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If s > t ≥ 1 with (s, -t) ≠ (2, -1) and n, r ∈ P then ⌊ ( ∑{k=n}^{∞} 1/{rk}{s,-t} )^{-1} ⌋ = {rn}{s,-t} - {r(n-1)}{s,-t} - 1. If t = -1 and s, n, r ∈ P then ⌊ ( ∑{k=n}^{∞} 1/{rk}{s,-1}^2 )^{-1} ⌋ = {rn}{s,-1}^2 - {r(n-1)}{s,-1}^2 - 1.

Let d: V → Z_0^+ be a symmetric function that satisfies d(∅) = 0 and ∀ X, Y ⊆ V (19) and (20). Let hatR: V → Z_0^+ be an even valued, symmetric, skew-supermodular function. Suppose that hatR(X) ≤ d(X) ∀ X ⊆ V. Then there exists a pairing M…

For any sequence s_0,...,s_n of non-negative integers satisfying \sum s_i=\sum i s_i=n, there exist i, j with 1\le i, j and i+j\le n so that n \left( {i+j \atop i} \right)s_{i+j}\geqslant e s_i s_j.

An interesting question is whether every red-blue coloring of a k-pseudorandom graph contains a monochromatic path of length \Omega(\frac{n}{\sqrt{k}}).

If T is a k-peripheral tree and G is a nontrivial connected graph, then aw(T square G,k)=k .

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.

if n is odd then there exist isometric embeddings of \frac{1}{2}H_{n} in the half-spin Grassmann graphs of \Pi whose images are not apartments.

If r is an integer that is not a part in either partitions λ or μ , then L_λ+(r),μ+(r)(q)≥ L_λ μ(q).

In particular, what is the value L_{k,S} = \lim_{n \to \infty} b_{k,S}(n)^{1/n}?

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).

We make the analogous conjecture here, that we may have both t and d in Theorem3.2, provided deg(d)=3.

In fact, what can be said (in general) about the connected components of these graphs: are they all paths or cliques ?

If ζ(α) = ζ(β), are α and β necessarily related by switching?

If L_S = 1, then S = \varnothing.

Possibly, however, it holds whenever G succcurlyeq H and H is transitive; this is not hard to verify when H is an edge.

The number of finite non-cyclic self 2-distance graphs with no induced subgraphs isomorphic to a square, a diamond, a complete graph with four vertices, or a butterfly is finite.

Is it true that the edges of any graph G with minimum degree d can be partitioned into pairwise disjoint sets, so that each set forms a spanning star forest of G in which every component is of size at least h(d), where h(d) tends to…

Let G be an edge-colored 2-connected graph on n vertices satisfying Fan's condition (see [9]), i.e., maxd(u), d(v) ≥ n/2 for every pair of vertices u, v of G with dist(u, v) = 2. Can G contain a compatible spanning circuit visiting each…

We further conjecture that an elliptic quadric is incident with m modulo q points of an m-ovoid of Q(4, q).