Why do graph degree sequences always have at least one number repeated?    $(1, 2, 2, 3)$ = Valid, as you can see, because the $2$ is repeated. $(1, 2, 3)$ = Not possible to construct a graph with because all numbers are unique.    I know that the graph degree sum must be even, but in this case, both are even.  Our professor said it had to do with Combinatorics, but we never touched that topic.  EDIT: Loops are not allowed.
Any help would be appreciated.  If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree?
If a function maps every connected set onto a connected set, is it necessarily continuous? I know the converse is true. Variance of dot product When reading a proof I came across the following step:  $$\operatorname{Var}(x^Ty) = x^T\operatorname{Var}(y)x$$ $x$ and $y$ are column vectors. How can you derive this? Shuffle an array with python, randomize array item order with python What's the easiest way to shuffle an array with python? After I installed plank as my dock panel, I removed the default Ubuntu dock by running    sudo apt-get remove gnome-shell-extension-ubuntu-dock   Then Ubuntu dock is not on my desktop anymore, but when I open the application menu or Activities overview it is there, how can I hide it?  Removed dock shows on the left: It was nice to discover the option for a room owner to add any user to a room, even if their rep is too low.     ...so I notified the user in question, and as instructed on the next screen, I had them got to chat.stackoverflow.com so that I'd be able to add them.    ...but when they did their part, and I did my part, I got the message:    Obviously both frustrated, the user's next logical question was:  "So I just have ask a couple questions and get someone to upvote them, right?"    Needless to say, we don't want to encourage new users to post questions for the sole purpose of bypassing a frustrating, misdocumented rule.  I realize this will likely quickly get closed since it has at least one  - which hasn't been touched in 5½ years.  However it would be nice if the issue was revisited:  Ideally, it would be cool if RO's did have the option of deciding who had access to the room they "own", but short of that, is it possible to remove the 6 "misleading" words from that page? How do I perform an IF...THEN in an SQL SELECT statement?  For example:  SELECT IF(Obsolete = 'N' OR InStock = 'Y' ? 1 : 0) AS Saleable, * FROM Product Is there an English term for "L'esprit de l'escalier"? Is there a criterion to differentiate "thriller" from "horror"? How can I test whether my camera is working after it was dropped? Issue : How can I evaluate my cameras performance synthetically?  Scenario : Bought a Nikon D3100 an year back. Photos looked good enough. The camera fell, zoom lens effected.  But works. Cleaned lens using soap, water etc.  Carefully. Now photos look much worse.  How do I know if Photos are really bad?  Is there a tool?    Sample Photo  Now (Bad)   Original (Good)   Please help me realize whether my camera has gone bad.  Can you tell your personal opinion also? How do I learn where the source file for a given Python module is installed? Is the method different on Windows than on Linux?  I'm trying to look for the source of the datetime module in particular, but I'm interested in a more general answer as well. How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$ p:commandbutton action doesn't work inside p:dialog If $M^n$ is a compact topological manifold (not necessarily with additional structure), is there an upper bound on the number of charts needed to cover $M$ ? Does this bound depend on the dimension of $M$ ?  Thanks in advance...  Cheers Terminal prompt not wrapping correctly I have an issue where if I type in very long commands in bash the terminal will not render what I'm typing correctly. I'd expect that if I had a command like the following:  username@someserver ~/somepath $ ssh -i /path/to/private/key myusername@something.someserver.com  The command should render on two lines. Instead it will often wrap around and start writing over the top of my prompt, somewhat like this:  myreallylongusername@something.somelongserver.comh -i /path/to/private/key  If I decide to go back and change some argument there's no telling where the cursor will show up, sometimes in the middle of the prompt, but usually on the line above where I'm typing.   Additional fun happens when when I Up to a previous command. I've tried this in both gnome-terminal and terminator and on i3 and Cinnamon. Someone suggested it was my prompt, so here that is:  \[\033[01;32m\]\u:\[\033[01;34m\] \W\033[01;34m \$\[\033[00m\]  Ctrll, reset, and clear all do what they say, but when I type the command back in or Up the same things happens.  I checked and checkwinsize is enabled in bash. This happens on 80x24 and other window sizes.  Is this just something I learn to live with? Is there some piece of magic which I should know? I've settled for just using a really short prompt, but that doesn't fix the issue. If I want to draw an overline over some text, not an equation, how can I do it? \overline seems to only work in math mode, and when used outside of math mode, it automatically says "Missing $ inserted". Debugging scripts, what is the difference between -x to set -euxo pipefail? The main way I know to debug scripts is adding -x to the shabang (#!/bin/bash -x).  I recently came across a new way, adding set -euxo pipefail right under the shabang, as in:  #!/bin/bash set -euxo pipefail   What is the main difference between the two ways of debugging? Are there times you would prefer one above the other?  As a freshman, , I couldn't extract such conclusion. Delete all files except files with the extension pdf in a directory I have a directory that contains the following:  x.pdf y.zip z.mp3 a.pdf   I want to delete all files apart from x.pdf and a.pdf. How do I do this from the terminal? There are no subdirectories so no need for any recursion. Removing files matching a numeric range regular expression I have a folder which has many files. Some of the data files are labelled 1, 2, 3, ..., 300  I want to remove these data files using the terminal. How do I do this?   I want to do something like   rm some-regular-expression-giving-numbers-from-0--300 In Volo's Guide it is suggested that a curse could transform someone into a race of 'monstrous origin' (p. 118). Does this apply to the Bestow Curse spell specifically? Note: this is a  (my apologies).  In film / lore, such magic does , ,   - or even  transformative curses.  Points &amp; Concerns:   this is within guidelines for a 'reasonable' curse, i.e. less harm than the listed set Bestow Curse writeup encourages DM-homebrewing &amp;/or interpretation. Great role-play possibility / RAI (hence most myths, fairy tales, cartoons, legends, etc) this could be subject to abuse, temptation to bestow a 'curse' as a blessing / boon one could also change: age, size, weight &amp;/or gender (all great curse ideas... but RAW-legal?)   Summation:  What are the limits of the 3rd lvl / Necromatic Bestow Curse spell for transformation? Why does light of high frequency appear violet?