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

If M and M / e are both non-degenerate, then Q_M(t) interlaces Q_M/e(t).

Specifically, is there an absolute positive constant c so that any connected graph with minimum degree at least d contains a spanning tree in which the degree of any non-leaf is at least cd/log d?

If χ(G-v)<χ(G) then α_*(G-v)≤α_*(G) .

What about the number of real-valued zonal spherical functions?

Under which conditions a)d(G_1,G_2)=q_1+q_2+|p_1-p_2|-2, b)d(G_1,G_2)=q_1+q_2+|p_1-p_2|-4 hold?

Let G be a graph. If r is an integer root of D_t(G, x), then r ∈ -3, -2, -1, 0.

If F = F_q is a finite field of odd size or F = F_∞ is an algebraically closed field of characteristic zero, then every graph Γ_F(f_2, f_3) of girth at least eight is isomorphic to Γ_3(F) = Γ_F(xy, x^2y).

It would be very interesting to prove the full version of Theorem 4.1 in a similar manner.

There exists a chain complex of finitely generated FI-modules C_* such that H_k(C_*) = H_k(K_p(S_•)).

Is it true, for example, that all values S(d, k) are even? In other words: If an odd subset of E_d is sliced by k hyperplanes, can one always add another edge to this set?

Let D be a k-arc-connected digraph and let l ≤ k. If (x_1, f_1, y_1), …, (x_l, f_l, y_l) are l triples such that x_1, …, x_l, y_1, …, y_l ∈ V(D) (not necessarily distinct) and f_i ∈ E^+(x_i) (respectively f_i ∈ E^-(y_i)), i = 1, …, l, then…

Suppose that k>ℓ . Then satex(n,P_k:m,P_ℓ) is attained asymptotically on the quasi-star or the quasi-clique.

How closely related are σ^2 and λ ? In particular, is it true that λ=Θ(σ^2) (that is, are there bounds on the ratios λ/σ^2 and σ^2/λ )?

If G is a subcubic planar graph drawn without any faces of length 5, then χ(G^2)≤ 6.

It is however conjectured that one of these constructions either “K-groupings” or “J-groupings” will yield a maximum independent set for a given I-graph.

Conjecture 21 states that the equality between the summations in Theorems 19 and 20 holds term-by-term.

There are no regular self 2-distance graphs of odd degree.

It would be interesting to see if a certain multivariate generating polynomial of Mahonian–Eulerian statistics, such as ∑{π∈𝔖_n} q^maj(π) ∏{i∈𝒟(π)} x_{π_i}, is stable.

For any n, r ∈ N with r ≥ 3 and n ≥ (r-1)^2 + 1, the set MUC(n, r) is not empty.