Riemann-Roch and Abel-Jacobi theory on a finite graph

It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.Comment: 35 pages. v3: Several minor changes made, mostly fixing typographical errors. This is the final version, to appear in Adv. Mat

Rank of divisors on tropical curves

We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, we confirm a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph, and construct an algorithm for computing the rank of a divisor on a tropical curve

Family-based Social Activism: Rethinking the Social Role of Families

This paper sheds light on family-based social activism, defined herein as the broad range of activities undertaken by family members to raise awareness and bring about policy reform about a serious condition or tragic event that affects or affected someone in their family. These activities constitute social activism because they are intended to foster social change. The activism is ‘family-based’ because the social change advocate is a family member responding to the serious or traumatic situation that affects or affected a family member. Mothers Against Drunk Driving, the Westray Families Group, and Family and Friends of Schizophrenics are just a few examples of the many family-based social activist groups which have formed in Canada. Family-based activism is not addressed as a specific subject of study in the sociology literature on families, social movements, social problems, or voluntary association membership. The first goal of this paper is to bring family-based social activism into focus and provide some empirical evidence of this phenomenon in Canada. The second goal is to situate family activism in the sociology literature by drawing theoretical inspirations from social memory literature and suggesting avenues for theoretical development in perspectives on families

K_6 minors in 6-connected graphs of bounded tree-width

We prove that every sufficiently big 6-connected graph of bounded tree-width either has a K_6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently big 6-connected graphs. Jorgensen conjectured that it holds for all 6-connected graphs.Comment: 33 pages, 8 figure