Tutte’s dichromate for signed graphs Goodall, Andrew; Litjens, Bart; Regts, Guus ...
Discrete Applied Mathematics,
01/2021, Letnik:
289
Journal Article
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We introduce the trivariate Tutte polynomial of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the ...number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the “dichromate” for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank–size generating function).
The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set — for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.
Introduction Goodall, Andrew
Journal of integrated care (Brighton, England),
08/2015, Letnik:
23, Številka:
4
Journal Article
The collaborative work that has been taking place between the health and social care sectors with clear foundations and progress, has dramatically helped the NHS at a time when the system is coming ...under pressure from a combination of an ageing population with increasingly complex needs and the impact of austerity which has taken its toll on all public services.Some of these challenges are unprecedented and are expected to grow in coming years....there is no doubt in my mind about the vital role we all will continue to play if we are to be successful in building a health and social care system that is fit for the future needs of the population.Integration and simplification of the law will also provide greater consistency and clarity to those involved in all aspects of social care as well a putting a renewed focus on prevention and early intervention - in line with the direction of NHS Wales and our policy for social care.The overarching aim of the Act is to promote health and social care in Wales to improve people's lives and I am confident that its implementation will make a significant difference to those that rely on these services.
We construct a new polynomial invariant of maps (graphs embedded in a closed surface, orientable or non-orientable), which contains as specializations the Krushkal polynomial, the Bollobás—Riordan ...polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial. Other evaluations include the number of local flows and local tensions taking non-identity values in a given finite group.
•45% of recreational athletes reported a successful outcome at nine-months post-operative.•Patient reported readiness was associated with a successful outcome after ACL reconstruction.•No physical ...measures were associated with a successful outcome after ACL reconstruction.•46% of participants failed to complete this study, an attrition rate of 46%.
Anterior cruciate ligament (ACL) injury and subsequent reconstruction is common and has a profound effect on health-related quality of life. There is currently limited understanding as to which variables are associated with a successful outcome post-ACL reconstruction (ACLR) in recreational athletes.
Explore the association between both patient-reported and performance-based measures, and successful outcome, post-ACLR in recreational athletes.
We sought to recruit recreational athletes within one month of a primary-ACLR for a prospective cohort study. A dichotomised patient specific functional scale of ≥9 points determined a successful outcome at nine-months post-operative. Secondary patient-reported and performance-based data were collected at baseline, three-, six-, and nine-months post-operative. The association between secondary data and the primary outcome was determined using binomial logistic regression, expressed using odds ratio (OR) and 95% confidence intervals (CI).
90 participants were recruited (males: 58, females: 32, mean age 32.8 years ±7.9, mean height 173.5 ±10.0, mean body mass 74.0 kg ±15.8), 87 consented to baseline measures. 47 participants completed full data collection and 21 (45%) reported a successful outcome. Higher knee osteoarthritis outcome score (OR range 1.07–1.12) and anterior cruciate ligament quality of life (ACL-QoL) scores (OR range 1.06–1.10) were associated with a successful outcome post-ACLR at various timepoints.
Patient-reported, rather than performance-based, measures were associated with successful outcome nine-months post-ACLR in recreational athletes. Both patient-reported and performance-based characteristics are advocated to guide optimal return to function in clinical practice.
Post-translational modifications (PTMs) expand the number of protein isoforms in eukaryotic proteome by orders of magnitude. Protein modification with isoprenoid lipids is a common PTM affecting ...hundreds of proteins controlling the transport of information and materials into, through, and out of the eukaryotic cell. In this modification, a soluble phosphoisoprenoid such as farnesyl (C15) or geranylgeranyl (C20) pyrophosphate moiety is recruited by one of three protein prenyltransferases to covalently modify a C-terminal cysteine(s) in a target protein. The three mammalian prenyltransferases are farnesyltransferase (FTase), geranylgeranyltransferase type I (GGTase I), and Rab geranylgeranyl transferase (also termed geranylgeranyltransferase type II - GGTase II). In this unit, synthetic isoprenoids conjugated to either a fluorophore or biotin group are used to assay the activity of protein prenyltransferases in vitro or to affinity tag prenylatable proteins in cell lysates. These protocols and their modifications can be used to study the mechanisms of protein prenylation, identify prenylation targets, and characterize inhibitors of protein prenyltransferases in vitro and in vivo.
Summary
Bacteria that inhabit the rhizosphere of agricultural crops can have a beneficial effect on crop growth. One such mechanism is the microbial‐driven solubilization and remineralization of ...complex forms of phosphorus (P). It is known that bacteria secrete various phosphatases in response to low P conditions. However, our understanding of their global proteomic response to P stress is limited. Here, exoproteomic analysis of Pseudomonas putida BIRD‐1 (BIRD‐1), Pseudomonas fluorescens SBW25 and Pseudomonas stutzeri DSM4166 was performed in unison with whole‐cell proteomic analysis of BIRD‐1 grown under phosphate (Pi) replete and Pi deplete conditions. Comparative exoproteomics revealed marked heterogeneity in the exoproteomes of each Pseudomonas strain in response to Pi depletion. In addition to well‐characterized members of the PHO regulon such as alkaline phosphatases, several proteins, previously not associated with the response to Pi depletion, were also identified. These included putative nucleases, phosphotriesterases, putative phosphonate transporters and outer membrane proteins. Moreover, in BIRD‐1, mutagenesis of the master regulator, phoBR, led us to confirm the addition of several novel PHO‐dependent proteins. Our data expands knowledge of the Pseudomonas PHO regulon, including species that are frequently used as bioinoculants, opening up the potential for more efficient and complete use of soil complexed P.
The number of homomorphisms from a finite graph F to the complete graph Kn 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).
Masbaum and Vaintrobʼs “Pfaffian matrix-tree theorem” implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of ...“3-Pfaffian” 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs – one of these is given by a forbidden subgraph characterization analogous to Littleʼs for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property of being 3-Pfaffian can be reduced to the property of an associated graph being Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are not 3-Pfaffian, none of which can be reduced to any other by deletion or contraction of triples.
We also find some necessary or sufficient conditions for the existence of a spanning tree of a 3-graph (much more succinct than can be obtained by the currently fastest polynomial-time algorithm of Gabow and Stallmann for finding a spanning tree) and a superexponential lower bound on the number of spanning trees of a Steiner triple system.
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