Structural Representation of Graphs

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Farhadian, Ameneh | 2022

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Type of Document: Ph.D. Dissertation
Language: Farsi
Document No: 55404 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Fanai, Hamid Reza
Abstract:
In this thesis, we have shown that unique subgraphs of a graph have a key role in structure of the graph. Using unique subgraph which is called “anchor” here, the reconstruction of graphs is explained. Using anchor, we have shown that almost every n-vertex graph is determined by its 3log(n)-vertex subgraphs. In the second part of the thesis, a novel randomized algorithm is proposed for the graph isomorphism problem which is very simple and fast. It solves this problem with running time O(n^{2.373} \log(n)) for any pair of $n$-vertex graphs whose adjacency matrices are not strongly co-det. Strongly co-det pair of matrices have very special symmetric structure which can be disarranged to be not strongly co-det by manipulating one element of the matrices
Keywords:
Reconstruction Conjecture ; Deck ; Graph Isomorphism ; Graph Anchor ; Almost Every Graphs ; Graph Reconstruction