Evaluations of topological Tutte polynomials

We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollob\'as and Riordan's ribbon graph polynomial, $R(G)$, and the topological Penrose polynomial, $P(G)$. The general framework provided by $Q(G)$ also leads to several other combinatorial interpretations these polynomials. In particular, we express $P(G)$, $R(G)$, and the Tutte polynomial, $T(G)$, as sums of chromatic polynomials of graphs derived from $G$; show that these polynomials count $k$-valuations of medial graphs; show that $R(G)$ counts edge 3-colourings; and reformulate the Four Colour Theorem in terms of $R(G)$. We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of $P(G)$ and $R(G)$.Comment: V2: major revision, several new results, and improved expositio

Feynman graph polynomials

The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.Comment: 35 pages, references adde

Bipartite partial duals and circuits in medial graphs

It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.Comment: v2: minor changes. To appear in Combinatoric

Irreducibility of the Tutte polynomial of an embedded graph

We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle 2008) employs a general logical framework and leads to an algorithm with running time f(k)*n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor improvements. 44 pages, 14 figure

The politics of the teaching of reading

Historically, political debates have broken out over how to teach reading in primary schools and infant classrooms. These debates and “reading wars” have often resulted from public concerns and media reportage of a fall in reading standards. They also reflect the importance placed on learning to read by parents, teachers, employers, and politicians. Public and media-driven controversies over the teaching of reading have resulted in intense public and professional debates over which specific methods and materials to use with beginning readers and with children who have reading difficulties. Recently, such debates have led to a renewed emphasis on reading proficiency and “standardized” approaches to teaching reading and engaging with literacy. The universal acceptance of the importance of learning to read has also led to vested interests in specific methods, reading programmes, and early literacy assessments amongst professional, business, commercial, and parental lobbying groups. This article traces these debates and the resulting growing support for a quantitative reductionist approach to early-reading programmes

On the Potts model partition function in an external field

We study the partition function of Potts model in an external (magnetic) field, and its connections with the zero-field Potts model partition function. Using a deletion-contraction formulation for the partition function Z for this model, we show that it can be expanded in terms of the zero-field partition function. We also show that Z can be written as a sum over the spanning trees, and the spanning forests, of a graph G. Our results extend to Z the well-known spanning tree expansion for the zero-field partition function that arises though its connections with the Tutte polynomial

Anthropogenic natal environmental effects on life histories in a wild bird population

Recent work suggests that the environment experienced in early life can alter life histories in wild populations [1, 2, 3, 4 and 5], but our understanding of the processes involved remains limited [6 and 7]. Since anthropogenic environmental change is currently having a major impact on wild populations [8], this raises the possibility that life histories may be influenced by human activities that alter environmental conditions in early life. Whether this is the case and the processes involved remain unexplored in wild populations. Using 23 years of longitudinal data on the Mauritius kestrel (Falco punctatus), a tropical forest specialist, we found that females born in territories affected by anthropogenic habitat change shifted investment in reproduction to earlier in life at the expense of late life performance. They also had lower survival rates as young adults. This shift in life history strategy appears to be adaptive, because fitness was comparable to that of other females experiencing less anthropogenic modification in their natal environment. Our results suggest that human activities can leave a legacy on wild birds through natal environmental effects. Whether these legacies have a detrimental effect on populations will depend on life history responses and the extent to which these reduce individual fitness