- Combinatorics and discrete mathematics.
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