Relative Tutte polynomials of tensor products of colored graphs

The tensor product $(G_1,G_2)$ of a graph $G_1$ and a pointed graph $G_2$ (containing one distinguished edge) is obtained by identifying each edge of $G_1$ with the distinguished edge of a separate copy of $G_2$, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to colored graphs and the generalized Tutte polynomial introduced by Bollob\'as and Riordan. In this paper we generalize the colored tensor product formula to relative Tutte polynomials of relative graphs, containing zero edges to which the usual deletion-contraction rules do not apply. As we have shown in a recent paper, relative Tutte polynomials may be used to compute the Jones polynomial of a virtual knot

A Möbius Identity Arising from Modularity in a Matroid Bilinear Form

The matrix for the bilinear form of the flag space of a matroid has (with respect to an appropriate basis) a tensor product structure when the matroid has a modular flat . When determinants are taken, an identity is obtained for the rho function (a certain product of the Möbius and beta functions) summed over flats with a fixed intersection with . When the identity is interpreted for Dowling lattices and finite projective spaces, identities with similar combinatorial proofs are obtained for binomial and Gaussian coefficients, respectively

Intersection theory for graphs

An intersection theory developed by the author for matroids embedded in uniform geometries is applied to the case when the ambient geometry is the lattice of partitions of a finite set so that the matroid is a graph. General embedding theorems when applied to graphs give new interpretations to such invariants as the dichromate of Tutte. A polynomial in n + 1 variables, the polychromate, is defined for graphs with n vertices. This invariant is shown to be strictly stronger than the dichromate, it is edge-reconstructible and can be calculated for proper graphs from the polychromate of the complementary graph. By using Tutte's construction for codichromatic graphs (J. Combinatorial Theory 16 (1974), 168–174), copolychromatic (and therefore codichromatic) graphs of arbitrarily high connectivity are constructed thereby solving a problem posed in Tutte's paper

Hyperplane reconstruction of the Tutte polynomial of a geometric lattice

An explicit construction is given which produces all the proper flats and the Tutte polynomial of a geometric lattice (or, more generally, a matroid) when only the hyperplanes are known. A further construction explicitly calculates the polychromate (a generalization of the Tutte polynomial) for a graph from its vertex-deleted subgraphs

Cultural eutrophication and the clam Macoma balthica: Evidence for trophic disruption and effects on blue crabs

Cultural eutrophication (CE) is the allochthonous input introduction of a quantity of matter, such as sediments, organic material, or nutrients, into a water body over the pre-anthropogenic (natural) levels. In most coastal estuaries CE has come to refer primarily to an increase in the concentration of phyto-nutrients. CE has been identified as the cause of very graphic phenomena such as hypoxia and fish kills. In this work I examine the potential for CE to alter the composition of the primary producer community and potentially alter or disrupt the benthic food web, using Macoma balthica as an indicator species. A series of surveys and experiments identified that clams in areas with greater than average nutrient concentrations had lower health, slower growth, and greater non-predatory mortality than clams in less eutrophic areas. Primary production, as estimated from chlorophyll a concentration, was greater at higher nutrient locations while the health and growth of clams was lower. The phytoplankton community in the more eutrophic areas had a lower proportion of diatoms relative to dinoflagellates. A biochemical analysis of clam tissue indicated that the clams from the less nutrient rich sites had a greater proportion of Eicosapentaenoic acid (EPA) relative to other fatty acids. Diatoms are rich in EPA compared to dinoflagellates. Thus, we hypothesize that CE induced shifts from diatom based production toward dinoflagellates may be limiting trophic transfer due to a lack of EPA. Using a series of models we were able to predict that trophic disruption could significantly reduce the scope for growth of the blue crab, Callenecties sapidus . Thus it is possible that the CE induced changes to primary producer community could disrupt the food web creating a trophic bottleneck

Several identities for the characteristic polynomial of a combinatorial geometry

In this paper we explore a research problem of Greene: to find inequalities for the Möbius function which become equalities in the presence of modularity. We replace these inequalities with identities and give combinatorial interpretations for the difference

Bioenergetic modeling of the blue crab (Callinectes sapidus) using the fish bioenergetics (3.0) computer program

To understand better the ecology and growth dynamics of the blue crab (Callinectes sapidus). we developed a bioenergetic model based upon the Fish Bioenergetics 3.0 computer program. We summarized and analyzed existing data from published studies on the ecology and physiology of both blue crab and closely related species to parameterize the model. The respiration and excretion components were estimated directly from published studies. Parts of the consumption component were estimated indirectly. The resulting model was evaluated for applicability against known growth trajectories from field and laboratory studies. The model predicted observed growth and consumption to a first approximation. Inspection of model results suggest that improvements in our knowledge of temperature- and size-dependant consumption are required before a more predictive model can be developed. However, at this point the model is sufficiently accurate to explore some fishery-related questions

Unavoidable parallel minors of regular matroids

This is the post-print version of the Article - Copyright @ 2011 ElsevierWe prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M (K_{3,k}), M(W_k), M(K_k), the cycle matroid of the graph obtained from K_{2,k} by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K_{3,k} by adding a complete graph on the vertex class with three vertices.This study is partially supported by a grant from the National Security Agency

Products of Linear Forms and Tutte Polynomials

Let \Delta be a finite sequence of n vectors from a vector space over any field. We consider the subspace of \operatorname{Sym}(V) spanned by \prod_{v \in S} v, where S is a subsequence of \Delta. A result of Orlik and Terao provides a doubly indexed direct sum of this space. The main theorem is that the resulting Hilbert series is the Tutte polynomial evaluation T(\Delta;1+x,y). Results of Ardila and Postnikov, Orlik and Terao, Terao, and Wagner are obtained as corollaries.Comment: Minor changes. Accepted for publication in European Journal of Combinatoric

Confinement of matroid representations to subsets of partial fields

Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' of P. We say that B confines M to P' if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled P'-matrix. We show that, under some conditions on the partial fields, on M, and on B, verifying whether B confines M to P' amounts to a finite check. A corollary of this result is Whittle's Stabilizer Theorem. A combination of the Confinement Theorem and the Lift Theorem from arXiv:0804.3263 leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields. We also use a combination of the Confinement Theorem and the Lift Theorem to prove a characterization, in terms of representability over partial fields, of the 3-connected matroids that have k inequivalent representations over GF(5), for k = 1, ..., 6. Additionally we give, for a fixed matroid M, an algebraic construction of a partial field P_M and a representation A over P_M such that every representation of M over a partial field P is equal to f(A) for some homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al.Comment: 45 page