I'm just curious because I saw on Wikipedia a single polynomial time solution to any NP-hard problem would imply there are polynomial time solutions for every single NP-hard problem. Also I assume there are a lot of NP-hard problems... So does that mean no one has ever been able to prove for any ...

This appears to be a plausible disproof of the Riemann hypothesis, but do you see any mistake ? Claim: Let $\Theta \in [\frac{1}{2}, 1]$ be the supremum of the real parts of the zeros of $\zeta$. Then $\Theta=1$. PROOF. Denote by $\Lambda$ the von Mangoldt function. For $x \geq 1$, define $\psi(...

So, if you have two graphs (both with a designated edge), you could take the direct sum of those two graphs, where you glue those two graphs to each other along the designated edge. I think different literature use different definitions. Anyways, see the picture below as an example of gluing two ...