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Topological combinatorics and simplicial networks
위상수학적 조합론과 고차원 네트워크

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dc.contributor.advisor국웅-
dc.contributor.author이강주-
dc.date.accessioned2019-10-21T03:38:58Z-
dc.date.available2019-10-21T03:38:58Z-
dc.date.issued2019-08-
dc.identifier.other000000158275-
dc.identifier.urihttps://hdl.handle.net/10371/162423-
dc.identifier.urihttp://dcollection.snu.ac.kr/common/orgView/000000158275ko_KR
dc.description학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2019. 8. 국웅.-
dc.description.abstract그래프의 일반화인 단체의 복합체는 고차원 조합론 분야에서 연구되는 대상이다. 여러 점들 사이의 상호작용을 고려하기 위해 고차원 네트워크가 등장하였다. 본 논문에서는 단체의 복합체의 조합론적인 측면과 네트워크 이론적 측면을 다룬다. 생성나무와 관련된 단체의 복합체를 연구하여 조합론적 공식을 얻는다. 전기 네트워크 이론을 이용하여 고차원 네트워크를 분석하기 위한 도구를 개발한다.-
dc.description.abstractSimplicial complexes as a generalization of graphs have been studied in high-dimensional combinatorics. Simplicial networks emerged due to demands for considering interactions among more than two vertices. This thesis covers combinatorial aspects and network-theoretic aspects of simplicial complexes. We present combinatorial formulas for simplicial complexes concerning spanning trees. We provide tools for analyzing simplicial networks based on electrical networks.-
dc.description.tableofcontents1 Introduction 1
1.1 Combinatorics of simplicial complexes . . . 1
1.2 Simplicial networks and electrical networks . . . 3
1.3 Organization of the thesis . . . 5
2 Mobius coinvariants and bipartite edge-rooted forests . . . 6
2.1 Introduction . . . 6
2.2 Background: Mobius coinvariant . . . 9
2.3 Combinatorial interpretations for \mu^{\bot}(Km+1) and \mu^{\bot}(Km+1;n+1) . . . 11
2.4 Formulas for \mu^{\bot}(Km+1) and \mu^{\bot}(Km+1;n+1) . . . 15
2.5 Homology of I(Km+1;n+1) . . . 22
2.6 The Mobius coinvariants of bi-coned graphs . . . 29
3 A formula for simplicial tree-numbers of matroid complexes . . . 31
3.1 Introduction . . . 31
3.2 Preliminaries . . . 33
3.3 The main result . . . 36
3.4 Examples . . . 39
4 A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes . . . 42
4.1 Introduction . . . 42
4.2 Preliminaries . . . 47
4.3 Proof of the main formula . . . 51
4.4 Another tree count for hypercubes . . . 53
5 Weighted tree-numbers of matroid complexes . . . 61
5.1 Introduction . . . 61
5.2 Weighted tree-numbers of simplicial complexes . . . 63
5.3 Weighted tree-numbers of matroid complexes . . . 72
5.4 Application: complete colorful complexes . . . . 79
6 Simplicial networks and effective resistance . . . 83
6.1 Introduction . . . 83
6.2 Preliminaries . . . 85
6.3 Simplicial networks and effective resistance . . . 89
6.4 Effective resistance via simplicial potential . . . 93
6.5 Effective resistance via high-dimensional trees . . . 97
7 High-dimensional networks and spanning forests . . . 104
7.1 Introduction . . . 104
7.2 Preliminaries . . . 107
7.3 An acyclization in codimension 1, and spanning forests in higher dimensions . . . 109
7.4 High-dimensional electrical networks and combinatorics . . . 111
8 Kirchhoff index of simplicial networks . . . 120
8.1 Introduction . . . 121
8.2 Preliminaries . . . 123
8.3 Electrical networks and Kirchhoff index . . . 126
8.4 Robustness of simplicial complexes . . . 132
8.5 An integral expression and a dynamical system . . . 135
8.6 An open problem: another proof of Theorem 8.3.4 . . . 139
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dc.language.isoeng-
dc.publisher서울대학교 대학원-
dc.subjectsimplicial complex-
dc.subjectLaplacian-
dc.subjectspanning tree-
dc.subjectmatroid-
dc.subjectnetwork-
dc.subjectcentrality-
dc.subject.ddc510-
dc.titleTopological combinatorics and simplicial networks-
dc.title.alternative위상수학적 조합론과 고차원 네트워크-
dc.typeThesis-
dc.typeDissertation-
dc.contributor.AlternativeAuthorKang-Ju Lee-
dc.contributor.department자연과학대학 수리과학부-
dc.description.degreeDoctor-
dc.date.awarded2019-08-
dc.identifier.uciI804:11032-000000158275-
dc.identifier.holdings000000000040▲000000000041▲000000158275▲-
Appears in Collections:
College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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