The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamilt...The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.展开更多
The rapid development of information technology and accelerated digitalization have led to an explosive growth of data across various fields.As a key technology for knowledge representation and sharing,knowledge graph...The rapid development of information technology and accelerated digitalization have led to an explosive growth of data across various fields.As a key technology for knowledge representation and sharing,knowledge graphs play a crucial role by constructing structured networks of relationships among entities.However,data sparsity and numerous unexplored implicit relations result in the widespread incompleteness of knowledge graphs.In static knowledge graph completion,most existing methods rely on linear operations or simple interaction mechanisms for triple encoding,making it difficult to fully capture the deep semantic associations between entities and relations.Moreover,many methods focus only on the local information of individual triples,ignoring the rich semantic dependencies embedded in the neighboring nodes of entities within the graph structure,which leads to incomplete embedding representations.To address these challenges,we propose Two-Stage Mixer Embedding(TSMixerE),a static knowledge graph completion method based on entity context.In the unit semantic extraction stage,TSMixerE leveragesmulti-scale circular convolution to capture local features atmultiple granularities,enhancing the flexibility and robustness of feature interactions.A channel attention mechanism amplifies key channel responses to suppress noise and irrelevant information,thereby improving the discriminative power and semantic depth of feature representations.For contextual information fusion,a multi-layer self-attentionmechanism enables deep interactions among contextual cues,effectively integrating local details with global context.Simultaneously,type embeddings clarify the semantic identities and roles of each component,enhancing the model’s sensitivity and fusion capabilities for diverse information sources.Furthermore,TSMixerE constructs contextual unit sequences for entities,fully exploring neighborhood information within the graph structure to model complex semantic dependencies,thus improving the completeness and generalization of embedding representations.展开更多
Let G be a non-complete graph such that its complement G is r-partite.In this paper,properties of the graph G are studied,including the Cohen-Macaulay property and the sequential Cohen-Macaulay property.For r=2,3,some...Let G be a non-complete graph such that its complement G is r-partite.In this paper,properties of the graph G are studied,including the Cohen-Macaulay property and the sequential Cohen-Macaulay property.For r=2,3,some constructions are established for G to be vertex decomposable and some sufficient conditions are provided for r≥4.展开更多
Let Z(λ,G)denote the zeta function of a graph G.In this paper the complement G^Cand the G^(xyz)-transformation G^(xyz)of an r-regular graph G with n vertices and m edges for x,y,z∈{0,1,+,-},are considerd.The relatio...Let Z(λ,G)denote the zeta function of a graph G.In this paper the complement G^Cand the G^(xyz)-transformation G^(xyz)of an r-regular graph G with n vertices and m edges for x,y,z∈{0,1,+,-},are considerd.The relationship between Z(λ,G)and Z(λ,G^C)is obtained.For all x,y,z∈{0,1,+,-},the explicit formulas for the reciprocal of Z(λ,G^(xyz))in terms of r,m,n and the characteristic polynomial of G are obtained.Due to limited space,only the expressions for G^(xyz)with z=0,and xyz∈{0++,+++,1+-}are presented here.展开更多
Let?G=(V,E)? be a graph. If φ is a function from the vertex set V(G) to the set of positive integers. Then two vertices?u, v ∈ V(G)? are?φ -equitable if|φ(u)-φ(v)|≤1.By the degree, equitable adjacency between ve...Let?G=(V,E)? be a graph. If φ is a function from the vertex set V(G) to the set of positive integers. Then two vertices?u, v ∈ V(G)? are?φ -equitable if|φ(u)-φ(v)|≤1.By the degree, equitable adjacency between vertices can be redefine almost all of the variants of the graphs. In this paper we study the degree equitability of the graph by defining equitable connectivity, equitable regularity, equitable connected graph and equitable complete graph. Some new families of graphs and some interesting results are obtained.展开更多
文摘The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.
基金supported by the National Natural Science Foundation of China(No.62267005)the Chinese Guangxi Natural Science Foundation(No.2023GXNSFAA026493)+1 种基金Guangxi Collaborative Innovation Center ofMulti-Source Information Integration and Intelligent ProcessingGuangxi Academy of Artificial Intelligence.
文摘The rapid development of information technology and accelerated digitalization have led to an explosive growth of data across various fields.As a key technology for knowledge representation and sharing,knowledge graphs play a crucial role by constructing structured networks of relationships among entities.However,data sparsity and numerous unexplored implicit relations result in the widespread incompleteness of knowledge graphs.In static knowledge graph completion,most existing methods rely on linear operations or simple interaction mechanisms for triple encoding,making it difficult to fully capture the deep semantic associations between entities and relations.Moreover,many methods focus only on the local information of individual triples,ignoring the rich semantic dependencies embedded in the neighboring nodes of entities within the graph structure,which leads to incomplete embedding representations.To address these challenges,we propose Two-Stage Mixer Embedding(TSMixerE),a static knowledge graph completion method based on entity context.In the unit semantic extraction stage,TSMixerE leveragesmulti-scale circular convolution to capture local features atmultiple granularities,enhancing the flexibility and robustness of feature interactions.A channel attention mechanism amplifies key channel responses to suppress noise and irrelevant information,thereby improving the discriminative power and semantic depth of feature representations.For contextual information fusion,a multi-layer self-attentionmechanism enables deep interactions among contextual cues,effectively integrating local details with global context.Simultaneously,type embeddings clarify the semantic identities and roles of each component,enhancing the model’s sensitivity and fusion capabilities for diverse information sources.Furthermore,TSMixerE constructs contextual unit sequences for entities,fully exploring neighborhood information within the graph structure to model complex semantic dependencies,thus improving the completeness and generalization of embedding representations.
基金Supported by the Natural Science Foundation of Shanghai(Grant No.19ZR1424100)the National Natural Science Foundation of China(Grant No.11971338)。
文摘Let G be a non-complete graph such that its complement G is r-partite.In this paper,properties of the graph G are studied,including the Cohen-Macaulay property and the sequential Cohen-Macaulay property.For r=2,3,some constructions are established for G to be vertex decomposable and some sufficient conditions are provided for r≥4.
基金National Natural Science Foundation of China(No.11671258)
文摘Let Z(λ,G)denote the zeta function of a graph G.In this paper the complement G^Cand the G^(xyz)-transformation G^(xyz)of an r-regular graph G with n vertices and m edges for x,y,z∈{0,1,+,-},are considerd.The relationship between Z(λ,G)and Z(λ,G^C)is obtained.For all x,y,z∈{0,1,+,-},the explicit formulas for the reciprocal of Z(λ,G^(xyz))in terms of r,m,n and the characteristic polynomial of G are obtained.Due to limited space,only the expressions for G^(xyz)with z=0,and xyz∈{0++,+++,1+-}are presented here.
文摘Let?G=(V,E)? be a graph. If φ is a function from the vertex set V(G) to the set of positive integers. Then two vertices?u, v ∈ V(G)? are?φ -equitable if|φ(u)-φ(v)|≤1.By the degree, equitable adjacency between vertices can be redefine almost all of the variants of the graphs. In this paper we study the degree equitability of the graph by defining equitable connectivity, equitable regularity, equitable connected graph and equitable complete graph. Some new families of graphs and some interesting results are obtained.
