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Connectivity of pseudomanifold graphs from an algebraic point of view
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2015 (English)In: Comptes Rendus Mathematiques de l'Academie des Sciences = Mathematical reports of the academy of science, ISSN 0706-1994, Vol. 353, no 12, 1061-1065 p.Article in journal (Refereed) Published
Abstract [en]

The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis, and Bjorner. In this note, we provide a unifying approach that allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley-Reisner rings.

Place, publisher, year, edition, pages
Elsevier, 2015. Vol. 353, no 12, 1061-1065 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-186129DOI: 10.1016/j.crma.2015.09.018ISI: 000366617600001ScopusID: 2-s2.0-84951310212OAI: oai:DiVA.org:kth-186129DiVA: diva2:925574
Note

QC 20160511

Available from: 2016-05-02 Created: 2016-05-02 Last updated: 2016-05-16Bibliographically approved
In thesis
1. Topological and Shifting Theoretic Methods in Combinatorics and Algebra
Open this publication in new window or tab >>Topological and Shifting Theoretic Methods in Combinatorics and Algebra
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of six papers related to combinatorics and commutative algebra.

In Paper A, we use tools from topological combinatorics to describe the minimal free resolution of ideals with a so called regular linear quotient. Our result generalises the pervious results by Mermin and by Novik, Postnikov & Sturmfels.

In Paper B, we describe the convex hull of the set of face vectors of coloured simplicial complexes. This generalises the Turan Graph Theorem and verifies a conjecture by Kozlov from 1997.

In Paper C, we use algebraic shifting methods to characterise all possible clique vectors of k-connected chordal graphs.

In Paper D, to every standard graded algebra we associate a bivariate polynomial that we call the Björner-Wachs polynomial. We show that this invariant provides an algebraic counterpart to the combinatorially defined h-triangle of simplicial complexes. Furthermore, we show that a graded algebra is sequentially Cohen-Macaulay if and only if it has a stable Björner-Wachs polynomial under passing to the generic initial ideal.

In Paper E, we give a numerical characterisation of the h-triangle of sequentially Cohen-Macaulay simplicial complexes; answering an open problem raised by Björner & Wachs in 1996. This generalise the Macaulay-Stanley Theorem. Moreover, we characterise the possible Betti diagrams of componentwise linear ideals.

In Paper F, we use algebraic and topological tools to provide a unifying approach to study the connectivity of manifold graphs. This enables us to obtain more general results.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2016. 152 p.
Series
TRITA-MAT-A, 2016:02
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-186136 (URN)978-91-7595-899-6 (ISBN)
Public defence
2016-06-07, F3, Lindstedtsvägen 26, Stockholm, 12:30 (English)
Opponent
Supervisors
Note

QC 20160516

Available from: 2016-05-16 Created: 2016-05-02 Last updated: 2016-05-16Bibliographically approved

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