Andrew Goodall

The ICTD Government Revenue Dataset

A major obstacle to cross-country research on the role of revenue and taxation in development has been the weakness of available data. This paper presents a new Government Revenue Dataset (GRD), developed through the International Centre for Tax and Development (ICTD). The dataset meticulously combines data from several major international databases, as well as drawing on data compiled from all available International Monetary Fund (IMF) Article IV reports. It achieves marked improvements in data coverage and accuracy, including a standardised approach to revenue from natural resources, and holds the promise of significant improvement in the credibility and robustness of research in this area. This paper sets out the issues with existing sources and explains the process of creating the new dataset, including a discussion of remaining limitations. It then presents data on tax and revenue trends over the past two decades, while a concluding section briefly considers potential strategies for, and barriers to, more effective data collection in future.

Polynomial graph invariants from homomorphism numbers

We give a new method of generating strongly polynomial sequences of graphs, i.e.,?sequences ( H k ) indexed by a tuple k = ( k 1 , ? , k h ) of positive integers, with the property that, for each fixed graph G , there is a multivariate polynomial p ( G ; x 1 , ? , x h ) such that the number of homomorphisms from G to H k is given by the evaluation p ( G ; k 1 , ? , k h ) . A classical example is the sequence of complete graphs ( K k ) , for which p ( G ; x ) is the chromatic polynomial of G . Our construction is based on tree model representations of graphs. It produces a large family of graph polynomials which includes the Tutte polynomial, the Averbouch-Godlin-Makowsky polynomial, and the Tittmann-Averbouch-Makowsky polynomial. We also introduce a new graph parameter, the branching core size of a simple graph, derived from its representation under a particular tree model, and related to how many involutive automorphisms it has. We prove that a countable family of graphs of bounded branching core size is always contained in the union of a finite number of strongly polynomial sequences.

A Tutte polynomial for non-orientable maps

We construct a new polynomial invariant of maps (graphs embedded in closed surfaces, not necessarily orientable). Our invariant is tailored to contain as evaluations the number of local flows and local tensions taking non-identity values in any given finite group. Moreover, it contains as specializations the Krushkal polynomial, the Bollobas-Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial of the under-lying graph of the map.

The tutte polynomial characterizes simple outerplanar graphs

We show that if G is a simple outerplanar graph and H is a graph with the same Tutte polynomial as G, then H is also outerplanar. Examples show that the condition of G being simple cannot be omitted.

Graph homomorphisms, the Tutte polynomial and “q-state Potts uniqueness”

Abstract We establish for which weighted graphs H homomorphism functions from multigraphs G to H are specializations of the Tutte polynomial of G , answering a question of Freedman, Lovasz and Schrijver. We introduce a new property of graphs called “ q -state Potts uniqueness” and relate it to chromatic and Tutte uniqueness, and also to “chromatic–flow uniqueness”, recently studied by Duan, Wu and Yu.

On the Parity of Colourings and Flows

We extend a result of Tarsi and show that the chromatic polynomial and flow polynomial evaluated at 1+k are up to sign the same modulo k2 for any integer k such that |k|?2.

Matroid invariants and counting graph homomorphisms

Abstract The number of homomorphisms from a finite graph F to the complete graph K n is the evaluation of the chromatic polynomial of F at n . Suitably scaled, this is the Tutte polynomial evaluation T ( F ; 1 − n , 0 ) and an invariant of the cycle matroid of F . De la Harpe and Jaeger [8] asked more generally when is it the case that a graph parameter obtained from counting homomorphisms from F to a fixed graph G depends only on the cycle matroid of F . They showed that this is true when G has a generously transitive automorphism group (examples include Cayley graphs on an abelian group, and Kneser graphs). Using tools from multilinear algebra, we prove the converse statement, thus characterizing finite graphs G for which counting homomorphisms to G yields a matroid invariant. We also extend this result to finite weighted graphs G (where to count homomorphisms from F to G includes such problems as counting nowhere-zero flows of F and evaluating the partition function of an interaction model on F ).

On the number of B-flows of a graph

We exhibit explicit constructions of contractors for the graph parameter counting the number of B-flows of a graph, where B is a subset of a finite Abelian group closed under inverses. These constructions are of great interest because of their relevance to the family of B-flow conjectures formulated by Tutte, Fulkerson, Jaeger, and others.

The Tutte polynomial characterizes simple outerplanar graphs

Abstract We show that if G is a simple outerplanar graph and H is a graph with the same Tutte polynomial as G , then H is also outerplanar. Examples show that the condition of G being simple cannot be omitted.