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≤…

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…

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.

For 0<α≤ 1 , among all trees, characterize the tree which has the maximum generalized distance spectral radius.

Let G' be a graph obtained from a (Δ,1) -bidegreed graph G by inserting a bouquet in an edge of a bouquet internal path. Then (i) if ρ(G)<1+√Δ-1, then ρ(G^′)>ρ(G); (ii) if ρ(G)>1+√Δ-1, then ρ(G^′)<ρ(G); (iii) if ρ(G)=1+√Δ-1, then…

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.

Let Y and Z be t-cross-intersecting sets in G_n whose sizes meet the bound in Theorem 1.2. If n is sufficiently large compared to t, then Y = Z and Y or Y^T is a t-coset.

(Brill-Noether Existence for ℝ-Divisors on Graphs) Let ρ(g,r,d)=g-(r+1)(g-d+r). Fix two real numbers r ≥0, 2 g-2 ≥d. If ρ(g,r,d)≥0 then there exists an ℝ-divisor of degree at most d and rank equal to r on G.

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.

(Covering Radius Conjecture) Let λ ∈ [1/g, g] (recall that g ≥ 1). The covering radius of N_G with respect to the polytope P_{1,λ} is at least √(g/λ)/n where n is the number of vertices of G.

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 .

Give a bound R depending on some invariants of the simplicial complex Δ such that for r ≥R the polynomial h^sd^r(Δ)(t) has only real roots.

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