Spanning Simplicial Complex of Certain Family of Graphs

Abstract

The spanning simplicial complex Ds(Gw) related to a family of graphs known as w-wing graphs is thoroughly examined in this thesis. The combinatorial essence of cycle-free edge subsets is captured by Ds(Gw), where each facet corresponds to a spanning tree of the graph Gw. We examine this complex’s combinatorial invariants, which together carry significant structural information. These invariants include the f -vector, Euler characteristic, and Hilbert series. In addition, we investigate the algebraic features of the Stanley–Reisner ring of Ds(Gw) and define and analyze the facet ideal. In addition to generalizing existing concepts, the results provide a tangible link between commutative algebra and combinatorial topology.