#### Degree Name

MS (Master of Science)

#### Program

Mathematical Sciences

#### Date of Award

8-2017

#### Committee Chair or Co-Chairs

Jeff Knisley

#### Committee Members

Debra Knisley, Michele Joyner

#### Abstract

Spectral clustering is often used to partition a data set into a specified number of clusters. Both the unweighted and the vertex-weighted approaches use eigenvectors of the Laplacian matrix of a graph. Our focus is on using vertex-weighted methods to refine clustering of observations. An eigenvector corresponding with the second smallest eigenvalue of the Laplacian matrix of a graph is called a Fiedler vector. Coefficients of a Fiedler vector are used to partition vertices of a given graph into two clusters. A vertex of a graph is classified as unassociated if the Fiedler coefficient of the vertex is close to zero compared to the largest Fiedler coefficient of the graph. We propose a vertex-weighted spectral clustering algorithm which incorporates a vector of weights for each vertex of a given graph to form a vertex-weighted graph. The proposed algorithm predicts association of equidistant or nearly equidistant data points from both clusters while the unweighted clustering does not provide association. Finally, we implemented both the unweighted and the vertex-weighted spectral clustering algorithms on several data sets to show that the proposed algorithm works in general.

#### Document Type

Thesis - Open Access

#### Recommended Citation

Masum, Mohammad, "Vertex Weighted Spectral Clustering" (2017). *Electronic Theses and Dissertations.* Paper 3266. http://dc.etsu.edu/etd/3266

#### Copyright

Copyright by the authors.

#### Included in

Discrete Mathematics and Combinatorics Commons, Other Mathematics Commons, Theory and Algorithms Commons