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My answer comes from the random graphs community. In the book Random Graphs, the quantity "edges minus vertices" is called the excess, which is quite standard terminology at least in random graphs. In …

For the special case of the complete graph $K_n$ which you mention in your post, Svante Janson answered your question in this paper; the answer is that the weighted diameter grows like $3 \log n$ in p …

Edit: This argument is wrong, as pointed out in the comments. I'll leave it up as a warning to others. I am going to respond to the question under the assumption that this is about infinite paper (as …

A version of the non-adaptive problem was studied by Uriel Feige, using slightly different language. In his paper, "On Sums of Independent Random Variables with Unbounded Variance, and Estimating the …

If you view the weights as edge lengths then you can view each graph as a metric space, and then use the Gromov-Hausdorff distance between the two metric spaces. This may not be at all suitable for yo …

Edit: I worked out the details of this exponential upper bound a bit more precisely. It is the case that $S(n) \leq 11*10^n$. I can only prove an exponential upper bound (rather than an exponential …

One whole family comes from considering properties that are monotone for connected graphs but can change when the connectivity changes. For example: the diameter of a graph -- defined to be the maximu …

Edit: Erdős got three things wrong. First of all, it wasn't Faudree, Shelp, and Rousseau, it was Faudree, Shelp, and Burr. Second, it wasn't "recently", it was in the future (with respect to the qu …

I think you can get the precise value (well, within an additive error of one) by a sort of limiting argument. Rather than a sequence of coins, for each $i=1,\ldots,n$ let $P_i$ be a Poisson process wi …

I haven't managed to find the answer to precisely your question but here are a couple of references that might be useful. Vershik and Yakubovich have a paper on The limit shape and fluctuations of ra …

Take two asymmetric $d$-regular graphs $H_1,H_2$, and let $G$ be their disjoint union. Then $d$ will be a repeated eigenvalue. If you want $G$ connected, take the complement of the graph obtained by …

This is an interesting question. For any fixed positive integer $d \geq 2$, write $T_d^{\infty}$ for the complete infinite rooted $d$-ary tree (by this I mean every node has exactly $d$ children). Lu …

I emailed Noga to ask him; here is his response (touched up slightly for MO; any errors in what I post are probably mine rather than Noga's). The only details not present are the required applications …

Given a rooted tree $T$ and an integer $k \geq 1$, let $N_k(T)$ be the number of subtrees of $T$ containing the root and having exactly $k$ nodes (take $N_k(T)=0$ if $T$ has less than $k$ nodes). Ne …

A colleague, Nicolas Broutin, who is not on MO, pointed me to a recent "preliminary report" of Gábor Pete, about joint work with Christophe Garban and Oded Schramm, on the scaling limit of the MST. In …