Problems

The dissipative barrier method suppresses spectral pollution when a differential operator is truncated, but can it hide genuine spectral points? Known as the graveyard problem, the question stayed open in dimension two and above for more…

Davis, Figiel, Johnson and Pełczyński showed their interpolation space admits a Schauder basis when the range space has a shrinking one. Can the DFJP space always be chosen with a basis whenever the range space has a basis? The paper…

The Elton–Odell theorem gives, in every infinite-dimensional normed space, a unit-sphere sequence with mutual distances at least 1+ε. Over C, identifying vectors differing by a unimodular scalar gives a toroidal distance. Does every…

Let p be a complex polynomial of degree n≥2 whose zeros all lie in the closed unit disk. For every zero a of p, there is a critical point ζ satisfying |ζ-a|<1, except when |a|=1 and p is a nonzero scalar multiple of z^n-a^n.

If a smooth bounded domain in R^n admits a Neumann eigenfunction of the Laplacian that is constant on the boundary, must the domain be a ball? Pompeiu posed an equivalent integral-equation form in 1929; Schiffer's 1957 reformulation via…

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≤…

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

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.

There exists t_0 ∈ N such that m_G_t(-∞, -2) is constant for all t ≥ t_0.

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 .