Almost all theorem in the textbooks assuming the graph as connected. Of course, it entails that it will work for a connected component. But then computed invariant would be referring to the connected component only. My concern is extending the results to disconnected graphs as well.
The problem I'm working on is disconnected bipartite graph. If I compute the adjacency matrix of the entire graph, and use its eigenvalues to compute the graph invariant, for examples Lovasz number, would the results still valid?
The other approach perhaps could be computing the adjacency matrix of each connected component, and use the eigenvalue to deduce the results for entire graph, if we could do so.
I'm self-studying the graph theory. If there is a good reference out there related to my concern (computing graph invariant of disconnected graph), please share.
It depends on the invariant. But as far as treating each part separately, you will get the same eigenvalues (the adjacency matrix of the whole graph can be written as block diagonal, with each block being the adjacency matrix for a connected component).
So if an invariant involves looking at the smallest eigenvalue, that will be the smallest eigenvalue of one of the components. How to interpret that will depend on the invariant.
If an invariant involves looking at multiplicity of an eigenvalue, you will need to consider whether an eigenvalues multiplicity comes from a component with a repeated eigenvalue, or if the eigenvalue is repeated for different blocks.
I don't think you will really find a reference to specifically deal with disconnected graphs, because the whole point is that you can consider the components separately (and interpret whether the invariant is the min of that for the components, the max, maybe product, will depend on context). The best reference will be the proof that the eigenvalue gives you information for a connected graph, and see how to interpret that.