arXiv subjects

Spencer Backman

Publications and source records attributed to Spencer Backman.

28 records · Page 2Linked to original sources

arXiv2017

A convolution formula for Tutte polynomials of arithmetic matroids and other combinatorial structures

In this note we generalize the convolution formula for the Tutte polynomial of Kook-Reiner-Stanton and Etienne-Las Vergnas to a more general setting that includes both arithmetic matroids and delta-matroids. As corollaries, we obtain new proofs of two positivity results for pseudo-arithmetic matroids and a combinatorial interpretation of the arithmetic Tutte polynomial at infinitely many points in terms of arithmetic flows and colorings. We also exhibit connections with a decomposition of Dahmen-Micchelli spaces and lattice point counting in zonotopes.

math.CO↗

arXiv2017

Riemann-Roch Theory for Graph Orientations

We develop a new framework for investigating linear equivalence of divisors on graphs using a generalization of Gioan's cycle--cocycle reversal system for partial orientations. An oriented version of Dhar's burning algorithm is introduced and employed in the study of acyclicity for partial orientations. We then show that the Baker--Norine rank of a partially orientable divisor is one less than the minimum number of directed paths which need to be reversed in the generalized cycle--cocycle reversal system to produce an acyclic partial orientation. These results are applied in providing new proofs of the Riemann--Roch theorem for graphs as well as Luo's topological characterization of rank-determining sets. We prove that the max-flow min-cut theorem is equivalent to the Euler characteristic description of orientable divisors and extend this characterization to the setting of partial orientations. Furthermore, we demonstrate that $Pic^{g-1}(G)$ is canonically isomorphic as a $Pic^{0}(G)$-torsor to the equivalence classes of full orientations in the cycle--cocycle reversal system acted on by directed path reversals. Efficient algorithms for computing break divisors and constructing partial orientations are presented.

math.CO↗

arXiv2015

Fourientations and the Tutte Polynomial

A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orientation of the graph, we investigate properties of cuts and cycles in fourientations which give trivariate generating functions that are generalized Tutte polynomial evaluations of the form \[(k+m)^{n-1}(k+l)^gT\left(\frac{αk + βl + m}{k+m},\frac{γk + l + δm}{k+l}\right)\] for $α,γ\in \{0,1,2\}$ and $β, δ\in \{0,1\}$. We introduce an intersection lattice of 64 cut-cycle fourientation classes enumerated by generalized Tutte polynomial evaluations of this form. We prove these enumerations using a single deletion-contraction argument and classify axiomatically the set of fourientation classes to which our deletion-contraction argument applies. This work unifies and extends earlier results for fourientations due to Gessel and Sagan, and results for partial orientations due to the first author, and the second author and David Perkinson, as well as results for total orientations due to many authors. We conclude by describing how these classes of fourientations relate to geometric, combinatorial, and algebraic objects including bigraphical arrangements, cycle-cocycle reversal systems, graphic Lawrence ideals, Riemann-Roch theory for graphs, zonotopal algebras, and the reliability polynomial.

math.CO↗

arXiv2015

Explicit Deformation of Lattice Ideals via Chip Firing Games on Directed Graphs

For a finite index sublattice $L$ of the root lattice $A_{n}$, we construct a deterministic algorithm to deform the lattice ideal $I_L$ to a nearby generic lattice ideal, answering a question posed by Miller and Sturmfels. Our algorithm is based on recent results of Perkinson, Perlman and Wilmes concerning commutative algebraic aspects of chip firing on directed graphs. As an application of our deformation algorithm, we construct a cellular resolution of the lattice ideal $I_L$ by degenerating the Scarf complex of its deformation.

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arXiv2015

Infinite Reduction of Divisors on Metric Graphs

We demonstrate that the greedy algorithm for reduction of divisors on metric graphs need not terminate by modeling the Euclidean algorithm in this context. We observe that any infinite reduction has a well defined limit allowing us to treat the greedy reduction algorithm as a transfinite algorithm and to analyze its running time via ordinal numbers. We provide lower and upper bounds which establish a worst case running time of $ω^{Θ({\rm deg}(D))}$.

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arXiv2015

Transfinite Ford-Fulkerson on a Finite Network

It is well-known that the Ford-Fulkerson algorithm for finding a maximum flow in a network need not terminate if we allow the arc capacities to take irrational values. Every non-terminating example converges to a limit flow, but this limit flow need not be a maximum flow. Hence, one may pass to the limit and begin the algorithm again. In this way, we may view the Ford-Fulkerson algorithm as a transfinite algorithm. We analyze the transfinite running-time of the Ford-Fulkerson algorithm using ordinal numbers, and prove that the worst case running-time is $ω^{Θ(|E|)}$. For the lower bound, we show that we can model the Euclidean algorithm via Ford-Fulkerson on an auxiliary network. By running this example on a pair of incommensurable numbers, we obtain a new robust non-terminating example. We then describe how to glue $k$ copies of our Euclidean example in parallel to obtain running-time $ω^k$. An upper bound of $ω^{|E|}$ is established via induction on $|E|$. We conclude by illustrating a close connection to transfinite chip-firing as previously investigated by the first author.

math.CO↗

arXiv2015

Partial Graph Orientations and the Tutte Polynomial

Gessel and Sagan investigated the Tutte polynomial, $T(x,y)$ using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is $2^gT(3,1/2)$. We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is $2^{n-1}T(1/2,3)$. We then prove that the number of partial orientations modulo cycle reversals is $2^gT(3,1)$ and the number of partial orientations modulo cut reversals is $2^{n-1}T(1,3)$. To prove these results, we introduce cut and cycle minimal partial orientations which provide distinguished representatives for partial orientations modulo cut and cycle reversals. These extend classes of total orientations introduced by Gioan, and Greene and Zaslavksy, and we highlight a close connection with graphic and cographic Lawrence ideals. We conclude with edge chromatic generalizations of the quantities presented, which allow for a new interpretation of the reliability polynomial for all probabilities, $p$ with $0 < p <1/2$.

math.CO↗

arXiv2012

A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees

Hereditary chip-firing models generalize the Abelian sandpile model and the cluster firing model to an exponential family of games induced by covers of the vertex set. This generalization retains some desirable properties, e.g. stabilization is independent of firings chosen and each chip-firing equivalence class contains a unique recurrent configuration. In this paper we present an explicit bijection between the recurrent configurations of a hereditary chip-firing model on a graph and its spanning trees.

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arXiv2011

Chip-Firing and Riemann-Roch Theory for Directed Graphs

We investigate Riemann-Roch theory for directed graphs. The Riemann-Roch criteria of Amini and Manjunath is generalized to all integer lattices orthogonal to some positive vector. Using generalized notions of a $v_0$-reduced divisor and Dhar's algorithm we investigate two chip-firing games coming from the rows and columns of the Laplacian of a strongly connected directed graph. We discuss how the "column" chip-firing game is related to directed $\vec{G}$-parking functions and the "row" chip-firing game is related to the sandpile model. We conclude with a discussion of arithmetical graphs, which after a simple transformation may be viewed as a special class of directed graphs which will always have the Riemann-Roch property for the column chip-firing game. Examples of arithmetical graphs are provided which demonstrate that either, both, or neither of the two Riemann-Roch conditions may be satisfied for the row chip-firing game.

math.CO↗

arXiv2009

Sum-product inequalities with perturbation

Suppose that A is a set of n real numbers, each at least 1 apart. Define the ``perturbed sum and product sets'' S and P to be the sums a + b + f(a,b) and products (a+g(a,b))(b+h(a,b)), where f, g, and h satisfy certain upper bounds in terms of the n, |a| and |b|. We develop almost best possible lower bounds on |P| + |S|, using the largest possible sizes of the ``perturbation parameters'' f(a,b), g(a,b) and h(a,b). Our proof uses Elekes's idea for bounding |A+A|+|A.A| from below, in combination with the Szemeredi-Trotter curve theorem (actually, a minor generalization of it) of Szekely, applied to certain polygonal arcs.

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