Problems

We have max_t m_H_t(2, ∞) = max_t m_G_t(2, ∞) and max_t m_H_t(-∞, -2) = max_t m_G_t(-∞, -2).

We believe that results similar to Theorem 4.3.5 can be proved for intersection graphs of other scalable objects. In particular, we conjecture that similar techniques apply to intersection graphs of (unit) regular hexagons.

For k,n,α ∈N , let G ∈ G_n,α have the minimum spectral radius in G_n,α . Then for sufficiently large n, (1) G ≅ F(k,k,k+1) for α=3,n=3k+1, (2) G ≅ F(k+1,k,k+1) for α=3 and n=3k+2, (3) G ≅ F(k,k,k,k+1) for α=4 and n=4k+1, (4) G ≅…

Characterize non-König-Egerváry graphs satisfying; • varrho_e(G) ≥ m(G) - ξ(G) + ε(G); • varrho_v(G) = n(G) - ξ(G) + ε(G); • varrho_v(G) = α(G) + μ(G) - ξ(G) + ε(G).

Let t_k,q(n) be defined by (5.1) t_k,q(j) = q^j, for 0 ≤ j ≤ k-1 t_k,q(n) = q ∑_l=0^k-2 (q-1)^l t_k,q(n-(l+2)), for n ≥ k. Then, S_F_q(T_2,3,…,k(n)) = t_k,q(n) for all values of n ≥ k.

Characterize the (m, n)-extremal graphs for all m and n.

We believe the assertion of Theorem 1.6 is true even for all smaller values of n, though we don't have a proof yet.

It would be interesting to determine whether there are schemes for which the fusing-relations graph and fusing-idempotents graph are not isomorphic.

For any integer n ≥ 6, we have β_b(K_1 ⊙ P_n) = ⌈ n/6 ⌉ + 1

Let C be a projective [n,k]_q three-weight code with non-zero weights w_1 < w_2 < w_3 satisfying w_1 + w_2 + w_3 = 3(1-1/q)n. Then w_2 = (1-1/q)n. Moreover, w_1 = w_2 - t and w_3 = w_2 + t, where t is a power of the characteristic p of F_q.

We conjecture that a version of this representation result holds even for convex geometries in which the empty set is not closed.

In particular, is it true that if the realization |Γ| of Γ through its direct complex Δ(Γ) is a manifold, then the realization of its partial dual |Γ^S| is also a manifold?

For any graph G on n ≥ 4 vertices, ▷ ∂{2}^{L}(G)≥n with equality if and only if G is the complete graph K{n} or K_{n} minus an edge; ▷ if n ≠7, then ∂{2}^{L}(G)≤∂{2}^{L}(P_{n}) with equality if and only if G is the path P_{n} ; ▷ if G is a…

Determination of ξ_G(λ_χ) for general chromatic characteristic polynomials of all 2-regular bipartite graphs is still in progress.

For the case k>5, Corollary 4.19 gives a necessary condition for what kind of cycles can appear in the promotion action on SYT(sc_k) . We do not know if this condition is sufficient.

Based upon the results generated from our Sage script, we submit as a conjecture that these graphs constructed be the smallest graphs (by order) that have characteristic-dependent well-covered dimension for any given characteristic.

When G is a tree, is it true that the Hilbert stratification of A_G consists of coordinate subspaces? In particular, does f=e^u give a general Hilbert sequence?

We don't think that our bound on the number of k-rich Möbius functions is tight. On the contrary, we conjecture that the right side can be replaced by O(n^4/k^3) in this case too.

This raises the question of how sparse a graph can be for the last Theorem to remain true.

Full characterization/ Examples of strongly minimal k-vertex-rigid graphs in three-space or higher dimensions.