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71.
72.
《Discrete Mathematics》2019,342(7):1966-1978
We study methods for computing the bridge number of a knot from a knot diagram. We prove equivalence between a geometric and a combinatorial definition of the bridge number of a knot diagram. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity. 相似文献
73.
The well known Zarankiewicz' conjecture is said that the crossing number of the complete bipartite graph Km,n (m≤ n) is Z(m,n), where Z(m,n)=\lfloor\frac{m}{2}\rfloor\lfloor\frac{m-1}{2}\rfloor\lfloor\frac{n}{2}\rfloor$\lfloor\frac{n-1}{2}\rfloor$ (for any real number x, $\lfloor x\rfloor$ denotes the maximal integer no more than x). Presently, Zarankiewicz' conjecture is proved true only for the case m≤ 6. In this article, the authors prove that if Zarankiewicz' conjecture holds for m≤9, then the crossing number of the complete tripartite graph K1,8,n is $Z(9, n)+ 12\lfloor\frac{n}{2}\rfloor$. 相似文献
74.
C(6,2)表示由圈C6增加边vivi 2(i=1,…,6,i 2(m od6))所得的图,把边vivi 2叫做C(6,2)的弦,B表示C(6,2)除去一条弦所得到的图,我们确定了B与Pn笛卡尔积的交叉数为5n-1. 相似文献
75.
K S Srivastava Shiv Singh Pratibha Gupta A K Srivastava V Kumar M Husain M K Prasad 《Pramana》1981,17(2):187-192
A possible explanation is given of the chemical shifts of x-ray K-absorption edges of metals when they undergo a chemical
combination and form compounds. It is proposed that when a metal forms a compound its Fermi edge changes. It explains the
numerical order as well as the nature of the chemical shifts. A fairly good agreement between the calculated and observed
values has been obtained. 相似文献
76.
Géza Tóth 《Discrete and Computational Geometry》2008,39(4):791-799
The crossing number
of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number
is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number
is the minimum number of pairs of edges that cross an odd number of times. Clearly,
. We construct graphs with
. This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte.
Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k
2/log 2
k).
G. Tóth’s research was supported by the Hungarian Research Fund grant OTKA-K-60427 and the Research Foundation of the City
University of New York. 相似文献
77.
In 15 , Thomassen proved that any triangle‐free k‐connected graph has a contractible edge. Starting with this result, there are several results concerning the existence of contractible elements in k‐connected graphs which do not contain specified subgraphs. These results extend Thomassen's result, cf., 2 , 3 , 9 - 13 . In particular, Kawarabayashi 12 proved that any k‐connected graph without K subgraphs contains either a contractible edge or a contractible triangle. In this article, we further extend these results, and prove the following result. Let k be an integer with k ≥ 6. If G is a k‐connected graph such that G does not contain as a subgraph and G does not contain as an induced subgraph, then G has either a contractible edge which is not contained in any triangle or a contractible triangle. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:97–109, 2008 相似文献
78.
C(n, k) is a graph obtained from n-cycle by adding edges v
i
v
i+k
(i = 1, 2,...,n, i + k (mod n)). There are several known results on the crossing numbers of the Cartesian products of C(n, k) (n ≤ 7) with paths, cycles and stars. In this paper we extend these results, and show that the crossing number of the Cartesian
product of C(8, 2) with P
n
is 8n.
Yuanqiu Huang: Research supported by NSFC (10771062) and New Century Excellent Talents in University (NCET-07-0276).
Jinwang Liu: Research supported by NSFC (10771058) and Hunan NSFC(O6jj20053). 相似文献
79.
Let Γ be a distance-regular graph of diameter d≥2 and a
1≠0. Let θ be a real number. A pseudo cosine sequence for θ is a sequence of real numbers σ
0,…,σ
d
such that σ
0=1 and c
i
σ
i−1+a
i
σ
i
+b
i
σ
i+1=θ
σ
i
for all i∈{0,…,d−1}. Furthermore, a pseudo primitive idempotent for
θ is E
θ
=s ∑
i=0
d
σ
i
A
i
, where s is any nonzero scalar. Let
be the characteristic vector of a vertex v∈VΓ. For an edge xy of Γ and the characteristic vector w of the set of common neighbours of x and y, we say that the edge xy is tight with respect to
θ whenever θ≠k and a nontrivial linear combination of vectors
,
and Ew is contained in
. When an edge of Γ is tight with respect to two distinct real numbers, a parameterization with d+1 parameters of the members of the intersection array of Γ is given (using the pseudo cosines σ
1,…,σ
d
, and an auxiliary parameter ε).
Let S be the set of all the vertices of Γ that are not at distance d from both vertices x and y that are adjacent. The graph Γ is pseudo 1-homogeneous with respect to
xy whenever the distance partition of S corresponding to the distances from x and y is equitable in the subgraph induced on S. We show Γ is pseudo 1-homogeneous with respect to the edge xy if and only if the edge xy is tight with respect to two distinct real numbers. Finally, let us fix a vertex x of Γ. Then the graph Γ is pseudo 1-homogeneous with respect to any edge xy, and the local graph of x is connected if and only if there is the above parameterization with d+1 parameters σ
1,…,σ
d
,ε and the local graph of x is strongly regular with nontrivial eigenvalues a
1
σ/(1+σ) and (σ
2−1)/(σ−σ
2). 相似文献
80.
Yishen Liu Prof. Meijia Gu Qihang Ding Zhiyun Zhang Wanxia Gong Yuncong Yuan Xiaofei Miao Prof. Huili Ma Prof. Xuechuan Hong Prof. Wenbo Hu Prof. Yuling Xiao 《Angewandte Chemie (International ed. in English)》2023,62(13):e202214875
Despite significant effort, a majority of heavy-atom-free photosensitizers have short excitation wavelengths, thereby hampering their biomedical applications. Here, we present a facile approach for developing efficient near-infrared (NIR) heavy-atom-free photosensitizers. Based on a series of thiopyrylium-based NIR-II (1000–1700 nm) dyads, we found that the star dyad HD with a sterically bulky and electron-rich moiety exhibited configuration torsion and significantly enhanced intersystem crossing (ISC) compared to the parent dyad. The electron excitation characteristics of HD changed from local excitation (LE) to charge transfer (CT)-domain, contributing to a ≈6-fold reduction in energy gap (ΔEST), a ≈10-fold accelerated ISC process, and a ≈31.49-fold elevated reactive oxygen species (ROS) quantum yield. The optimized SP@HD-PEG2K lung-targeting dots enabled real-time NIR-II lung imaging, which precisely guided rapid pulmonary coronavirus inactivation. 相似文献