Problems

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

Can we find the limit?

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≤…

A graph is irreducible by Y-Δ moves, pendant removal, self-edge removal, parallel reductions, series reductions, antenna jumping, and antenna absorption if and only if it has three medial strands which pairwise intersect twice, there is a…

Conjecture 16. The core pattern μ_n(n=8,10,12,...) is copied in the second and third subsegments of the pattern μ_n+2 .

Suppose that G ∈ G^r . Is λ^(p)(G) continuously differentiable for p>r ? Is λ^(p)(G) continuously differentiable for p \ne k, k=2, ..., r ?

Let G be a connected non-transmission-regular graph with n vertices. Then D_1 - λ_1(D) > 1/n+1.

Another problem worth mentioning is whether the lower bound for c_2(G) still holds without the regularity assumptions, i.e. if we only assume that the graph has large girth and the degree of each vertex is greater than 2.

Clearly a(v) \le \bar{a}(v) and we conjecture that a(v) = \bar{a}(v) based on empirical observations.

Let (G, p) be a generic framework in R^d. If (G, p) is globally (d, k)-rigid and G is not complete, then there exists σ ∈ ker DR_k(G, p)^T such that rank Ω(σ) = |V| − d + k − 1.

Let Γ be a connected t-valenced graph with two main and two plain eigenvalues. There exists a positive integer C such that if t ≥C, then Γ is a strong graph.

Theorem 5.1, which produces infinitely many graphs with Δ≥4, leaves two open problems. The first is to determine whether the inequality in the theorem could be improved to a statement of equality.

Let G be a connected graph of order n. If 1/2<α<1 , then λ_n(A_α(G))≥ λ_n(A_α(K_1,n-1)),the equality holds if and only if G ≅ K_1,n-1 .

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.