On The Topology of Graph Complexes

Balanarayan, Mukund (2025) On The Topology of Graph Complexes. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Mukund Balanarayan (20MS155)) 20MS155_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

This thesis explores several families of simplicial complexes built from graphs, with the aim of uncovering how their combinatorial structure reflects deeper topological invariants. One of our primary goals is to demonstrate techniques that are commonly used in practice, including homology, discrete Morse theory, and a brief exposition on Stanley-Reisner theory. Next we discuss extensions of Froberg’s theorem on edge ideals. Properties of the proposed generalization by Margaret Bayer. Particularly, we investigate the homotopy type of cylindrical graphs. Finally, we define the Non-Zero forcing complex of a graph and compute the homotopy types of various classes of graphs. 1

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