I ran across the following formula on Wikipedia. If $G = (V,E)$ is a graph, then viewing the graph as a simplicial complex (vertices are $0$-cells and edges are $1$-cells), we have the formula $$H_1(G) \cong {\bf Z}^{|E|-|V|+1}.$$ Is there a higher-dimensional analogue to this? In particular, is there a formula for $H_{r-1}(G)$ when $G$ is an $r$-uniform hypergraph? (This would make all the maximal faces have dimension $r-1$.) If there isn't a formula in general, then perhaps just for complete graphs there is a generalisation? For $K_n^{(2)} = K_n$, the formula above works out to give a free abelian group of rank ${n-1\choose 2}$, and I wonder if the formula would generalise to ${n-1\choose r}$ or something similar, but I'm having trouble reasoning in higher dimensions.
Thanks in advance!!
Edit. I'd still be interested in any kind of generalisation of the formula above, but I think I have done something wrong, which may be why my conjectured formula does not work for higher $r$. It is not complete graphs I'm studying. For some $r\ge 1$ fixed, I'm looking at the simplicial complex that has as vertex set $[n]^{(r)}$, with a face joining any set of vertices that have nonempty intersection. So when $r=1$ this is a disjoint union of $n$ points. I would like to know the $(r-1)$st reduced homology of the complex. When $r=1$, this is $n-1$ because there are $n$ connected components (and the "reduced" in reduced homology makes it decrease by $1$). When $r=2$, I drew some pictures and it seems to me (though I have had some trouble proving it rigorously) that the complex is homotopy equivalent to $K_{n}$, since there are $n$ simplices, once for each distinct integer.
I have computational reason to believe that the rank of the $(r-1)$st homology group of this complex is ${n-1\choose r}$, but am not able to even see why for $r>2$, let alone prove it. But the description of this simplicial complex seems natural enough that I suspect it may have a name/has been studied before. Is this the case? And if not, there is enough symmetry in its structure that perhaps finding this homology group shouldn't be too difficult, but I am a bit stumped. I would appreciate any pointers!