An H-covering(resp.decomposition) of a graph G is a set of subgraphs of G,{H1,H2,…,Hk} such that H1 is isomorphic to H for eachi,and each edge of G belongs to at least(resp.exactly) one of the subgraphs Hi,fox 1≤i≤...An H-covering(resp.decomposition) of a graph G is a set of subgraphs of G,{H1,H2,…,Hk} such that H1 is isomorphic to H for eachi,and each edge of G belongs to at least(resp.exactly) one of the subgraphs Hi,fox 1≤i≤k.An H-covering(resp.decomposition) of a graph G is a magic H-covering(resp.decomposition) if there is a bijection f:V(G)∪E(G)→{1,...,|V(G)|+|E(G)|} such that the sum of labels of edges and the vertices of each copy of H in the decomposition is a constant.If G admits a unique H-covering H and H is a magic H-covering of G,then G is H-magic.In this paper,we show that if G admits a magic H-covering(resp.decomposition),and satisfies some other conditions,then a union of k vertex joint graph G,kG,and a graph obtained from kG,Gk admit a magic H-covering or decomposition.展开更多
Pentacoordinated Al(Al~Ⅴ)species in silica-alumina are promising to promote the formation of acid sites or act as surface defects for tailoring single-atom catalysts.However,pentahedral coordination(Al~Ⅴ)is rarely o...Pentacoordinated Al(Al~Ⅴ)species in silica-alumina are promising to promote the formation of acid sites or act as surface defects for tailoring single-atom catalysts.However,pentahedral coordination(Al~Ⅴ)is rarely observed in conventionally prepared silica-alumina.Here,we show that high population and dispersion of Al~Ⅴ species on the surface of amorphous silica-alumina(ASA)can be achieved by means of flame spray pyrolysis.High resolution TEM/EDX,high magnetic-field NMR and DFT calculations are employed to characterize the structure of as-prepared ASAs.Solid-state ^(27)Al multi-quantum MAS NMR experiments show that most of the Al~Ⅴspecies are formed independently from the alumina phase and are accessible for vip molecules on the surface.Upon water adsorption,these Al~Ⅴ species are transformed to Al~Ⅵ species,structurally similar to surface Al~Ⅳ species,as confirmed by DFT calculations.The outstanding catalytic activity of as-synthesized ASA is demonstrated using the in situ H/D exchange reaction with deuterated benzene as an example.The Al~Ⅴ-rich ASA provides a much lower activation energy(～30 kJ/mol)than that reported for zeolite H-ZSM-5(～60 kJ/mol).The superior catalytic performance is attributed to the high Al~Ⅴcontent promoting the surface active sites in ASA.The knowledge gained on the synthesis of Al~Ⅴ-rich ASAs and the nature of aluminum coordination in these materials could pave the way to more efficient silica-alumina based catalysts.展开更多
基金Supported by the National Natural Science Foundation of China(11702094)the Fundamental Research Funds for the Central Universities(3142015043)North China Institute of Science and Technology Applied Mathematics Innovation Team(3142018059)。
文摘An H-covering(resp.decomposition) of a graph G is a set of subgraphs of G,{H1,H2,…,Hk} such that H1 is isomorphic to H for eachi,and each edge of G belongs to at least(resp.exactly) one of the subgraphs Hi,fox 1≤i≤k.An H-covering(resp.decomposition) of a graph G is a magic H-covering(resp.decomposition) if there is a bijection f:V(G)∪E(G)→{1,...,|V(G)|+|E(G)|} such that the sum of labels of edges and the vertices of each copy of H in the decomposition is a constant.If G admits a unique H-covering H and H is a magic H-covering of G,then G is H-magic.In this paper,we show that if G admits a magic H-covering(resp.decomposition),and satisfies some other conditions,then a union of k vertex joint graph G,kG,and a graph obtained from kG,Gk admit a magic H-covering or decomposition.
基金the financial supports by Australian Research Council Discovery Projects (DP150103842)Discovery Earlier Career Research Project (DE190101618)+1 种基金the Faculty's MCR Scheme, Energy and Materials Clusters at the University of Sydneythe support by the National Natural Science Foundation of China (21522310, 21473244 and 21210005)
文摘Pentacoordinated Al(Al~Ⅴ)species in silica-alumina are promising to promote the formation of acid sites or act as surface defects for tailoring single-atom catalysts.However,pentahedral coordination(Al~Ⅴ)is rarely observed in conventionally prepared silica-alumina.Here,we show that high population and dispersion of Al~Ⅴ species on the surface of amorphous silica-alumina(ASA)can be achieved by means of flame spray pyrolysis.High resolution TEM/EDX,high magnetic-field NMR and DFT calculations are employed to characterize the structure of as-prepared ASAs.Solid-state ^(27)Al multi-quantum MAS NMR experiments show that most of the Al~Ⅴspecies are formed independently from the alumina phase and are accessible for vip molecules on the surface.Upon water adsorption,these Al~Ⅴ species are transformed to Al~Ⅵ species,structurally similar to surface Al~Ⅳ species,as confirmed by DFT calculations.The outstanding catalytic activity of as-synthesized ASA is demonstrated using the in situ H/D exchange reaction with deuterated benzene as an example.The Al~Ⅴ-rich ASA provides a much lower activation energy(～30 kJ/mol)than that reported for zeolite H-ZSM-5(～60 kJ/mol).The superior catalytic performance is attributed to the high Al~Ⅴcontent promoting the surface active sites in ASA.The knowledge gained on the synthesis of Al~Ⅴ-rich ASAs and the nature of aluminum coordination in these materials could pave the way to more efficient silica-alumina based catalysts.
