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A complete solution with references can be found in Section 8.1 of Brualdi, Combinatorial Matrix Classes, Cambridge University Press, 2006. Here is how to make an extreme point, and all extreme points …

This paper has a reference to a positive answer to the first question.

(Too long for a comment). Note that there are two different ways to define "all the embeddings" of a graph, both completely natural. I'll illustrate by example. Suppose a graph consists of two triangl …

Given that an upper bound is requested, here is one. By symmetry one can replace $\frac1{k_1}+\cdots+\frac1{k_m}$ by $\frac1{k_1}$ and multiply everything by $m$. Now do the sum over $k_1$ (which I'll …

In Ref 1 we called these things semi-Eulerian circular designs or semi-Eulerian quasigroups. They exist for all odd $n\ge 7$. I'll state the problem again. Find an Eulerian circuit in $K_n$ which (co …

If I understand it correctly, there is a counterexample on page 1518 of this paper. Then again, I might not have read it carefully enough.

The complement of a matching generalises. Take $1<k<|X|$ and identify $Y$ with the $k$-subsets of $X$. Let $R(x)$ be the $k$-subsets containing $x$. Note that the automorphism group acts as the symmet …

About 58% of the graphs on 12 vertices have diameter 3, so filtering a complete generation will be as fast as any. On 20 vertices the fraction has dropped to about 31% but the total is so vast that g …

There will be strongly-regular graphs of the same parameters with equal values of all those invariants. Since the parameters determine the eigenvalues, all the invariants determined by the spectrum (s …

All simple non-empty regular graphs of even degree have a two factor, see here . So you are just asking when they have a 1-factor. In addition to having an even number of vertices, the conditions are …

As $k\to\infty$, $\binom{2k-2}{k-1}\sim 2^{2k-2}/\sqrt{\pi k}$. As $N-k\to\infty$, $\binom{2N-2k}{N-1}\sim 2^{2N-2k}/\sqrt{\pi (N-k)}$. Now approximate the sum by an integral: $$\int_0^N \frac{\ln k} …

(A comment rather than an answer.) Here is a plot of $a_n/n^5$ (red) and $b_n/n^5$ (blue). It might not go far enough to show the asymptotic behaviour, but a possibility is that $a_n$ and $b_n$ are as …

(This is a comment, not an answer.) If $f_n(x)$ is your polynomial, starting with $f_0(x)=1$, then $$ \sum_{n=0}^\infty f_n(x) y^n = \frac{1-xy+x^2y^2+x^2y^3}{(1+xy^2)(1-xy-xy^2)} = 1 + \frac{x …

Let $q$ be a prime power and let $P$ be a projective plane of order $q$. It has $q^2+q+1$ points and $q^2+q+1$ lines. Each point lies on $q+1$ lines, and each line has $q+1$ points. Each pair of lines …

As far as I am aware, there is no such program. Also, it needs care to interpret gradpart's claims. Gradpart can make counts greater than one graph per machine instruction, which proves that it doesn' …