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Find a nontrivial lower bound or upper bound of QMX(n).

Ehrhart equivalence is a necessary and sufficient condition for (not necessarily finite or rational) discrete equidecomposability.

Is it true that for every nonnegative integer k, there exists a connected graph G satisfying φ(G) − κ(G) + 1 = k?

Is there a nice combinatorial proof for the number of interior lattice points of P_n(132,312) ?

We discuss ... including some new ones that we present in this last section (in particular Conjectures 11.3 and 11.5).

φ(n, m) = n(m-1) + 1.

For k a non-negative integer and s, t ≥ k, the coefficient of x^{2s-k}e^{tx} in f_{s+t-1}(x) is given by [x^{2s-k}e^{tx}] f_{s+t-1}(x) = (-1)^{s} \frac{t^{2s+2t-2k-2}}{2^{s-k}\cdot (s-k)! (t-k)!} \cdot Q_{k}(s,t), where Q_{k}(s,t) is a…

Can Theorem 4.2 be true for dimension ≥ 4 ?

We conjecture that the necessary conditions are sufficient in general, except eventually for a few values (for example it can be shown that K_{4,4,4,1} cannot be decomposed into K_4's).

The complement of multicone graphs K_w ∇ L(P) are DS with respect to their signless Laplacian spectrum.

W(n,k,4)=0 if either (i) n<14 or (ii) n=14,k<14.

Can we find the limit?

Do either the height 1 or Hilbert basis extensions generate the same poset Cones(d)?

Let G ∈ G_3(n,n-3) be a graph of order n ≥ 6. Then the following cases hold: i) if β<α<0<γ<ρ , then G is Seidel equivalent to K_i,j∪ barK_p ; ii) if ρ<γ<0<α<β , then G is Seidel equivalent to overlineK_i,j∪ barK_p ,where 1≤ i ≤[n/3], i≤ j≤…

We conjecture that if a Γ -degree sequence d' has a tree realization then each such a realization of d' has the same number of pendant vertices.

If D is a digraph of order n, then d_I(D) + d_I(barD) ≤ n + 1.

For all primitive digraphs G such that G ≠ K_n^*, exp(G)/l(G) ≥ 2.

The construction described above has Ω(n^6) crossings. Does there exist a cycle of small area in every drawing of K_n such that every pair of edges intersect a constant number of times?

By (10) they occur in inverse pairs, with 1 an eigenvalue for all odd n; how big is the largest?

The +1 in (3) can be replaced by +1 / 2 (which is best possible).