If so, could you provide examples and specify the conditions under which this occurs? Thank you in advance



Do there exist graphs whose adjacency matrix is positive semi-definite?
If you don't allow self-loops in the graph, then the trace is $0$. If the adjacency matrix is PSD then $0$ is the only eigenvalue. Since your adjacency matrix is symmetric, it must equal to $0$. So you essentially have the empty graph. 


If you allow self-loops then you can also get a PSD adjacency matrix by adding some diagonal matrix. This corresponds to adding some self-loops in your original graph. 


