Báo cáo toán học: "Eigenvectors and Reconstruction" pps

... [HE1] for A and B. Without loss of generality, suppose T = (t 1 , t 2 ) ⊆ R − . Let t ∈ T and let µ l (t) be the µ l in Theorem 3 for A and B. Notice that the lowest eigenvectors of A + tJ and B + ... matrices. Suppose that A and B are hypomorphic. Let λ i be a simple eigenvalue of A and B. Let p i = (p 1,i , p 2,i , . . . , p n,i ) t be a unit vector in eigen λ i (A) and q i =...

Ngày tải lên: 07/08/2014, 13:21

Ngày tải lên: 05/08/2014, 15:20

... common prefix of ϕ (x )and (y) has the same length as the maximal common prefix of x and y. In particular, the vertices x and y are adjacent (have distance 1) if and only if k = l +1andy = x 0,i 1 , ... |z| where z is the maximal common prefix of x and y. With this distance, we find that two vertices x and y are connected by an edge if and only if d Γ (x, y) = 1, and in this case, w...

Ngày tải lên: 07/08/2014, 08:22

... at z =0, and upper and lower subgraphs G upper (at z>0)andG lower (at z<0). The edges on the central cycle are longer than all other edges. upper subgraph G upper (with z>0) and in the ... n–gon, and for inner vertices because of the convexity assumption. Lemma 3. Let v be a vertex of G. There is a right rising and a left rising path from v to a positive pole, and also bo...

Ngày tải lên: 07/08/2014, 13:21

... A. V. Kostochka and D. R. Woodall, Choosability conjectures and multicircuits, Discrete Math. 240 (2001), 123–143. [9] W. Wang and K W. Lih, Choosability, edge choosability, and total choosability ... 1(b), where only x and y are incident with edges of G that are not shown (so that w, and v if present, have degree 2 in G and not just in B; that is, z 0 /∈ {u, w} in Fig. 1(a) an...

Ngày tải lên: 07/08/2014, 13:21

... By the symmetry between y and z, we can suppose that α ∈ ¯ϕ(y). By (a) and (b), there is a colour β ∈ ¯ϕ(x) and an edge e  of colour β joining y and z. Uncolour e  and colour e z with β. This ... Partitions and Edge Colourings of Multigraphs Alexandr V. Kostochka ∗ and Michael Stiebitz † Submitted: May 23, 2007; Accepted: Jul 1, 2008; Published: Jul 6, 2008 Abstract Erd˝os an...

Ngày tải lên: 07/08/2014, 15:22

... · u k and v = v 1 · · · v ℓ of respective lengths k and ℓ. If alph(u)∩ alph(v) = ∅, then u ≺ v = σ ≺ τ (106) where σ = std(u) and τ = std(v)[k] if u k < v ℓ , and σ = std(u)[ℓ] and τ = ... k or µ −1 (k + τ i−k ) otherwise, and where x i and y i are defined, for i  k, as the x i and y i associated to σ as in Lemma 4.7, and, for i > k, as x i−k and y i−k associated...

Ngày tải lên: 08/08/2014, 01:20

... graph K and vertex set S ⊆ V (G), we say that S is (G, +v, K) maximal if and only if K  G[S] and K ⊆ G[S ∪ {v}], for every vertex v ∈ V (G) − S; we say that G is (+e, K) maximal if and only ... , 9}] ∼ = C 9 . Graphs F 1 and F 2 are shown in Figures 1 and 2 . Our computational approach is based on the following lemmas and observations. Lemma 1. For r  2 and positive integers...

Ngày tải lên: 08/08/2014, 11:20

... (18.4) ↓ = 13.2 and (18.4.67) ↓ = 15.2.34. The coproduct ∆ and counit ε on N are given, respectively, by ∆(m A ) =  B · ∪C=A m B ↓ ⊗ m C ↓ and ε(m A ) = δ A,∅ , where B · ∪C = A means that B and C form ... a, b, c, and d are non-splittable, then ac = bd if and only if a = b and c = d. Together with the r emarks on A ∪∪ B following (23), this implies that if {u 1 , u 2 , . . . ,...

Ngày tải lên: 08/08/2014, 12:23

... v) with corners x, u, y and v, which is just the subgraph of G induced by the union of all those vertices on four geodesics connecting x and u, u and y, y and v, and v and x, respectively. Let ... Theorems 2 and 4. Corollary 5 Every 4-chordal gra ph must be 1-hyperbolic and it has hyperbolicity one if and only i f it contains one of C 4 , H 1 and H 2 as an i sometric subg...

Ngày tải lên: 08/08/2014, 12:23