Problems

The characteristic set of a path-star tree contains an edge.

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).

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 ≅…

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?

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.

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

Suppose that the graphs Γ_1 and Γ_2 have the same refined spectra. Are their complements cospectral?

It is natural to conjecture that ker(f^*)=Z_d+2^2⊕ Z_d(d+2)^β(G)-2 holds for non-bipartite graphs in general.

Let k ≥5 be an odd integer and G be a (n,d,λ) -graph satisfying d^k-1≫ λ^k-2 . Then G has global resilience (1 / 4+o(1)) n d with respect to being C_k -free.

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)

No two non-isomorphic H-shape trees are L-cospectral.

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].

Let S be the parameter matrix of a k-transversal in a d-uniform r-regular hyper graph G. Then the characteristic polynomial of S is φ(λ)=λ^d-2∏_i=j^d(λ-ξ^jkr), where ξ is a d-th primitive root of unity.