Given a simple (no loops, no multi-edges) undirected graph $G$ on $n$-vertices, one can assign an abstract simplicial complex known as the independence complex (https://en.m.wikipedia.org/wiki/Independence_complex) on $n$-vertices whose faces are exactly the independent sets of the graph $G$.
My question is: Is every abstract simplicial complex the independence complex of a simple graph?
Best Answer
If $X$ is a subset of the vertex set of a graph and if every $2$-element subset of $X$ is independent, then $X$ itself is independent. So the simplicial complex consisting of all the $\leq2$-element subsets of $\{1,2,3\}$ is not the independence complex of any graph.