Consider a graph  G on 200 vertices, created by adding the following edges $(v_1, v_{101}), (v_2, v_{102})$ to the disjoint union of two complete graphs $K_{100}$ with respective vertices $v_1,...v_{100}$ and $v_{101},...,v_{200}$. What is the number of spanning trees for graph G?   I know that $t(K_n) = n^{n-2}$ according to Cayley's formula.   But I'm not sure about the disjoint union, does that mean that after performing a disjoint union, I get a discontinuous graph with 2 components, where each component is a complete graph $K_{100}$, that is then made continuous by the edge addition?  And if that is the case, would the answer then be $t(G) = 2\cdot(t(K_{100})^2)$?
I was doing graph theory homework. I was counting the spaning trees of graph G. Such that G is two complete graphs on n vertices connected by two edges. I got to this expression: $$2n^{2\left(n-2\right)}+2\left(n^{\left(n-2\right)}\sum_{i=0}^{n-2}\binom{n-2}{i}\left(1+i\right)^{\left(i-1\right)}\left(n-i-1\right)^{\left(n-i-3\right)}\right)$$ I know that this expression gives the correct result, however, I was told by my teacher that it is possible to express that number in closed form e.g. without the summation. However, I have yet to find a solution.
Proof that the coefficients in an OLS model follow a t-distribution with (n-k) degrees of freedom Categories other than "off-topic" should allow custom close reasons Custom close reasons are a great boon to Stack Exchange sites, allowing the community to provide more tailored advice to site-specific situations. Their big problem (aside from being limited to three by default) is that they are all lumped under "off-topic". This is often confusing.  I suggest that all close categories but "exact duplicate" should support customized close reasons &mdash; not just the "off-topic" category.  For example, the recommendation close reason     Questions asking us to recommend or find a tool, library or favorite off-site resource are off-topic for Stack Overflow as they tend to attract opinionated answers and spam.   This would make more sense as a child of primarily opinion-based.    Similarly, the "why doesn't this work" close reason     Questions seeking debugging help ("why isn't this code working?") must include the desired behavior, a specific problem or error and the shortest code necessary to reproduce it in the question itself. Questions without a clear problem statement are not useful to other readers. See: .   This would make more sense as a child of unclear what you're asking.  Similar examples can be given for pretty much any of the Stack Exchange sites. Does Two-Face die at the end of The Dark Knight? Does the Two-Face die in the final scene of the Dark Knight? If he does, then how does he become one of the long lasting rival of Batman? Does it have or has? Show alert popup on standard page layout on click of standard button "Save" Residues of $z^2\sin(\frac{1}{z})$ Which is better: side quests after killing the warrior or Playthrough 2? 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