By a graph I will understand an undirected graph without multiple edges or loops. By a morphism of graphs I will understand a map $f$ between the underlying sets of vertices, such that if $x$ and $y$ are adjacent, then $f(x)$ and $f(y)$ are either adjacent or equal.
Let $G$ be a finite graph. One can realise $G$ as a CW-complex $|G|$ and look at topological invariants, such as singular homology. But this captures only very little information about $G$, because except from $H_0(|G|)$ and $H_1(|G|)$ all homology groups are zero.
Consider the following alternative construction: Let us write $\Delta_n$ for the complete graph on $n$ vertices, and let us re-baptise this graph by the name "standard $n$-simplex". There are obvious codegeneracy and coface maps between standard simplices, so that we obtain a cosimplicial object $\Delta_\bullet$ in the category of graphs. Now, proceed as usual: Morphisms $\Delta_\bullet \to G$ form a simplicial set, applying the free group construction yields then simplicial group, and the associated chain complex is the one whose homology $H_i^{\mathrm{sing}}(G)$ I shall call "singular homology of $G$".
Obvious properties of $H_i^{\mathrm{sing}}(G)$ are: It is a finitely generated commutative group ($G$ is finite), covariantly functorial in $G$. In particular, if we work with coefficients in a field, we obtain representations of the automorphism group of $G$. The homology of the point is $\mathbb Z$ in degree $0$ and trivial in higher degrees. We can define singular cohomology accordingly, and get then a natural pairing between homology and cohomology.
The list of all natural questions one must ask after making such a definition is long, so I will not ask everything.
(a) Is there a comparison map $H_i(|G|) \to H_i^{\mathrm{sing}}(G)$, maybe even on the level of chain complexes? Is there some more elaborate CW-complex $||G||$ one can naturally associate with $G$ such that $H_i(||G||)$ gives back singular homology of $G$? In that case, one would ask for a natural map $|G| \to ||G||$.
(b) Given a graph $G$, is there a largest integer $i$ such that $H_i^{\mathrm{sing}}(G)$ is nonzero? Assuming yes, is this integer less or equal the size of the largest complete subgraph of $G$.
(c) Is there a Künneth morphism in singular cohomology? –is there a natural ring strucure on cohomology?
(d) What is a homotopy between morphisms of graphs? Given an answer to that, do homotopically equivalent morphisms induce the same maps in homology?
(z) Can you give an example of a graph with nontrivial $H_2^{\mathrm{sing}}(G)$?
Best Answer
You talk about morphisms from $\Delta_\bullet$ to a graph $G$. I presume a morphism from $\Delta_n$ to $G$ is just an embedding of $\Delta_n$ in $G$, that is an $(n+1)$-clique in $G$ with a labelling of its vertices from $0$ to $n$. It seems to me that defining (co)homology in this way will be the same as the standard simplicial/singular (co)homology of the space $\widehat G$ obtained from $G$ by "filling in" each $(n+1)$-clique with an $(n+1)$-simplex.
As a example consider the graph $G$ consisting of the vertices and edges of a regular icosahedron. Then its $3$-cliques correspond to the faces of the icosahedron and it has no $4$-cliques. Therefore $\widehat G$ will consist of the boundary of the solid icosahedron in $\mathbb{R}^3$ and so your second homology group will be nonzero.
I hasten to add that I have not checked any details, and admit in advance that my thoughts here may be msiguided or just complete nonsense.