A{(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4.It follows from Euler s formula that the number of triangular faces is eight.A set H of disjoint quadrangular faces of S i...A{(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4.It follows from Euler s formula that the number of triangular faces is eight.A set H of disjoint quadrangular faces of S is called resonant pattern if S has a perfect matching M such that every quadrangular face in H is M-alternating.Let k be a positive integer,S is k-resonant if any i≤k disjoint quadrangular faces of S form a resonant pattern.Moreover,if graph S is k-resonant for any integer k,then S is called maximally resonant.In this paper,we show that the maximally resonant{(3,4),4}-fullerene graphs are S_6,S_8,S_(10)^(2),S_(12)^(2),S_(12)^(4),S_(12)^(5),S_(14)^(3),S_(14)^(5),S_(16)^(3),S_(18)^(5),S_(24)as shown in Fig.1.As a corollary,it is shown that if a{(3,4),4}-fullerene graph is 4-resonant,then it is also maximally resonant.展开更多
A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C_4-free k-cyclic graphs of ...A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C_4-free k-cyclic graphs of order n. Furthermore, we determine the first three unicycles and bicyclic, C_4-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the(combinatorial)展开更多
In this paper we consider the problem of finding the smallest number such that any graph G of order n admits a decomposition into edge disjoint copies of C4 and single edges with at most elements. We solve this proble...In this paper we consider the problem of finding the smallest number such that any graph G of order n admits a decomposition into edge disjoint copies of C4 and single edges with at most elements. We solve this problem for n sufficiently large.展开更多
Machine learning(ML)integrated with density functional theory(DFT)calculations have recently been used to accelerate the design and discovery of single-atom catalysts(SACs)by establishing deep structure–activity rela...Machine learning(ML)integrated with density functional theory(DFT)calculations have recently been used to accelerate the design and discovery of single-atom catalysts(SACs)by establishing deep structure–activity relationships.The traditional ML models are always difficult to identify the structural differences among the single-atom systems with different modification methods,leading to the limitation of the potential application range.Aiming to the structural properties of several typical two-dimensional MA_(2)Z_(4)-based single-atom systems(bare MA_(2)Z_(4) and metal single-atom doped/supported MA_(2)Z_(4)),an improved crystal graph convolutional neural network(CGCNN)classification model was employed,instead of the traditional machine learning regression model,to address the challenge of incompatibility in the studied systems.The CGCNN model was optimized using crystal graph representation in which the geometric configuration was divided into active layer,surface layer,and bulk layer(ASB-GCNN).Through ML and DFT calculations,five potential single-atom hydrogen evolution reaction(HER)catalysts were screened from chemical space of 600 MA_(2)Z_(4)-based materials,especially V_(1)/HfSn_(2)N_(4)(S)with high stability and activity(Δ_(GH*)is 0.06 eV).Further projected density of states(pDOS)analysis in combination with the wave function analysis of the SAC-H bond revealed that the SAC-dz^(2)orbital coincided with the H-s orbital around the energy level of−2.50 eV,and orbital analysis confirmed the formation ofσbonds.This study provides an efficient multistep screening design framework of metal single-atom catalyst for HER systems with similar two-dimensional supports but different geometric configurations.展开更多
基金Supported by NSFC(Grant Nos.11801148 and 11626089)the Foundation for the Doctor of Henan Polytechnic University(Grant No.B2014-060)。
文摘A{(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4.It follows from Euler s formula that the number of triangular faces is eight.A set H of disjoint quadrangular faces of S is called resonant pattern if S has a perfect matching M such that every quadrangular face in H is M-alternating.Let k be a positive integer,S is k-resonant if any i≤k disjoint quadrangular faces of S form a resonant pattern.Moreover,if graph S is k-resonant for any integer k,then S is called maximally resonant.In this paper,we show that the maximally resonant{(3,4),4}-fullerene graphs are S_6,S_8,S_(10)^(2),S_(12)^(2),S_(12)^(4),S_(12)^(5),S_(14)^(3),S_(14)^(5),S_(16)^(3),S_(18)^(5),S_(24)as shown in Fig.1.As a corollary,it is shown that if a{(3,4),4}-fullerene graph is 4-resonant,then it is also maximally resonant.
基金Supported by the National Natural Science Foundation of China（11171273） Supported by the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical Uni- versity（Z2016170）
文摘A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C_4-free k-cyclic graphs of order n. Furthermore, we determine the first three unicycles and bicyclic, C_4-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the(combinatorial)
基金support from FCT—Fundacao para a Ciencia e a Tecnologia(Portugal),through Projects PTDC/MAT/113207/2009PEst-OE/MAT/UI0297/2011(CMA).
文摘In this paper we consider the problem of finding the smallest number such that any graph G of order n admits a decomposition into edge disjoint copies of C4 and single edges with at most elements. We solve this problem for n sufficiently large.
基金supported by the National Key R&D Program of China(2021YFA1500900)National Natural Science Foundation of China(U21A20298,22141001).
文摘Machine learning(ML)integrated with density functional theory(DFT)calculations have recently been used to accelerate the design and discovery of single-atom catalysts(SACs)by establishing deep structure–activity relationships.The traditional ML models are always difficult to identify the structural differences among the single-atom systems with different modification methods,leading to the limitation of the potential application range.Aiming to the structural properties of several typical two-dimensional MA_(2)Z_(4)-based single-atom systems(bare MA_(2)Z_(4) and metal single-atom doped/supported MA_(2)Z_(4)),an improved crystal graph convolutional neural network(CGCNN)classification model was employed,instead of the traditional machine learning regression model,to address the challenge of incompatibility in the studied systems.The CGCNN model was optimized using crystal graph representation in which the geometric configuration was divided into active layer,surface layer,and bulk layer(ASB-GCNN).Through ML and DFT calculations,five potential single-atom hydrogen evolution reaction(HER)catalysts were screened from chemical space of 600 MA_(2)Z_(4)-based materials,especially V_(1)/HfSn_(2)N_(4)(S)with high stability and activity(Δ_(GH*)is 0.06 eV).Further projected density of states(pDOS)analysis in combination with the wave function analysis of the SAC-H bond revealed that the SAC-dz^(2)orbital coincided with the H-s orbital around the energy level of−2.50 eV,and orbital analysis confirmed the formation ofσbonds.This study provides an efficient multistep screening design framework of metal single-atom catalyst for HER systems with similar two-dimensional supports but different geometric configurations.
