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On a class of commutative algebras associated to graphs
Stockholm University, Faculty of Science, Department of Mathematics.
2016 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In 2004 Alexander Postnikov and Boris Shapiro introduced a class of commutative algebras for non-directed graphs. There are two main types of such algebras, algebras of the first type count spanning trees and algebras  of the second type count spanning forests. These algebras have a number of interesting properties including an explicit formula for their Hilbert series. In this thesis we mainly work with the second type of algebras, we discover more properties of the original algebra and construct a few generalizations. In particular we prove that the algebra counting forests depends only on graphical matroid of the graph and converse. Furthermore, its "K-theoretic" filtration reconstructs the whole graph. We introduse $t$ labelled algebras of a graph, their Hilbert series contains complete information about the Tutte polynomial of the initial graph. Finally we introduce similar algebras for hypergraphs. To do this, we define spanning forests and trees of a hypergraph and the corresponding "hypergraphical" matroid.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2016. , 40 p.
National Category
Discrete Mathematics Algebra and Logic
Identifiers
URN: urn:nbn:se:su:diva-132987OAI: oai:DiVA.org:su-132987DiVA: diva2:956005
Presentation
2016-09-05, Sal 14, hus 5, Kräftriket, Stockholm, 13:00 (English)
Opponent
Supervisors
Available from: 2016-11-03 Created: 2016-08-28 Last updated: 2016-11-03Bibliographically approved
List of papers
1. On Postnikov-Shapiro Algebras and their generalizations
Open this publication in new window or tab >>On Postnikov-Shapiro Algebras and their generalizations
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A.Postnikov and B.Shapiro introduced a class of commutative algebras which enumerate forests and trees of graphs. Our main result is that the algebra counting forests depends only on graphical matroid and converse.

Also we generalize algebras for a hypergraph. For this, we define spanning forests and trees of a hypergraph and the corresponding "hypergraphical" matroid.

Keyword
Tutte polynomial, graphical matroid, spanning forest
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-132531 (URN)
Available from: 2016-08-15 Created: 2016-08-15 Last updated: 2016-11-03Bibliographically approved
2. "K-theoretic" analog of Postnikov-Shapiro algebra distinguishes graphs
Open this publication in new window or tab >>"K-theoretic" analog of Postnikov-Shapiro algebra distinguishes graphs
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we study a filtered "K-theoretical" analog of a graded algebra associated to any loopless graph G which was introduced in \cite{PS}. We show that two such filtered algebras are isomorphic if and only if their graphs are isomorphic. We also study a large family of filtered generalizations of the latter graded algebra which includes the above "K-theoretical" analog.

National Category
Discrete Mathematics
Identifiers
urn:nbn:se:su:diva-132988 (URN)
Available from: 2016-08-28 Created: 2016-08-28 Last updated: 2016-10-18Bibliographically approved

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Nenashev, Gleb
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