PhD thesis, University of Warwick. Request Changes to record. In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties.
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The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs.
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We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes.
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This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general.
We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. In mathematics , combinatorial topology was an older name for algebraic topology , dating from the time when topological invariants of spaces for example the Betti numbers were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes.
After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups.http://l0ad57da.justinstalledpanel.com/12303.php
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This point of view is often attributed to Emmy Noether ,  and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf ,  who was influenced by Noether, and to Leopold Vietoris and Walther Mayer , who independently defined homology. A fairly precise date can be supplied in the internal notes of the Bourbaki group.
While topology was still combinatorial in , it had become algebraic by Azriel Rosenfeld proposed digital topology for a type of image processing that can be considered as a new development of combinatorial topology.