Let's suppose that $n$ is a natural number representing the number of vertices in a graph. How would we show that the number of spanning trees of the complete graph $K_n$ is $n^{n−2}$? I tested out a few values on $n$ and it makes sense logically, but how would we prove this?
I predicted the formula by finding total number of non-isomorphic labelled trees of order 1 is 1 , order 2 is 1,order 3 is 3,order 4 is 16,order 5 is 125.But how do i prove it ? I am beginner in graph theory so i will be very thankful if i get some simple approach to solve the problem(Not by using some difficult theorems).
How can I change wallpaper automatically? I have found a guide for that but doesn't sure if it will work or it is safe to use. The guide:  I am asking if is these introductions safe and will work? If not, is there any other solutions for that? I am using Ubuntu 20.04 LTS Thank you. Importing an .obj file and nothing is showing How can I find the source of a mysterious bad smell? We have recently moved home, and all is fine appart from a mystery smell which I'm unable to track down. It's driving me crazy (actually, my wife thinks I am crazy as she can't smell it).  I'm hoping someone on here could offer some tips on how to track it down, or what it might be.  Here's some info:  Background: The house is a two floor (plus basement) block/concrete construction.  There is a wide hallway on the groundfloor connecting the rooms (including kitchen, toilet/shower room, stairwell).  The stairs are wood-board on concrete.  The floor in the hall and kitchen  is tiles over concrete, the rest woodblock/parquet over concrete.  Location of the smell: I usually smell it when coming down the stairs, seems to be strongest about half way down.  However sometimes it doesn't smell on the stairs at all, but rather in the ground floor hallway.  Seems to move around quite a lot. Occasionally I have felt it upstairs, but assume this is it drifting. The basement smells fine.  The Smell: The best description of the smell I can give is something like old rubbish.  What I've investigated so far:  1) Something died somewhere?  This was my initial thought, and since I usually notice the smell on the stairs I thought I'd try there first.  I've taken up some of the wood on the stairs (not all yet) but not looking promising, as there really isn't much space between the wood boards and the concrete.  Not really sure where else something could have gotten, as everything seems to be tight against the concrete.  2) Smell from kitchen moving into rest of house? I never really detected a bad smell in the kitchen, but thought I'd rule it out anyway. I thought maybe it was the bin under the kitchen sink that might be causing it.  So I moved the bin to a small storage room at the back of the kitchen (and kept the door closed) and thoroughly cleaned inside the cupboard under the sink. No change.  I also looked behind all of the cabinets for a problem, but could not see anything.  3) Gas from drains? The sink doesn't smell in the kitchen, and it's got a P-bend trap on the pipe - so I can't see a problem there.  Cant detect a smell from the plugholes in the bathroom either.  The bathroom sink waste pipe has a bottle-trap on it.   Don't know if there is a trap on the shower as it's a sealed unit, but I'm guessing yes (and can't smell anything).  Can any one think of something I might have missed, or give any advice? I couldn't find much on the new Lucene-based search, and don't know much about the engine in general, so I'm asking this question.  Does anybody have a description of how search results are ranked?  Does the rank calculation include:   rep of post's author? views? unique referers? PageRank of referers? rank of questions that link to the post? whether a question is closed? vote count? answer count? How do I install drivers for Realtek RTL8812BU? Check the uniform convergence of series of functions Check that given series converges uniformly for $x\in(-\pi,\pi)$ or not  1) $\sum_{n=1}^{\infty} (\frac{x}{n})^n$  2) $\sum_{n=1}^{\infty}\frac{1}{((x+\pi)n)^2}$  first series is power series so I find the radius of convergence that comes $R=e$ which is inside the given set so I am not able to give the final conclusion. after this I were thinking to apply $M_n$ test , but I am not able to find $M_n$. Please help to solve this problem. I have an array of integers, how do I use each one in a mysql query (in php)? I'm working on a VAR model and am doing a log transformation of the raw data.   x = log(x)   After differencing, running various tests and running a VAR(2) model, I add the predicted differences cumulatively to the original time series (logged) like this:  prd.diffs &lt;- as.ts(prd$mean$d.x1) U &lt;- tail(undiff,2) cumu &lt;- cumsum(prd.diffs) + U[1]   However, while the output now looks as it should, it is still in logged values.  My question is as follows:  How does one return the logged output to original values? I'm under the impression that using exp() will not give the correct results, and having tried some variations it hasn't worked for me.   Does the fact that it is a VAR($p$) model (and possibly what lag it has) influence the method, and how should that be implemented to get the correct output?  P.S. I realise that the above code does not constitute a replicable example as I didn't deem that necessary for the question, but should that be desired I can add some more code. When I press the power button a dialog opens, but I want to shutdown the system without receiving "Do you really really want to shutdown?". How can I choose a direct shutdown action on a pressed power button?   I had the same problem with the 11.04 version, but I don't remember the solution and yesterday I upgraded to Ubuntu 11.10. Sign of Work and potential energy in electrostatics Conceptual question:  Suppose we have a configuration of point charges. If the potential of the energy of the system is negative, this means work is positive.   I'm kind of rusty with my mechanics, does the fact that the work is positive mean that to arrange the configuration you have to do work against the electric field? I.e. supply energy via some external force?  In terms of electrostatics, is there a general way of interpreting the sign of potential energy and work? What is the sign of the  done on the system and by the system?  My chemistry book says when work is done on the system, it is positive. When work is done by the system, it is negative.  My physics book says the opposite. It says that when work is done on the system, it is negative. When work is done by the system, it is positive.  Why do they differ? Can an html element have multiple ids? What does the above Error mean? What is an installation candidate?  I was trying to do  sudo apt-get install munin-memcached     and I get this message:  Reading package lists... Done Building dependency tree        Reading state information... Done Package munin-memcached is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source E: Package munin-memcached has no installation candidate   I googled for the problem and someone said to do a apt-get upgrade but it still did not solve my problem. Consider the partial sum of the divergent Harmonic series $H_n = \sum\limits_{k = 1}^{n}\frac{1}{k}$. I recently saw a question which required finding out the asymptotic bounds of $H_n$. Now, I could not find any bound closer than $O(n)$. So I looked up the series and found  Wikipedia article which says that $H_n - \ln n$ approaches a constant (the ). This clearly shows that $H_n = \Theta(\log n)$.  However, considering the question was in an beginners undergraduate algorithms course in CS, I was wondering if there is a simpler proof of this? Is there some clever proof requiring elementary mathematics to show $H_n = \Theta(\log n)$?  PS: This is not homework in the sense that it is not my homework, and the actual question does not ask for a proof, but rather asks a lot of functions to be arranged asymptotically. I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?   The standard rule you always hear:     “If a person knows which item you are talking about then use "the"   . . . doesn’t clear things up for me, as I have no idea whether or not they know. Is there PvP in Diablo 3? How does it work? How can Obito use his Mangekyo so often? Right-click on Touchpad not working after update to 18.04 LTS Where's the Rainbow Jetpack gone in Jetpack Joyride? Same question listed twice in reputation list This might not be a bug but I see the same question list twice in my reputation display, I know it is grouping things by a six hour difference, but is this really expected?    The first list says 2 events but shows all 5, similarly the second list says 3 events and shows all five.