Rickard Fors: Independence Complexes of Certain Families of Graphs

The focus will be on independence complexes of different sequences of graphs. For such independence complexes, we will study generating functions and closed formulae for the Euler characteristics and f-polynomials, as well as homology groups of different degrees. All of these can be computed by hand, although this quickly becomes tedious as well as really difficult to do, hence recursive methods will be used instead. The generating functions, bounded formulae and recursive equations will be compared to known number sequences, and where possible bijections to other problems will be established. For the independence complexes of each graph sequence, formulae will be given for where the homology groups are nonzero, as well as in some cases formulae for the exact dimension of the homology groups for each complex.