Kriel, Christo2018-07-182018-07-182017Kriel, Christo Willem (2017) Graphs and graphs polynomials, University of the Witwatersrand, Johannesburg, https://hdl.handle.net/10539/25012https://hdl.handle.net/10539/25012A dissertation submitted to the School of Mathematics in fulfilment of the requirements for the degree of Master of Science School of Mathematics University of the Witwatersrand, October 2017In this work we study the k-defect polynomials of a graph G. The k defect polynomial is a function in λ that gives the number of improper colourings of a graph using λ colours. The k-defect polynomials generate the bad colouring polynomial which is equivalent to the Tutte polynomial, hence their importance in a more general graph theoretic setting. By setting up a one-to-one correspondence between triangular numbers and complete graphs, we use number theoretical methods to study certain characteristics of the k-defect polynomials of complete graphs. Specifically we are able to generate an expression for any k-defect polynomial of a complete graph, determine integer intervals for k on which the k-defect polynomials for complete graphs are equal to zero and also determine a formula to calculate the minimum number of k-defect polynomials that are equal to zero for any complete graph.Online resource (ix, 90 leaves)enPolynomialsGraph theoryThesis