-polynomial.  This will appear in the chapter referred to in the above paragraph, but a preliminary version of this is in Section 8 of an article available on arXiv.  arXiv

Once more with reference to the first topic above, one would like to devise natural step-by-step processes by which a graph evolves.  In joint work with Marius Tiba, we show how by a sequence of edge shuffles which diminishes an energy functional, a graph achieves an almost regular configuration.  A version of this work is currently on arXiv, with a revision underway.  arXiv