arXiv subjects
Publications and source records attributed to Demet Taylan.
We provide lower bounds on the connectivity of the independence complexes of hypergraphs. Additionally, we compute the homotopy types of the independence complexes of $d$-uniform properly-connected triangulated hypergraphs.
For a simplicial complex $X$, the $d$-clique complex $Δ_d(X)$ is the simplicial complex having all subsets of vertices whose $(d + 1)$-subsets are contained by $X$ as its faces. We prove that if $p = n^α$, with $α< \max\{\frac{-1}{k-d +1},-\frac{d+1}{\binom{k}{d}}\}$ or $α> \frac{-1}{\binom{2k+2}{d}}$, then the $k$-th reduced homology group of the random $d$-clique complex $Δ_d(G_d(n,p))$ is asymptotically almost surely vanishing, and if $\frac{-1}{t} < α< \frac{-1}{t+1}$ where $t = (\frac{(d+1)(k+1)}{\binom{(d+1)(k+1)}{d+1}-(k+1)})^{-1}$, then the $(kd + d -1)$-st reduced homology group of $Δ_d(G_d(n,p))$ is asymptotically almost surely nonvanishing. This provides a partial answer to a question posed by Eric Babson.
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, which are simplicial complexes parametrized by graphs. Additionally, we compute the homotopy type of dominance complexes of chordal graphs.