Search Results

An answer which seems snide (but isn't!) is $$P(G)=x^k$$ where $k$ is the number of connected components. Another which is possible but maybe not as reasonable is the constant polynomial $Q(G)=k.$ Re …

I think that you need to formulate a more specific question. For fixed $k$, the $n$ is fairly irrelevant. Let $g(n)$ be a non-decreasing function which increases to infinity but exceedingly slowly (su …

This is an interesting question and more subtle than it first appears. One question is the behavior of minimal cost spanning tree for the complete graph. For the question at hand it is worth first con …

What is the purpose of the variable $n$ in your notation? If I understand correctly, we could say that you have a bipartite graph with two vertex classes of size $n$ and each of the $n^2$ possible ed …

I think you would need a condition something like $|E|<(1+(1-\epsilon)\ln(V))V$. If $|E|=3|V|$ then it could be that every vertex is on $6$ $3$-cycles. That is only one such graph but I would expect t …

I think perhaps the problem with the variance is not the overlap of the different trees but the fact that the number of spanning trees can be much much larger than 1 but not much less. With $p=\frac …

I will assume that $\Gamma$ is regular of degree $1$ (so $n=2m$ for some $m$.) You say that you don't care specifically about the uniform distribution on $S_n=S_{2m}.$ However I will deal mainly with …