what graph have exactly one positive eigenvalue

Question

I know a Complete k-Partite Graph have exactly one positive eigenvalue. can I say a graph with a exactly one positive eigenvalue is Complete k-Partite Graph? and is the answer yes? how to proof it ?

Answer 1

If we adjoin isolated vertices to a complete multipartite graph, there is still only one positive eigenvalue. So we aim to prove a connected graph $X$ with exactly one positive eigenvalue is complete multipartite. Observe that if there is an induced $P_4$ then, by interlacing, $X$ has at least two positive eigenvalues. Therefore $X$ has diameter at most two. If the diameter is one, we’re complete, and we’re done.

Suppose $a$ and $b$ are two non-adjacent vertices. If $a$ has a neighbour not adjacent to $b$ then this neighbour, along with an $ab$-path of length two gives an induced $P_4$. So any two non-adjacent vertices have exactly the same neighbours. Now it is easy to prove $X$ is complete multipartite.