The Jacobian of a finite graph is a finite abelian group whose cardinality is equal to the number of spanning trees of . In this earlier post, I discussed a family of combinatorial bijections between elements of and the set of spanning trees of . I also discussed the fact that for plane graphs, these Bernardi bijections come from a natural simply transitive action of on (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 on 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 as an intermediate object rather than . The latter is canonically isomorphic (as a -torsor) to the set of break divisors on ; the former is isomorphic to the circuit-cocircuit reversal system, which we now introduce.
In this post I’ll talk about another favorite recreational math puzzle, the (in)famous “Pentagon Problem”. First, though, I wanted to provide a solution to the Ghost Bugs problem from my last blog post. The puzzle is the following:
You are given four lines in a plane in general position (no two parallel, no three intersecting in a common point). On each line a ghost bug crawls at some constant velocity (possibly different for each bug). Being ghosts, if two bugs happen to cross paths they just continue crawling through each other uninterrupted. Suppose that five of the possible six meetings actually happen. Prove that the sixth does as well.
Here is the promised solution. The idea (like in Einstein’s theory of general relativity) is to add an extra dimension corresponding to time. We thus lift the problem out of the page and replace the four lines by the graph of the bugs’ positions as a function of time. Since each bug travels at a constant speed, each of the four resulting graphs is a straight line. By construction, two lines and intersect if and only if the corresponding bugs cross paths.
Suppose that every pair of bugs cross paths except possibly for bugs 3 and 4. Then the lines each intersect one another (in distinct points) and therefore they lie in a common plane. Since line intersects lines and in distinct points, it must lie in the same plane. The line cannot be parallel to , since their projections to the page (corresponding to forgetting the time dimension) intersect. Thus and must intersect, which means that bugs 3 and 4 do indeed cross paths.
Cool, huh? As I mentioned in my last post, I can still vividly remember how I felt in the AHA! moment when I discovered this solution more than 15 years ago.
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 consists of reversing all the edges in a directed cycle in . Similarly, a cocycle flip consists of reversing all the edges in a directed cocycle in , 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
In an earlier post, I described a graph-theoretic analogue of the Riemann-Roch theorem and some of its applications. In this post, I’d like to discuss a proof of that theorem which is a bit more streamlined than the one which Norine and I gave in our original paper [BN]. Like our original proof, the one we’ll give here is based on the concept of reduced divisors. Continue reading
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 denotes the set of constraints and is the dimension of the space of analytic functions satisfying those constraints, then the Riemann-Roch theorem asserts that
where is the genus (“number of holes”) of the Riemann surface , is the total number of constraints, and is the “canonical divisor” on . 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