Problems

Moreover, we conjecture that the only finite singularities of Φ_q(t) are of the form q^m/(q-1), m ≥ 1.

Let G be a graph with order n and size m. Then λ_n(A_1/2(G))≥ m/n-1-n-2/2.

Let G* be a maximum bipartite minor of a graph G as defined in Thm. 3.29. Is there a generalized Laplacian matrix M(G) such that an eigenfunction of M(G) has |V(G*)| weak nodal domains?

If G is a connected graph of order n ≥ 4 and ρ_ABC(G) ≤ √2, then G ∈ P_n, C_n, S_4.

For any 1 ≤ α ≤ 6, p_∞^(α) := lim_n → ∞ p_n^(α) exists and is given by: p_∞^(1) = 1/π (6.1) p_∞^(2) = p_∞^(5) = 1/2 - 1/π (6.2) p_∞^(3) = p_∞^(4) = 2/π - 1/2 (6.3) p_∞^(6) = 1 - 3/π (6.4)

It would be interesting to know if there exists a regular/vertex-transitive self-complementary graph Γ on n vertices with the second eigenvalue in the bounds frac√n(n-4)-12 < λ_2 ≤ n-7/2 - 2cos(π(n-1)/n).

Possibly, however, it holds whenever G succcurlyeq H and H is transitive; this is not hard to verify when H is an edge.

The case when k is odd was posed as an open problem by Nikiforov [53].