An L（3, 2, 1）-labeling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that ｜f（u）-f（v）｜≥3 if dG（u,v） = 1, ｜f（u）-f（v）｜≥2 if dG（u,v） = 2, and ｜f（u...An L（3, 2, 1）-labeling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that ｜f（u）-f（v）｜≥3 if dG（u,v） = 1, ｜f（u）-f（v）｜≥2 if dG（u,v） = 2, and ｜f（u）-f（v）｜≥1 if dG（u,v） = 3. The L（3, 2,1）-labeling problem is to find the smallest number λ3（G） such that there exists an L（3, 2,1）-labeling function with no label greater than it. This paper studies the problem for bipartite graphs. We obtain some bounds of λ3 for bipartite graphs and its subclasses. Moreover, we provide a best possible condition for a tree T such that λ3（T） attains the minimum value.展开更多
Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are...Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.展开更多
Given a graph G and a positive integer d, an L( d, 1) -labeling of G is afunction / that assigns to each vertex of G a non-negative integer such that |f(u)-f (v) | >=d ifd_c(u, v) =1;|f(u)-f(v) | >=1 if d_c(u, v...Given a graph G and a positive integer d, an L( d, 1) -labeling of G is afunction / that assigns to each vertex of G a non-negative integer such that |f(u)-f (v) | >=d ifd_c(u, v) =1;|f(u)-f(v) | >=1 if d_c(u, v) =2. The L(d, 1)-labeling number of G, lambda_d(G) is theminimum range span of labels over all such labelings, which is motivated by the channel assignmentproblem. We consider the question of finding the minimum edge span beta_d( G) of this labeling.Several classes of graphs such as cycles, trees, complete k-partite graphs, chordal graphs includingtriangular lattice and square lattice which are important to a telecommunication problem arestudied, and exact values are given.展开更多
L (2, 1)-labeling number, λ(G( Z , D)) , of distance graph G( Z , D) is studied. For general finite distance set D , it is shown that 2D+2≤λ(G( Z , D))≤D 2+3D. Furthermore, λ(G( Z , D)) ≤8 when...L (2, 1)-labeling number, λ(G( Z , D)) , of distance graph G( Z , D) is studied. For general finite distance set D , it is shown that 2D+2≤λ(G( Z , D))≤D 2+3D. Furthermore, λ(G( Z , D)) ≤8 when D consists of two prime positive odd integers is proved. Finally, a new concept to study the upper bounds of λ(G) for some special D is introduced. For these sets, the upper bound is improved to 7.展开更多
An L(2, 1, 1)-labeling of a graph G is an assignment of non-negative integers(labels) to the vertices of G such that adjacent vertices receive labels with difference at least 2, and vertices at distance 2 or 3 receive...An L(2, 1, 1)-labeling of a graph G is an assignment of non-negative integers(labels) to the vertices of G such that adjacent vertices receive labels with difference at least 2, and vertices at distance 2 or 3 receive distinct labels. The span of such a labeling is the difference between the maximum and minimum labels used, and the minimum span over all L(2, 1, 1)-labelings of G is called the L(2, 1, 1)-labeling number of G, denoted by λ_(2,1,1)(G). In this paper, we investigate the L(2, 1, 1)-labelings of caterpillars. Some useful sufficient conditions for λ_(2,1,1)(T) = △_(2)(T) =maxuv∈E(T)(d(u) + d(v))) are given. Furthermore, we show that the sufficient conditions we provide are also necessary for caterpillars with △_(2)(T) = 6.展开更多
An L(3,2,1)-labeling of a graph G is a function f from the vertex set V(G)to the set of all non-negative integers(labels)such that|f(u)-f(v)|≥3 if d(u,v)=1,|f(u)-f(v)≥2 if d(u,v)=2 and|f(u)-f(v)|≥1 if d(u,v)=3.For ...An L(3,2,1)-labeling of a graph G is a function f from the vertex set V(G)to the set of all non-negative integers(labels)such that|f(u)-f(v)|≥3 if d(u,v)=1,|f(u)-f(v)≥2 if d(u,v)=2 and|f(u)-f(v)|≥1 if d(u,v)=3.For a non-negative integer k,a k-L(3,2,1)-labeling is an L(3,2,1)-labeling such that no label is greater than k.The L(3,2,1)-labeling number of G,denoted byλ3,2,1(G),is the smallest number k such that G has a k-L(3,2,1)-labeling.In this article,we characterize the L(3,2,1)-labeling numbers of trees with diameter at most 6.展开更多
A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this pape...A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this paper, we show that for or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when or 11. Moreover, we show that? if 1) either (mod 6), m is odd, r = 3, or 2) (mod 3), m is even (mod 2), r = 0.展开更多
An L(h,k)-labeling of a graph G is an assignment of non-negative integers to the vertices such that if two vertices u and v are adjacent then they receive labels that differ by at least h, and when u and v are not adj...An L(h,k)-labeling of a graph G is an assignment of non-negative integers to the vertices such that if two vertices u and v are adjacent then they receive labels that differ by at least h, and when u and v are not adjacent but there is a two-hop path between them, then they receive labels that differ by at least k. The span λ of such a labeling is the difference between the largest and the smallest vertex labels assigned. Let λ<sub>h</sub>k</sup> ( G )denote the least λ such that G admits an L(h,k) -labeling using labels from {0,1,...λ}. A Cayley graph of group is called circulant graph of order n, if the group is isomorphic to Z<sub>n.</sub> In this paper, initially we investigate the L(h,k) -labeling for circulant graphs with “large” connection sets, and then we extend our observation and find the span of L(h,k) -labeling for any circulants of order n. .展开更多
Both straw incorporation and irrigation practices affect biological nitrogen(N)fixation(BNF),but it is still unclear how straw incorporation impacts BNF under continuous(CFI)or intermittent(IFI)flooding irrigation in ...Both straw incorporation and irrigation practices affect biological nitrogen(N)fixation(BNF),but it is still unclear how straw incorporation impacts BNF under continuous(CFI)or intermittent(IFI)flooding irrigation in a rice cropping system.A15N2-labeling chamber system was placed in a rice field to evaluate BNF with straw incorporation under CFI or IFI for 90 d.The nif H(gene encoding the nitrogenase reductase subunit)DNA and c DNA in soil were amplified using real-time quantitative polymerase chain reaction,and high-throughput sequencing was applied to the nif H gene.The total fixed N in the straw incorporation treatment was 14.3 kg ha^(-1)under CFI,being 116%higher than that under IFI(6.62 kg ha^(-1)).Straw incorporation and CFI showed significant interactive effects on the total fixed N and abundances of nif H DNA and c DNA.The increase in BNF was mainly due to the increase in the abundances of heterotrophic diazotrophs such as Desulfovibrio,Azonexus,and Azotobacter.These results indicated that straw incorporation stimulated BNF under CFI relative to IFI,which might ultimately lead to a rapid enhancement of soil fertility.展开更多
The L(2,1)-labelling number of distance graphs G(D), denoted by λ(D), isstudied. It is shown that distance graphs satisfy λ(G) ≤Δ~2. Moreover, we prove λ({1,2, ..., k})=2k +2 and λ({1,3,..., 2k -1}) =2k + 2 for ...The L(2,1)-labelling number of distance graphs G(D), denoted by λ(D), isstudied. It is shown that distance graphs satisfy λ(G) ≤Δ~2. Moreover, we prove λ({1,2, ..., k})=2k +2 and λ({1,3,..., 2k -1}) =2k + 2 for any fixed positive integer k. Suppose k, a ∈ N and k,a≥2. If k≥a, then λ({a, a + 1,..., a + k - 1}) = 2(a + k-1). Otherwise, λ({a, a + 1, ..., a + k- 1}) ≤min{2(a + k-1), 6k -2}. When D consists of two positive integers,6≤λ(D)≤8. For thespecial distance sets D = {k, k + 1}(any k ∈N), the upper bound of λ(D) is improved to 7.展开更多
An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference betw...An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an L(0,1)-labelling is the maximum label number assigned to any vertex of G. The L(0,1)-labelling number of a graph G, denoted by λ0.1(G) is the least integer k such that G has an L(0,1)-labelling of span k. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph by L(0,1)-labelling and have shown that, △-1≤λ0.1(G)≤△ for a cactus graph, where △ is the degree of the graph G.展开更多
An L（j, k）-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive integers which are at least j apart, and vertices at distance two receive integers w...An L（j, k）-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive integers which are at least j apart, and vertices at distance two receive integers which are at least k apart. Given an L（j, k）-labeling f of G, define the L（j, k） edge span of f, βj,k（G,f） =max{ ｜f（x）-f（y）｜： {x,y}∈E（G）}. The L（j,k） edge span of G, βj,k （G） is min βj,k（ G, f）, where the minimum runs over all L（j, k）-labelings f of G. The real L（.j, k）-labeling of a graph G is a generalization of the L（j, k）-labeling. It is an assignment of nonnegative real numbers to the vertices of G satisfying the same distance one and distance two conditions. The real L（j, k） edge span of a graph G is defined accordingly, and is denoted by βj,k（G）. This paper investigates some properties of the L（j, k） edge span and the real L（j, k） edge span of graphs, and completely determines the edge spans of cycles and complete t-partite graphs.展开更多
L（ s, t）-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L （s, t）-labeling of a graph G is an assig...L（ s, t）-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L （s, t）-labeling of a graph G is an assignment of integers to the vertices of G such that adjacent vertices receive integers which differ by at least s, and vertices that are at distance of two receive integers which differ by at least t. Given an L（s, t） -labeling f of a graph G, the L（s, t） edge span of f, βst （ G, f） = max { ｜f（u） -f（v）｜： （ u, v） ∈ E（G） } is defined. The L（ s, t） edge span of G, βst（G）, is minβst（G,f）, where the minimum runs over all L（s, t）-labelings f of G. Let T be any tree with a maximum degree of △≥2. It is proved that if 2s≥t≥0, then βst（T） =（ [△/2 ] - 1）t ＋s; if 0≤2s 〈 t and △ is even, then βst（T） = [ （△ - 1） t/2 ] ; and if 0 ≤2s 〈 t and △ is odd, then βst（T） = （△ - 1） t/2 ＋ s. Thus, the L（s, t） edge spans of the Cartesian product of two paths and of the square lattice are completely determined.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
AIM: To evaluate the role of fluorine-18-labeled fluorodeoxyglucose positron emission tomography (18F-FDG PET) in various rheumatic diseases and its potential in the early assessment of treatment response in a limited...AIM: To evaluate the role of fluorine-18-labeled fluorodeoxyglucose positron emission tomography (18F-FDG PET) in various rheumatic diseases and its potential in the early assessment of treatment response in a limited number of patients. METHODS: This study involved 28 newly diagnosed patients, of these 17 had rheumatoid arthritis (RA) and 11 had seronegative spondyloarthropathy (SSA). In the SSA group, 7 patients had ankylosing spondylitis, 3 had psoriatic arthritis, and one had non-specific SSA. Patients with RA were selected as per the American College of Rheumatology criteria. One hour after FDG injection, a whole body PET scan was performed from the skull vertex to below the knee joints using a GE Advance dedicated PET scanner. Separate scans were acquired for both upper and lower limbs. Post-treatment scans were performed in 9 patients in the RA group (at 6-9 wk from baseline) and in 1 patient with psoriatic arthropathy. The pattern of FDG uptake was analysed visually and quantified as maximum standardized uptake value (SUVmax) in a standard region of interest. Metabolic response on the scan was assessed qualitatively and quantitatively and was correlated with clinical assessment. RESULTS: The qualitative FDG uptake was in agreement with the clinically involved joints, erythrocyte sedimentation rate, C-reactive protein values and the clinical assessment by the rheumatologist. All 17 patients in the RA group showed the highest FDG avidity in painful/swollen/tender joints. The uptake pattern was homogeneous, intense and poly-articular in distribution. Hypermetabolism in the regional nodes (axillary nodes in the case of upper limb joint involvement and inguinal nodes in lower limb joints) was a constant feature in patients with RA. Multiple other extra-articular lesions were also observed including thyroid glands (in associated thyroiditis) and in the subcutaneous nodules. Treatment response was better appreciated using SUVmax values than visual interpretation, when compared with clinical evaluation. Four patients showed a favourable response, while 3 had stable disease and 2 showed disease progression. The resolution of regional nodal uptake (axillary or inguinal nodes based on site of joint involvement) in RA following disease modifying anti-rheumatoid drugs was noteworthy, which could be regarded as an additional parameter for identifying responding patients. In the SSA group, uptake in the affected joint was heterogeneous, low grade and nonsymmetrical. In particular, there was intense tendon and muscular uptake corresponding to symptomatic joints. The patients with psoriatic arthritis showed intense FDG uptake in the joints and soft tissue. CONCLUSION: 18F-FDG PET accurately delineates the ongoing inflammatory activity in various rheumatic diseases (both at articular and extra-articular sites) and relates well to clinical symptoms. Different metabolic patterns on FDG-PET scanning in RA and SSA can have important implications for their diagnosis and management in the future with the support of larger studies. FDG-PET molecular imaging is also a sensitive tool in the early assessment of treatment response, especially when using quantitative information. With these benefits, FDG-PET could play a pivotal clinical role in the management of inflammatory joint disorders in the future.展开更多
Background: Attention deficit hyperactivity disorder (ADHD) was long considered to be limited to children and adolescents, but it is now known that ADHD symptoms may persist into adulthood. It is plausible that the et...Background: Attention deficit hyperactivity disorder (ADHD) was long considered to be limited to children and adolescents, but it is now known that ADHD symptoms may persist into adulthood. It is plausible that the etiology of ADHD is not one-dimensional, but instead involves various neuroanatomical and neurochemical systems, with the causes of the main abnormalities believed to be catecholaminergic. Iodine-123-labeled meta-iodobenzylguanidine (123I-MIBG) is a physiological analogue of norepinephrine (NE). To the best of our knowledge, there are no reports about the abnormalities of MIBG scintigraphy in patients with ADHD. The cases presented are male adult patients with different comorbid psychiatric disorders. Case presentation: The cases presented are male adult patients with different psychiatric disorders. Case 1 was a 52-year-old male residential construction foreman, who had been diagnosed with acute schizophrenia-like psychotic disorder. 123I-MIBG planar and SPECTs of the studies were performed at 20 min (early phase) and 3 h (late phase) after intravenous injection of 111 MBq. Planar images were processed to determine the heart (H) to mediastinum (M) ratio (H/M). The early and late H/M ratios were 1.22 and 1.07, respectively. Case 2 was a 52-year-old male civil servant who was diagnosed with adjustment disorder. The early and late H/M ratios were 1.98 and 2.08, respectively. Conclusion: Clinical studies make it clear that symptoms of ADHD are more heterogenous and subtle in adults than children. Adult ADHD is significantly commorbid with wide range of other 12-month disorders. MIBG imaging can be useful to diagnose patients with preexisting psychiatric disorders and ADHD. It might be possible to diagnose objectively though the psychiatric statuses resemble or coexist with other psychiatric disorders. Needless to say, further research is important.展开更多
Objective To study the biochemecal and immunological characterization of the 200 kD schistosomulum surface antigen Method and results A very high molecular weight schistosomulum surface antigen of Mr】200kD was identi...Objective To study the biochemecal and immunological characterization of the 200 kD schistosomulum surface antigen Method and results A very high molecular weight schistosomulum surface antigen of Mr】200kD was identified and characterized using monoclonal antibodies. Carbohydrate modification experiments followed by radioimmunobinding assays demonstrated that the epitope recognised by the mAbs on this antigen was carbohydrate in nature, while protein digestion experiments followed by SDS-PAGE indicated that this antigen also contained protein. Immunoprecipitation of <sup>125</sup>I-labelled cercarial, schistosomulum, adult worm and miracidial surface antigens followed by gel analysis showed the carbohydrate epitope to be present on 5 cercarial, 2 schistosomulum and 5 miracidial surface molecules, and suggested a possible ecological function involved in adapting the parasite to the aquatic free-living stages of its life cycle and possibly also in protecting the early schistosomula from host immune damage. The 5 cercarial surfacs antigens proved to be associated with the CHR, since all the mAbs which recognised those antigens could induce a strong CHR. A kinetic investigation of the carbohydrate epitope on schistosomula of different ages demonstrated a gradual and possibly irreversible loss during the culture period. The epitope completely disappeared from the surface of adult worms. Conclusion To demonstrate an epitope common to a number of surface molecules of various developmental stages of schistosome and therefore explains the immunological cross-reactivity among different stages at the molecular level.展开更多
Given two non-negative integers h and k, an L（h, k）-labeling of a graph G = （V, E） is a function from the set V to a set of colors, such that adjacent nodes take colors at distance at least h, and nodes at distanc...Given two non-negative integers h and k, an L（h, k）-labeling of a graph G = （V, E） is a function from the set V to a set of colors, such that adjacent nodes take colors at distance at least h, and nodes at distance 2 take colors at distance at least k. The aim of the L（h, k）-labeling problem is to minimize the greatest used color. Since the decisional version of this problem is NP-complete, it is important to investigate particular classes of graphs for which the problem can be efficiently solved. It is well known that the most common interconnection topologies, such as Butterfly-like, Beneg, CCC, Trivalent Cayley networks, are all characterized by a similar structure： they have nodes organized as a matrix and connections are divided into layers. So we naturally introduce a new class of graphs, called （l × n）-multistage graphs, containing the most common interconnection topologies, on which we study the L（h, k）-labeling. A general algorithm for L（h, k）-labeling these graphs is presented, and from this method an efficient L（2, 1）-labeling for Butterfly and CCC networks is derived. Finally we describe a possible generalization of our approach.展开更多
An L（2, 1）-labelling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that │f（u） - f（v）│≥2 if dG（u, v） = 1 and │f（u） - f（v）│ ≥ 1 if dG（u, v） = 2. Th...An L（2, 1）-labelling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that │f（u） - f（v）│≥2 if dG（u, v） = 1 and │f（u） - f（v）│ ≥ 1 if dG（u, v） = 2. The L（2, 1）-labelling problem is to find the smallest number, denoted by A（G）, such that there exists an L（2, 1）-labelling function with no label greater than it. In this paper, we study this problem for trees. Our results improve the result of Wang [The L（2, 1）-labelling of trees, Discrete Appl. Math. 154 （2006） 598-603].展开更多
基金The NSF (60673048) of China the NSF (KJ2009B002,KJ2009B237Z) of Education Ministry of Anhui Province.
文摘An L（3, 2, 1）-labeling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that ｜f（u）-f（v）｜≥3 if dG（u,v） = 1, ｜f（u）-f（v）｜≥2 if dG（u,v） = 2, and ｜f（u）-f（v）｜≥1 if dG（u,v） = 3. The L（3, 2,1）-labeling problem is to find the smallest number λ3（G） such that there exists an L（3, 2,1）-labeling function with no label greater than it. This paper studies the problem for bipartite graphs. We obtain some bounds of λ3 for bipartite graphs and its subclasses. Moreover, we provide a best possible condition for a tree T such that λ3（T） attains the minimum value.
基金The National Natural Science Foundation of China(No.10971025)
文摘Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.
文摘Given a graph G and a positive integer d, an L( d, 1) -labeling of G is afunction / that assigns to each vertex of G a non-negative integer such that |f(u)-f (v) | >=d ifd_c(u, v) =1;|f(u)-f(v) | >=1 if d_c(u, v) =2. The L(d, 1)-labeling number of G, lambda_d(G) is theminimum range span of labels over all such labelings, which is motivated by the channel assignmentproblem. We consider the question of finding the minimum edge span beta_d( G) of this labeling.Several classes of graphs such as cycles, trees, complete k-partite graphs, chordal graphs includingtriangular lattice and square lattice which are important to a telecommunication problem arestudied, and exact values are given.
文摘L (2, 1)-labeling number, λ(G( Z , D)) , of distance graph G( Z , D) is studied. For general finite distance set D , it is shown that 2D+2≤λ(G( Z , D))≤D 2+3D. Furthermore, λ(G( Z , D)) ≤8 when D consists of two prime positive odd integers is proved. Finally, a new concept to study the upper bounds of λ(G) for some special D is introduced. For these sets, the upper bound is improved to 7.
基金Supported by the National Natural Science Foundation of China (Grant No. 11601265)the Scientific Research Foundation of Jimei University (Grant No. Q202201)。
文摘An L(2, 1, 1)-labeling of a graph G is an assignment of non-negative integers(labels) to the vertices of G such that adjacent vertices receive labels with difference at least 2, and vertices at distance 2 or 3 receive distinct labels. The span of such a labeling is the difference between the maximum and minimum labels used, and the minimum span over all L(2, 1, 1)-labelings of G is called the L(2, 1, 1)-labeling number of G, denoted by λ_(2,1,1)(G). In this paper, we investigate the L(2, 1, 1)-labelings of caterpillars. Some useful sufficient conditions for λ_(2,1,1)(T) = △_(2)(T) =maxuv∈E(T)(d(u) + d(v))) are given. Furthermore, we show that the sufficient conditions we provide are also necessary for caterpillars with △_(2)(T) = 6.
基金Supported by the National Natural Science Foundation of China(Grant No.11601265)the High-Level Talent Innovation and Entrepreneurship Project of Quanzhou City(Grant No.2017Z033).
文摘An L(3,2,1)-labeling of a graph G is a function f from the vertex set V(G)to the set of all non-negative integers(labels)such that|f(u)-f(v)|≥3 if d(u,v)=1,|f(u)-f(v)≥2 if d(u,v)=2 and|f(u)-f(v)|≥1 if d(u,v)=3.For a non-negative integer k,a k-L(3,2,1)-labeling is an L(3,2,1)-labeling such that no label is greater than k.The L(3,2,1)-labeling number of G,denoted byλ3,2,1(G),is the smallest number k such that G has a k-L(3,2,1)-labeling.In this article,we characterize the L(3,2,1)-labeling numbers of trees with diameter at most 6.
文摘A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this paper, we show that for or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when or 11. Moreover, we show that? if 1) either (mod 6), m is odd, r = 3, or 2) (mod 3), m is even (mod 2), r = 0.
文摘An L(h,k)-labeling of a graph G is an assignment of non-negative integers to the vertices such that if two vertices u and v are adjacent then they receive labels that differ by at least h, and when u and v are not adjacent but there is a two-hop path between them, then they receive labels that differ by at least k. The span λ of such a labeling is the difference between the largest and the smallest vertex labels assigned. Let λ<sub>h</sub>k</sup> ( G )denote the least λ such that G admits an L(h,k) -labeling using labels from {0,1,...λ}. A Cayley graph of group is called circulant graph of order n, if the group is isomorphic to Z<sub>n.</sub> In this paper, initially we investigate the L(h,k) -labeling for circulant graphs with “large” connection sets, and then we extend our observation and find the span of L(h,k) -labeling for any circulants of order n. .
基金supported by the National Natural Science Foundation of China(Nos.42177333 and 31870500)the National Special Program for Key Basic Research of the Ministry of Science and Technology of China(No.2015FY110700)the Jiangsu Agriculture Science and Technology Innovation Fund,China(No.JASTIFCX(20)2003)。
文摘Both straw incorporation and irrigation practices affect biological nitrogen(N)fixation(BNF),but it is still unclear how straw incorporation impacts BNF under continuous(CFI)or intermittent(IFI)flooding irrigation in a rice cropping system.A15N2-labeling chamber system was placed in a rice field to evaluate BNF with straw incorporation under CFI or IFI for 90 d.The nif H(gene encoding the nitrogenase reductase subunit)DNA and c DNA in soil were amplified using real-time quantitative polymerase chain reaction,and high-throughput sequencing was applied to the nif H gene.The total fixed N in the straw incorporation treatment was 14.3 kg ha^(-1)under CFI,being 116%higher than that under IFI(6.62 kg ha^(-1)).Straw incorporation and CFI showed significant interactive effects on the total fixed N and abundances of nif H DNA and c DNA.The increase in BNF was mainly due to the increase in the abundances of heterotrophic diazotrophs such as Desulfovibrio,Azonexus,and Azotobacter.These results indicated that straw incorporation stimulated BNF under CFI relative to IFI,which might ultimately lead to a rapid enhancement of soil fertility.
基金National Natural Science Foundation of China(No.10671074 and No.60673048)Natural Science Foundation of Education Ministry of Anhui Province(No.KJ2007B124 and No.2006KJ256B)
文摘The L(2,1)-labelling number of distance graphs G(D), denoted by λ(D), isstudied. It is shown that distance graphs satisfy λ(G) ≤Δ~2. Moreover, we prove λ({1,2, ..., k})=2k +2 and λ({1,3,..., 2k -1}) =2k + 2 for any fixed positive integer k. Suppose k, a ∈ N and k,a≥2. If k≥a, then λ({a, a + 1,..., a + k - 1}) = 2(a + k-1). Otherwise, λ({a, a + 1, ..., a + k- 1}) ≤min{2(a + k-1), 6k -2}. When D consists of two positive integers,6≤λ(D)≤8. For thespecial distance sets D = {k, k + 1}(any k ∈N), the upper bound of λ(D) is improved to 7.
文摘An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an L(0,1)-labelling is the maximum label number assigned to any vertex of G. The L(0,1)-labelling number of a graph G, denoted by λ0.1(G) is the least integer k such that G has an L(0,1)-labelling of span k. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph by L(0,1)-labelling and have shown that, △-1≤λ0.1(G)≤△ for a cactus graph, where △ is the degree of the graph G.
基金The National Natural Science Foundation of China (No10971025)
文摘An L（j, k）-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive integers which are at least j apart, and vertices at distance two receive integers which are at least k apart. Given an L（j, k）-labeling f of G, define the L（j, k） edge span of f, βj,k（G,f） =max{ ｜f（x）-f（y）｜： {x,y}∈E（G）}. The L（j,k） edge span of G, βj,k （G） is min βj,k（ G, f）, where the minimum runs over all L（j, k）-labelings f of G. The real L（.j, k）-labeling of a graph G is a generalization of the L（j, k）-labeling. It is an assignment of nonnegative real numbers to the vertices of G satisfying the same distance one and distance two conditions. The real L（j, k） edge span of a graph G is defined accordingly, and is denoted by βj,k（G）. This paper investigates some properties of the L（j, k） edge span and the real L（j, k） edge span of graphs, and completely determines the edge spans of cycles and complete t-partite graphs.
基金The National Natural Science Foundation of China(No10671033)Southeast University Science Foundation ( NoXJ0607230)
文摘L（ s, t）-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L （s, t）-labeling of a graph G is an assignment of integers to the vertices of G such that adjacent vertices receive integers which differ by at least s, and vertices that are at distance of two receive integers which differ by at least t. Given an L（s, t） -labeling f of a graph G, the L（s, t） edge span of f, βst （ G, f） = max { ｜f（u） -f（v）｜： （ u, v） ∈ E（G） } is defined. The L（ s, t） edge span of G, βst（G）, is minβst（G,f）, where the minimum runs over all L（s, t）-labelings f of G. Let T be any tree with a maximum degree of △≥2. It is proved that if 2s≥t≥0, then βst（T） =（ [△/2 ] - 1）t ＋s; if 0≤2s 〈 t and △ is even, then βst（T） = [ （△ - 1） t/2 ] ; and if 0 ≤2s 〈 t and △ is odd, then βst（T） = （△ - 1） t/2 ＋ s. Thus, the L（s, t） edge spans of the Cartesian product of two paths and of the square lattice are completely determined.
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AIM: To evaluate the role of fluorine-18-labeled fluorodeoxyglucose positron emission tomography (18F-FDG PET) in various rheumatic diseases and its potential in the early assessment of treatment response in a limited number of patients. METHODS: This study involved 28 newly diagnosed patients, of these 17 had rheumatoid arthritis (RA) and 11 had seronegative spondyloarthropathy (SSA). In the SSA group, 7 patients had ankylosing spondylitis, 3 had psoriatic arthritis, and one had non-specific SSA. Patients with RA were selected as per the American College of Rheumatology criteria. One hour after FDG injection, a whole body PET scan was performed from the skull vertex to below the knee joints using a GE Advance dedicated PET scanner. Separate scans were acquired for both upper and lower limbs. Post-treatment scans were performed in 9 patients in the RA group (at 6-9 wk from baseline) and in 1 patient with psoriatic arthropathy. The pattern of FDG uptake was analysed visually and quantified as maximum standardized uptake value (SUVmax) in a standard region of interest. Metabolic response on the scan was assessed qualitatively and quantitatively and was correlated with clinical assessment. RESULTS: The qualitative FDG uptake was in agreement with the clinically involved joints, erythrocyte sedimentation rate, C-reactive protein values and the clinical assessment by the rheumatologist. All 17 patients in the RA group showed the highest FDG avidity in painful/swollen/tender joints. The uptake pattern was homogeneous, intense and poly-articular in distribution. Hypermetabolism in the regional nodes (axillary nodes in the case of upper limb joint involvement and inguinal nodes in lower limb joints) was a constant feature in patients with RA. Multiple other extra-articular lesions were also observed including thyroid glands (in associated thyroiditis) and in the subcutaneous nodules. Treatment response was better appreciated using SUVmax values than visual interpretation, when compared with clinical evaluation. Four patients showed a favourable response, while 3 had stable disease and 2 showed disease progression. The resolution of regional nodal uptake (axillary or inguinal nodes based on site of joint involvement) in RA following disease modifying anti-rheumatoid drugs was noteworthy, which could be regarded as an additional parameter for identifying responding patients. In the SSA group, uptake in the affected joint was heterogeneous, low grade and nonsymmetrical. In particular, there was intense tendon and muscular uptake corresponding to symptomatic joints. The patients with psoriatic arthritis showed intense FDG uptake in the joints and soft tissue. CONCLUSION: 18F-FDG PET accurately delineates the ongoing inflammatory activity in various rheumatic diseases (both at articular and extra-articular sites) and relates well to clinical symptoms. Different metabolic patterns on FDG-PET scanning in RA and SSA can have important implications for their diagnosis and management in the future with the support of larger studies. FDG-PET molecular imaging is also a sensitive tool in the early assessment of treatment response, especially when using quantitative information. With these benefits, FDG-PET could play a pivotal clinical role in the management of inflammatory joint disorders in the future.
文摘Background: Attention deficit hyperactivity disorder (ADHD) was long considered to be limited to children and adolescents, but it is now known that ADHD symptoms may persist into adulthood. It is plausible that the etiology of ADHD is not one-dimensional, but instead involves various neuroanatomical and neurochemical systems, with the causes of the main abnormalities believed to be catecholaminergic. Iodine-123-labeled meta-iodobenzylguanidine (123I-MIBG) is a physiological analogue of norepinephrine (NE). To the best of our knowledge, there are no reports about the abnormalities of MIBG scintigraphy in patients with ADHD. The cases presented are male adult patients with different comorbid psychiatric disorders. Case presentation: The cases presented are male adult patients with different psychiatric disorders. Case 1 was a 52-year-old male residential construction foreman, who had been diagnosed with acute schizophrenia-like psychotic disorder. 123I-MIBG planar and SPECTs of the studies were performed at 20 min (early phase) and 3 h (late phase) after intravenous injection of 111 MBq. Planar images were processed to determine the heart (H) to mediastinum (M) ratio (H/M). The early and late H/M ratios were 1.22 and 1.07, respectively. Case 2 was a 52-year-old male civil servant who was diagnosed with adjustment disorder. The early and late H/M ratios were 1.98 and 2.08, respectively. Conclusion: Clinical studies make it clear that symptoms of ADHD are more heterogenous and subtle in adults than children. Adult ADHD is significantly commorbid with wide range of other 12-month disorders. MIBG imaging can be useful to diagnose patients with preexisting psychiatric disorders and ADHD. It might be possible to diagnose objectively though the psychiatric statuses resemble or coexist with other psychiatric disorders. Needless to say, further research is important.
文摘Objective To study the biochemecal and immunological characterization of the 200 kD schistosomulum surface antigen Method and results A very high molecular weight schistosomulum surface antigen of Mr】200kD was identified and characterized using monoclonal antibodies. Carbohydrate modification experiments followed by radioimmunobinding assays demonstrated that the epitope recognised by the mAbs on this antigen was carbohydrate in nature, while protein digestion experiments followed by SDS-PAGE indicated that this antigen also contained protein. Immunoprecipitation of <sup>125</sup>I-labelled cercarial, schistosomulum, adult worm and miracidial surface antigens followed by gel analysis showed the carbohydrate epitope to be present on 5 cercarial, 2 schistosomulum and 5 miracidial surface molecules, and suggested a possible ecological function involved in adapting the parasite to the aquatic free-living stages of its life cycle and possibly also in protecting the early schistosomula from host immune damage. The 5 cercarial surfacs antigens proved to be associated with the CHR, since all the mAbs which recognised those antigens could induce a strong CHR. A kinetic investigation of the carbohydrate epitope on schistosomula of different ages demonstrated a gradual and possibly irreversible loss during the culture period. The epitope completely disappeared from the surface of adult worms. Conclusion To demonstrate an epitope common to a number of surface molecules of various developmental stages of schistosome and therefore explains the immunological cross-reactivity among different stages at the molecular level.
基金Sapienza University of Rome(project"Parallel and Distributed Codes")
文摘Given two non-negative integers h and k, an L（h, k）-labeling of a graph G = （V, E） is a function from the set V to a set of colors, such that adjacent nodes take colors at distance at least h, and nodes at distance 2 take colors at distance at least k. The aim of the L（h, k）-labeling problem is to minimize the greatest used color. Since the decisional version of this problem is NP-complete, it is important to investigate particular classes of graphs for which the problem can be efficiently solved. It is well known that the most common interconnection topologies, such as Butterfly-like, Beneg, CCC, Trivalent Cayley networks, are all characterized by a similar structure： they have nodes organized as a matrix and connections are divided into layers. So we naturally introduce a new class of graphs, called （l × n）-multistage graphs, containing the most common interconnection topologies, on which we study the L（h, k）-labeling. A general algorithm for L（h, k）-labeling these graphs is presented, and from this method an efficient L（2, 1）-labeling for Butterfly and CCC networks is derived. Finally we describe a possible generalization of our approach.
基金Supported by the National Natural Science Foundation of China (No. 10971248,11101057)Anhui Provincial Natural Science Foundation (No. 10040606Q45)Postdoctoral Science Foundation of Jiangsu Provinc (No.1102095C)
文摘An L（2, 1）-labelling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that │f（u） - f（v）│≥2 if dG（u, v） = 1 and │f（u） - f（v）│ ≥ 1 if dG（u, v） = 2. The L（2, 1）-labelling problem is to find the smallest number, denoted by A（G）, such that there exists an L（2, 1）-labelling function with no label greater than it. In this paper, we study this problem for trees. Our results improve the result of Wang [The L（2, 1）-labelling of trees, Discrete Appl. Math. 154 （2006） 598-603].
