For positive numbers j and k, an L（j,k）-labeling f of G is an assignment of numbers to vertices of G such that ｜f（u）-f（v）｜≥j if uv∈E（G）, and ｜f（u）-f（v）｜≥k if d（u,v）=2. Then the span of f is the di...For positive numbers j and k, an L（j,k）-labeling f of G is an assignment of numbers to vertices of G such that ｜f（u）-f（v）｜≥j if uv∈E（G）, and ｜f（u）-f（v）｜≥k if d（u,v）=2. Then the span of f is the difference between the maximum and the minimum numbers assigned by f. The L（j,k）-number of G, denoted by λj,k（G）, is the minimum span over all L（j,k）-labelings of G. In this paper, we give some results about the L（j,k）-number of the direct product of a path and a cycle for j≤k.展开更多
Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i...Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i） = ｜f＊^-1（i）｜. A labeling f is called friendly if ｜vf（1） - vf（0）｜ ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if（G） = e（1） - el（0）. The set [if（G） ｜ f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI（G）. In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.展开更多
An L（2, 1）-labeling of a graph G is a function f from the vertex set V（G） to the set of all nonnegative integers such that ｜f（x） - f（y）｜ 〉 2 if d（x, y） = 1 and ｜f（x）-f（y）｜ ≥ 1 ifd（x, y） = 2. The ...An L（2, 1）-labeling of a graph G is a function f from the vertex set V（G） to the set of all nonnegative integers such that ｜f（x） - f（y）｜ 〉 2 if d（x, y） = 1 and ｜f（x）-f（y）｜ ≥ 1 ifd（x, y） = 2. The L（2, 1）-labeling number λ（G） of G is the smallest number k such that G has an L（2, 1）-labeling with max{f（v） ： v ∈ V（G）} = k. We study the L（3, 2, 1）-labeling which is a generalization of the L（2, 1）-labeling on the graph formed by the （Cartesian） product and composition of 3 graphs and derive the upper bounds of λ3（G） of the graph.展开更多
The main purpose of this paper is to develop and simplify the general conditions for an s-stage explicit canonical difference scheme of q-th order, while the simplified order conditions for canonical RKN methods which...The main purpose of this paper is to develop and simplify the general conditions for an s-stage explicit canonical difference scheme of q-th order, while the simplified order conditions for canonical RKN methods which are applied to a special kind of second order ordinary differential equations are also obtained here.展开更多
L(2,1)-labeling number of the product and the join graph on two fans are discussed in this paper, we proved that L(2,1)-labeling number of the product graph on two fans is?λ(G) ≤ Δ+3 , L(2,1)-labeling number of the...L(2,1)-labeling number of the product and the join graph on two fans are discussed in this paper, we proved that L(2,1)-labeling number of the product graph on two fans is?λ(G) ≤ Δ+3 , L(2,1)-labeling number of the join graph on two fans is?λ(G) ≤ 2Δ+3.展开更多
While elucidating the inhibitory mechanism of a hydrolytic enzyme by aldehyde-containing natural product,we discovered a reaction involving a rapid 6π-azaelectrocyclization of azatrienes generated from aldehyde with ...While elucidating the inhibitory mechanism of a hydrolytic enzyme by aldehyde-containing natural product,we discovered a reaction involving a rapid 6π-azaelectrocyclization of azatrienes generated from aldehyde with lysine residues.The electrocyclic reaction of the 1-azatriene system,a cyclization precursor,exhibited a significant substituent effect.Asymmetric chiral piperidine synthesis and a one-pot library synthesis of pyridines on solid-supports were applied to synthesize pyridine/indole alkaloid-type natural product.Additionally,we developed lysine-based labeling and engineering of biomolecules and living cells based on the rapid 6π-azaelectrocyclization.Both labels and oligosaccharide structures were introduced efficiently and selectively into surface lysines under the mild conditions;notable effects of N-glycans on proteins and living cells were visualized for the first time by PET and noninvasive fluorescence imaging.展开更多
基金Supported by Faculty Research Grant, Hong Kong Baptist University
文摘For positive numbers j and k, an L（j,k）-labeling f of G is an assignment of numbers to vertices of G such that ｜f（u）-f（v）｜≥j if uv∈E（G）, and ｜f（u）-f（v）｜≥k if d（u,v）=2. Then the span of f is the difference between the maximum and the minimum numbers assigned by f. The L（j,k）-number of G, denoted by λj,k（G）, is the minimum span over all L（j,k）-labelings of G. In this paper, we give some results about the L（j,k）-number of the direct product of a path and a cycle for j≤k.
基金Supported by FRG/07-08/II-08 Hong Kong Baptist University
文摘Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i） = ｜f＊^-1（i）｜. A labeling f is called friendly if ｜vf（1） - vf（0）｜ ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if（G） = e（1） - el（0）. The set [if（G） ｜ f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI（G）. In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.
文摘An L（2, 1）-labeling of a graph G is a function f from the vertex set V（G） to the set of all nonnegative integers such that ｜f（x） - f（y）｜ 〉 2 if d（x, y） = 1 and ｜f（x）-f（y）｜ ≥ 1 ifd（x, y） = 2. The L（2, 1）-labeling number λ（G） of G is the smallest number k such that G has an L（2, 1）-labeling with max{f（v） ： v ∈ V（G）} = k. We study the L（3, 2, 1）-labeling which is a generalization of the L（2, 1）-labeling on the graph formed by the （Cartesian） product and composition of 3 graphs and derive the upper bounds of λ3（G） of the graph.
文摘The main purpose of this paper is to develop and simplify the general conditions for an s-stage explicit canonical difference scheme of q-th order, while the simplified order conditions for canonical RKN methods which are applied to a special kind of second order ordinary differential equations are also obtained here.
文摘L(2,1)-labeling number of the product and the join graph on two fans are discussed in this paper, we proved that L(2,1)-labeling number of the product graph on two fans is?λ(G) ≤ Δ+3 , L(2,1)-labeling number of the join graph on two fans is?λ(G) ≤ 2Δ+3.
基金supported by Grants-in-Aid for Scientific Research No.19681024 and 19651095 from the Japan Society for the Promotion of ScienceGrant-in-Aid for Science Research on Priority Areas 16073222 from the Ministry of Education,Culture,Sports,Science and Technology(MEXT)+2 种基金Molecular Imaging Research Program from MEXT,Matching Fund Subsidy for a Private University,Collaborative Development of Innovative Seeds from Japan Science and Technology Agency(JST)New Energy and Industrial Technology Development Organization(NEDO,project ID:07A01014a)a Research Grants from Yamada Science Foundation
文摘While elucidating the inhibitory mechanism of a hydrolytic enzyme by aldehyde-containing natural product,we discovered a reaction involving a rapid 6π-azaelectrocyclization of azatrienes generated from aldehyde with lysine residues.The electrocyclic reaction of the 1-azatriene system,a cyclization precursor,exhibited a significant substituent effect.Asymmetric chiral piperidine synthesis and a one-pot library synthesis of pyridines on solid-supports were applied to synthesize pyridine/indole alkaloid-type natural product.Additionally,we developed lysine-based labeling and engineering of biomolecules and living cells based on the rapid 6π-azaelectrocyclization.Both labels and oligosaccharide structures were introduced efficiently and selectively into surface lysines under the mild conditions;notable effects of N-glycans on proteins and living cells were visualized for the first time by PET and noninvasive fluorescence imaging.
