The Circuit-Cocircuit Reversal System and Torsor Structures on Spanning Trees

The Jacobian of a finite graph $G$ is a finite abelian group whose cardinality is equal to the number of spanning trees of $G$.  In this earlier post, I discussed a family of combinatorial bijections between elements of ${\rm Jac}(G)$ and the set ${\mathcal T}(G)$ of spanning trees of $G$.  I also discussed the fact that for plane graphs, these Bernardi bijections come from a natural simply transitive action of ${\rm Jac}(G)$ on ${\mathcal T}(G)$ (or, more precisely, a natural isomorphism class of such actions).

In the present post, I’d like to discuss a different family of simply transitive actions of ${\rm Jac}(G)$ on ${\mathcal T}(G)$ discovered by myself, Spencer Backman (a former student of mine), and Chi Ho Yuen (a current student of mine).  One virtue of our construction is that it generalizes in a natural way from graphs to regular matroids.  (We will mostly stick to the graphical case in this post, but will insert some comments about extensions to regular and/or oriented matroids.  A regular matroid can be thought of, rather imprecisely, as the smallest natural class of objects which contain graphs and admit a duality theory generalizing duality for planar graphs. Regular matroids are special cases of the more general concept of oriented matroids.)

One of the main new ideas in [BBY] (as we will henceforth refer to our paper) is to use the torsor ${\rm Pic}^{g-1}(G)$ as an intermediate object rather than ${\rm Pic}^{g}(G)$.  The latter is canonically isomorphic (as a ${\rm Jac}(G)$-torsor) to the set of break divisors on $G$; the former is isomorphic to the circuit-cocircuit reversal system, which we now introduce.

Riemann-Roch theory for graph orientations

In this post, I’d like to sketch some of the interesting results contained in my Ph.D. student Spencer Backman’s new paper “Riemann-Roch theory for graph orientations”.

First, a bit of background.  In a 2007 paper, Emeric Gioan introduced the cycle-cocycle reversal system on a (finite, connected, unoriented) graph G, which is a certain natural equivalence relation on the set of orientations of G.  Recall that an orientation of G is the choice of a direction for each edge.  A cycle flip on an orientation ${\mathcal O}$ consists of reversing all the edges in a directed cycle in ${\mathcal O}$.  Similarly, a cocycle flip consists of reversing all the edges in a directed cocycle in ${\mathcal O}$, where a directed cocycle (also called a directed cut) is the collection of all oriented edges connecting a subset of vertices of G to its complement.  The cycle-cocycle reversal system is the equivalence relation on the set of orientations of G generated by all cycle and cocycle flips.  In his paper, Gioan proves (via a deletion-contraction recursion) the surprising fact that the number of equivalence classes equals the number of spanning trees in G.  A bijective proof of this result was subsequently obtained by Bernardi. Continue reading

Riemann-Roch for Graphs and Applications

I plan to write several posts related to the Riemann-Roch Theorem for Graphs, which was published several years ago in this paper written jointly with Serguei Norine.  In this post I want to explain the statement of the theorem, give some anecdotal background, and mention a few applications which have been discovered in recent years.

The Riemann-Roch Theorem

The (classical) Riemann-Roch Theorem is a very useful result about analytic functions on compact one-dimensional complex manifolds (also known as Riemann surfaces).  Given a set of constraints on the orders of zeros and poles, the Riemann-Roch Theorem computes the dimension of the space of analytic functions satisfying those constraints.  More precisely, if $D$ denotes the set of constraints and $r(D)$ is the dimension of the space of analytic functions satisfying those constraints, then the Riemann-Roch theorem asserts that

$r(D) - r(K-D) = {\rm deg}(D) + 1 - g$

where $g$ is the genus (“number of holes”) of the Riemann surface $X$, ${\rm deg}(D)$ is the total number of constraints, and $K$ is the “canonical divisor” on $X$.  See the Wikipedia page for much more information.

Before formulating the combinatorial analogue of this result which Norine and I discovered, I want to briefly reminisce about how this result came about.  In the summer of 2006, my Georgia Tech REU (Research Experience for Undergraduates) student Dragos Ilas worked on a graph-theoretic conjecture which I had made some time earlier.  Dragos spent eight weeks working on the problem and compiled a lot of experimental evidence toward my conjecture.  He gave a talk about the problem one Friday toward the end of the summer in an REU Mini-Conference that I was organizing at Georgia Tech.  Serguei Norine (then a postdoc working with my colleague Robin Thomas) was in the audience.  On Monday morning, Serguei knocked on my office door and showed me an extremely clever proof of my conjecture.  I told Serguei about my real goal, which was to prove a graph-theoretic analogue of the Riemann-Roch theorem.  I outlined what I had in mind and within a week, we had exactly the kind of Riemann-Roch formula that I had hoped for… thanks in large part to Serguei’s amazing combinatorial mind! Continue reading