# Microsoft Office Professional Plus 2016 X86 X64 Fully Activated

Microsoft Office Professional Plus 2016 X86 X64 Fully Activated

reg and Save it in Windows\System32\Macromed\Flash.. Step 4. Now in case you want to activate this version. Open “Office2016.reg” in Word and you will see a prompt to import the file. Click OK. The Office 2016 version will be activated. If you face any problem in activating the Office. Please leave a comment below.

Office 2016 Licensing :

There is a difference in licensing between the Business and the Home edition of Office. There is a limit of how many devices can be licensed and a minimum purchase of Office 365 is required to use the Home edition. With Office 365, you will need to pay per device to get office 365 license.Q:

Given a graph with $n$ vertices and $m$ edges, is there a bound for the number of connected components?

Given a simple graph on $n$ vertices with $m$ edges, is there a known upper bound for the number of connected components? I have seen this claimed in some algorithms, but I would like a more general result.

A:

If $G$ is a graph with $m$ edges, then the chromatic number $\chi(G)$ of $G$ satisfies $\chi(G) \leq \frac{m}{2} + 1$. Therefore, if $G$ has $n$ vertices and $m$ edges, then $\chi(G) \leq \frac{m}{2} + 1 \leq \frac{n}{2} + 1$.

We know that $\chi(G) \geq \frac{n}{2} + 1$ for all $G$ with $n$ vertices, since $G$ is disconnected if and only if it is bipartite and therefore has chromatic number at most $\frac{n}{2}$.

Therefore, if $G$ has $n$ vertices and $m$ edges, then $\chi(G) \leq \frac{n}{2} + 1$.

If you are looking for $m$ edges and $n$ vertices, then you have $2^n$ ways to choose the vertex set and $m \times 2^n$ ways to choose the edge set. Each connected component corresponds to a choice of pair of vertices. The answer then depends on how many of these are singletons.

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