The Graph Isomorphism Problem

The graph isomorphism ($\mathsf{GI}$) problem is one of the few known problems in computational complexity that are in the complexity class $\mathsf{NP}$, but neither known to be in $\mathsf{P}$ nor in $\mathsf{NP}$-complete. So it is a very good candidate for a so called $\mathsf{NP}$-intermediate problem. Another well known candidate is factorization.
For many types of graphs the $\mathsf{GI}$ problem could be solved efficiently, i.e., using only poly($n$) many resources, whereof $n$ is the number of vertices in the graph. However, one is able to generate hard instances where all the algorithms fail and an exponential amount of resources is needed in order to solve the problem. But finding such hard instances is not easy. In particular for random graphs, the isomorphism problem is almost always easy. So the average complexity of $\mathsf{GI}$ is low, but the worst case complexity is hard.

Definition [Graph Isomorphism] Two graphs $G=(V_1,E_1)$ and $H=(V_2,E_2)$ are isomorphic if there is a bijective mapping $\sigma: V_1 \rightarrow V_2$ such that if there is an edge between the vertices $u$ and $v$ in $G$, i.e., $(u,v) \in E_1$, then there is an edge between the vertices $\sigma(u)$ and $\sigma(v)$ in $H$, i.e. $(\sigma(u),\sigma(v)) \in E_2$.

The actual description of the problem is easy and understandable even to people that perhaps see graphs first their first time. In Figure 1 the undirected graph $G$ is shown.

Figure 1 - The graph $G$

$G$ has $8$ vertices and each one is of degree $3$. In Figure 2 you can see the two graphs $H_1$ and $H_2$. One of these is isomorphic to $G$ and the other one is not.

Figure 2 - On the left it is graph $H_1$ and on the right $H_2$

Question: Suppose you could drag each vertex with your mouse. Which set of vertices can be rearranged such that the graph $G$ appears? That means, is $G \cong H_1$ or $G \cong H_2$?

The sum of integers is -1/12 ... or not?

In the current days it seems that people picked up an old discussion and brought it to new attention in their forums. In heated debates they discuss and exchange their point of views, sometimes rather impolite than objective.

They talk about infinite sums of integers, in particular divergent sums, and show how these sums can be manipulated to be equal to a finite value. This causes trouble, since it seems on the first sight rather unintuitive and wrong.

The most used example for their offending object is the simple sum $$1 + 2 + 3 + 4 + ... = \sum^\infty_{i=1} i$$, which can be manipulated such that it is finally equal to $-1/12$. The first one who showed this curiosity was the famous mathematician Srinivasa Ramanujan. And from the fact that a credible mathematician wrote down those things, there should arise some doubt that maybe perhaps there is some truth in these formulas and that the misunderstandings come from the fact, that the formulas are probably not correctly cited and used.