Demonstrations on the Simplicial complex of Graph

Question

where I cannot understand $F\in\Gamma\land G\subseteq F\Rightarrow G\in\Gamma$. I would like to see an example about the simplicial complex of a graph such as a cycle graph $C_3$.

What are demonstrations such as $\Gamma (C_3)$ about $\Gamma$?

Answer 1

The elements called faces, sometimes called facets, of the simplicial complex $\Gamma$ form a subset of the powerset $\mathcal P(V)$. Each element of $\Gamma (1-2-3)$ and $\Gamma (C_3)$ is called a face, for example the element $1$ in $\Gamma (1-2-3)$ is a face.

Example on a path graph

Consider the path graph $1-2-3$ where the $\mathcal P({1,2,3})=\{\emptyset,1,2,3,12,23,13,123\}$. Now $1,2,3,12,23\in\Gamma$ but $13\not\in\Gamma$ so $123\not\in\Gamma$ so

$\Gamma (1-2-3) =\{\emptyset, 1,2,3,12,23\}\subset\mathcal P(1,2,3)$

where six elements and $\text{dim } \Gamma = \text{dim }(12)=1$.

Example on $C_3$

$\Gamma (C_3) =\{\emptyset,1,2,3,12,23,31,123\}=\mathcal P(1,2,3)$

where eight elements (all elements of the powerset) and $\text{dim } \Gamma = 2$.