Masters Degrees (Mathematical Sciences)
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Browsing Masters Degrees (Mathematical Sciences) by Subject "Adjacency Matrix"
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- ItemSpectral radii of matrices associated with graphs(Stellenbosch : Stellenbosch University, 2015-12) Dadedzi, Kenneth; Wagner, Stephan; Stellenbosch University. Faculty of Science. Department Mathematical Sciences (Mathematics)ENGLISH ABSTRACT : The spectral radius of a graph is defined as the largest absolute value of the eigenvalues of a matrix associated with the graph. In this thesis, we study the spectral radii of the adjacency matrix, the distance matrix and a matrix related to the distance matrix associated to simple graphs. For the adjacency matrix, we determine the spectral radii of some classes of graphs. We prove that the spectral radius can be used to estimate the number of walks and closed walks in the graph. Furthermore, we present bounds on the spectral radius in terms of some graph parameters. We then proceed to show that the greedy tree, the Volkmann tree and the extended star graph maximise the spectral radius among all trees with prescribed degree sequence, maximum degree and number of leaves respectively. We also collect results on the spectral radii of the distance matrices of some classes of graphs. We investigate a matrix that is related to the distance matrix where we proved that the greedy tree, the Volkmann tree and the extended star graph maximise its spectral radius among all trees with prescribed degree sequence, maximum degree and number of leaves respectively.