Let G be a graph, an independent set Y in G is called an essential independent set (or essential set for simplicity), if there is {y 1,y 2} Y such that dist (y 1,y 2)=2. In this paper, we wi...Let G be a graph, an independent set Y in G is called an essential independent set (or essential set for simplicity), if there is {y 1,y 2} Y such that dist (y 1,y 2)=2. In this paper, we will use the technique of the vertex insertion on l connected ( l=k or k+1,k≥2 ) claw free graphs to provide a unified proof for G to be hamiltonian or 1 hamiltonian, the sufficient conditions are expressed by the inequality concerning ∑ki=0N(Y i) and n(Y) for each essential set Y={y 0,y 1,...,y k} of G , where Y i={y i,y i-1 ,...,y i-(b-1) }Y for i∈{0,1,...,k} (the subscriptions of y j ’s will be taken modulo k+1 ), b ( 0【b【k+1 ) is an integer, and n(Y)={v∈V(G): dist (v,Y)≤2 }.展开更多
The emerging two-dimensional MXene-derived quantum dots(MQDs)have garnered considerable research interest owing to their abundant active edge atoms,excellent electrical conductivity,and remarkable optical properties.C...The emerging two-dimensional MXene-derived quantum dots(MQDs)have garnered considerable research interest owing to their abundant active edge atoms,excellent electrical conductivity,and remarkable optical properties.Compared with their two-dimensional(2D)counterpart MXene,MQDs with forceful size and quantum confinement effects exhibit more unparalleled properties and have considerably contributed to the advanced photocatalysis,detection,energy storage,and biomedicine fields.This critical review summarizes the fundamental properties of MQDs in terms of structure,electricity,and optics.The mechanism,characteristics,and comparisons of two typical synthesis strategies(traditional chemical method and novel fluorine-free or chemical-free method)are also presented.Furthermore,the similarities and differences between MQDs and 2D MXenes are introduced in terms of their functional groups,light absorption capacity,energy band structure,and other properties.Moreover,recent advances in the applications of MQD-based materials for energy conversion and storage(ECS)are discussed,including photocatalysis,batteries,and supercapacitors.Finally,current challenges and future opportunities for advancing MQD-based materials in the promising ECS field are presented.展开更多
文摘Let G be a graph, an independent set Y in G is called an essential independent set (or essential set for simplicity), if there is {y 1,y 2} Y such that dist (y 1,y 2)=2. In this paper, we will use the technique of the vertex insertion on l connected ( l=k or k+1,k≥2 ) claw free graphs to provide a unified proof for G to be hamiltonian or 1 hamiltonian, the sufficient conditions are expressed by the inequality concerning ∑ki=0N(Y i) and n(Y) for each essential set Y={y 0,y 1,...,y k} of G , where Y i={y i,y i-1 ,...,y i-(b-1) }Y for i∈{0,1,...,k} (the subscriptions of y j ’s will be taken modulo k+1 ), b ( 0【b【k+1 ) is an integer, and n(Y)={v∈V(G): dist (v,Y)≤2 }.
文摘The emerging two-dimensional MXene-derived quantum dots(MQDs)have garnered considerable research interest owing to their abundant active edge atoms,excellent electrical conductivity,and remarkable optical properties.Compared with their two-dimensional(2D)counterpart MXene,MQDs with forceful size and quantum confinement effects exhibit more unparalleled properties and have considerably contributed to the advanced photocatalysis,detection,energy storage,and biomedicine fields.This critical review summarizes the fundamental properties of MQDs in terms of structure,electricity,and optics.The mechanism,characteristics,and comparisons of two typical synthesis strategies(traditional chemical method and novel fluorine-free or chemical-free method)are also presented.Furthermore,the similarities and differences between MQDs and 2D MXenes are introduced in terms of their functional groups,light absorption capacity,energy band structure,and other properties.Moreover,recent advances in the applications of MQD-based materials for energy conversion and storage(ECS)are discussed,including photocatalysis,batteries,and supercapacitors.Finally,current challenges and future opportunities for advancing MQD-based materials in the promising ECS field are presented.
