Main Eigenvalues of Graphs and Signed Graphs

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Kamali, Sara | 2020

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 53305 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Akbari, Saeed; Ghorbani, Ebrahim
Abstract:
Let G be a simple graph. An eigenvalue of G, is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. A signed graph is a graph with a sign to each edge. If in the adjacency matrix of background graph change elements that corresponded by -1, set -1 and in the other elements don’t make any change, then we reach the sign matrix of a signed graph. By an eigenvalue of a signed graph, we mean an eigenvalue of its sign matrix. In this research, we study main eigenvalues of graphs and signed graphs. At first, we present the necessary and sufficient conditions for any graph which has exactly m main eigenvalues. Then, by introducing and creating regular graphs and semi-regular graphs, we indentify graphs with at most 2 main eigenvalues
Keywords:
Signed Graph ; Regular Graph ; Graph Eigenvalue ; Eigen Vectors ; Eigen Values ; Semi Regular Graphs ; Adjacency Matrix