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On Face Vectors and Resolutions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consist of the following three papers.

  • Convex hull of face vectors of colored complexes. In this paper we verify a conjecture by Kozlov (Discrete ComputGeom18(1997) 421–431), which describes the convex hull of theset of face vectors ofr-colorable complexes onnvertices. As partof the proof we derive a generalization of Turán’s graph theorem.
  • Cellular structure for the Herzog–Takayama Resolution. Herzog and Takayama constructed explicit resolution for the ide-als in the class of so called ideals with a regular linear quotient.This class contains all matroidal and stable ideals. The resolu-tions of matroidal and stable ideals are known to be cellular. Inthis note we show that the Herzog–Takayama resolution is alsocellular.
  • Clique Vectors ofk-Connected Chordal Graphs. The clique vectorc(G)of a graphGis the sequence(c1,c2,...,cd)inNd, whereciis the number of cliques inGwithivertices anddis the largest cardinality of a clique inG. In this note, we usetools from commutative algebra to characterize all possible cliquevectors ofk-connected chordal graphs.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. , ix, 21 p.
Series
TRITA-MAT. MA, ISSN 1401-2278 ; 2014:07
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-145029ISBN: 978-91-7595-153-9 (print)OAI: oai:DiVA.org:kth-145029DiVA: diva2:715779
Presentation
2014-05-30, Rum 3721, Matematik, Lindstedtsvägen 25, plan 7, KTH, Stockholm, 13:15 (English)
Opponent
Supervisors
Note

QC 20140513

Available from: 2014-05-13 Created: 2014-05-06 Last updated: 2014-05-14Bibliographically approved
List of papers
1. Convex hull of face vectors of colored complexes
Open this publication in new window or tab >>Convex hull of face vectors of colored complexes
2014 (English)In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 36, 247-250 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we verify a conjecture by Kozlov [D.N. Kozlov, Convex Hulls of f- and beta-vectors, Discrete Comput. Geom. 18 (1997) 421-431], which describes the convex hull of the set of face vectors of r-colorable complexes on n vertices. As part of the proof we derive a generalization of Turn's graph theorem.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-145026 (URN)10.1016/j.ejc.2013.07.004 (DOI)000328869800022 ()2-s2.0-84882950186 (Scopus ID)
Note

QC 20140514

Available from: 2014-05-06 Created: 2014-05-06 Last updated: 2017-12-05Bibliographically approved
2. Cellular structure for the Herzog–Takayama resolution
Open this publication in new window or tab >>Cellular structure for the Herzog–Takayama resolution
2014 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Article in journal (Refereed) Published
Abstract [en]

Herzog and Takayama constructed an explicit resolution for the ideals with a regular linear quotient. These ideals include all matroidal and stable ideals. The resolutions of matroidal and stable ideals are known to be cellular. In this note, we show that the Herzog–Takayama resolution is also cellular.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-145027 (URN)10.1007/s10801-014-0524-7 (DOI)000349247100002 ()2-s2.0-84901735112 (Scopus ID)
Note

QC 20140514

Available from: 2014-05-06 Created: 2014-05-06 Last updated: 2017-12-05Bibliographically approved
3. Clique vectors of k-connected chordal graphs
Open this publication in new window or tab >>Clique vectors of k-connected chordal graphs
2015 (English)In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 132, 188-193 p.Article in journal (Refereed) Published
Abstract [en]

The clique vector c(G) of a graph G is the sequence (c(1), c(2), ..., c(d)) in N-d, where c(i) is the number of cliques in G with i vertices and d is the largest cardinality of a clique in G. In this note, we use tools from commutative algebra to characterize all possible clique vectors of k-connected chordal graphs.

Keyword
Clique vectors, Chordal graphs, Graph connectivity
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-145028 (URN)10.1016/j.jcta.2015.01.001 (DOI)000350528900009 ()2-s2.0-84921502318 (Scopus ID)
Note

QC 20150408. Updated from submitted to published.

Available from: 2014-05-06 Created: 2014-05-06 Last updated: 2017-12-05Bibliographically approved

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Citation style
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