165 research outputs found
Evaluations of topological Tutte polynomials
We find new properties of the topological transition polynomial of embedded
graphs, . We use these properties to explain the striking similarities
between certain evaluations of Bollob\'as and Riordan's ribbon graph
polynomial, , and the topological Penrose polynomial, . The general
framework provided by also leads to several other combinatorial
interpretations these polynomials. In particular, we express , ,
and the Tutte polynomial, , as sums of chromatic polynomials of graphs
derived from ; show that these polynomials count -valuations of medial
graphs; show that counts edge 3-colourings; and reformulate the Four
Colour Theorem in terms of . 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 and .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
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
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
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
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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
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