Graphs, compositions, polynomials and applications

Abstract

In this thesis, we study graph compositions of graphs and two graph polynomials, the k-defect polynomials and the Hosoya polynomials. This study was motivated by the fact that it is known that the number of compositions for certain graphs can be extracted from their k-defect polynomials, for example trees and cycles. We want to investigate if these results can be extended to other classes of graphs, in particular to theta and multibridge graphs. Furthermore we want to investigate if we can mimic these results of k-defect polynomials to Hosoya polynomials of graphs. In particular, investigating if the Hosoya polynomials of graphs can be computed using, similar methods to k-defect polynomials. We start the investigation by improving the upper bound for the number of graph compositions of any graph. Thereafter, we give the exact number of graph composi- tion of theta and 4-bridge graphs. We then nd explicit expressions of the k-defect polynomials of a theta graph via its bad coloring polynomial. Furthermore, we nd explicit expressions for the Hosoya polynomials of multibridge graphs and q-vertex joins of graphs with diameter 1 and 2.