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
The induced matching partition number of graph G is the minimum integer k such that there exists a k-partition(V1,V2,…,Vk) of V(G)such that,for each i(1≤i≤k),G[Vi] is 1-regular.In this paper,we study the induced m...The induced matching partition number of graph G is the minimum integer k such that there exists a k-partition(V1,V2,…,Vk) of V(G)such that,for each i(1≤i≤k),G[Vi] is 1-regular.In this paper,we study the induced matching partition number of product graphs.We provide a lower bound and an upper bound for the induced matching partition number of product graphs,and exact results are given for some special product graphs.展开更多
The acquisition of valuable design knowledge from massive fragmentary data is challenging for designers in conceptual product design.This study proposes a novel method for acquiring design knowledge by combining deep ...The acquisition of valuable design knowledge from massive fragmentary data is challenging for designers in conceptual product design.This study proposes a novel method for acquiring design knowledge by combining deep learning with knowledge graph.Specifically,the design knowledge acquisition method utilises the knowledge extraction model to extract design-related entities and relations from fragmentary data,and further constructs the knowledge graph to support design knowledge acquisition for conceptual product design.Moreover,the knowledge extraction model introduces ALBERT to solve memory limitation and communication overhead in the entity extraction module,and uses multi-granularity information to overcome segmentation errors and polysemy ambiguity in the relation extraction module.Experimental comparison verified the effectiveness and accuracy of the proposed knowledge extraction model.The case study demonstrated the feasibility of the knowledge graph construction with real fragmentary porcelain data and showed the capability to provide designers with interconnected and visualised design knowledge.展开更多
Textile production has received considerable attention owing to its significance in production value,the complexity of its manufacturing processes and the extensive reach of its supply chains.However,textile industry ...Textile production has received considerable attention owing to its significance in production value,the complexity of its manufacturing processes and the extensive reach of its supply chains.However,textile industry consumes substantial energy and materials and emits greenhouse gases that severely harm the environment.In addressing this challenge,the concept of sustainable production offers crucial guidance for the sustainable development of the textile industry.Low-carbon manufacturing technologies provide robust technical support for the textile industry to transition to a low-carbon model by optimizing production processes,enhancing energy efficiency and minimizing material waste.Consequently,low-carbon manufacturing technologies have gradually been implemented in sustainable textile production scenarios.However,while research on low-carbon manufacturing technologies for textile production has advanced,these studies predominantly concentrate on theoretical methods,with relatively limited exploration of practical applications.To address this gap,a thorough overview of carbon emission management methods and tools in textile production,as well as the characteristics and influencing factors of carbon emissions in key textile manufacturing processes is presented to identify common issues.Additionally,two new concepts,carbon knowledge graph and carbon traceability,are introduced,offering strategic recommendations and application directions for the low-carbon development of sustainable textile production.Beginning with seven key aspects of sustainable textile production,the characteristics of carbon emissions and their influencing factors in key textile manufacturing process are systematically summarized.The aim is to provide guidance and optimization strategies for future emission reduction efforts by exploring the carbon emission situations and influencing factors at each stage.Furthermore,the potential and challenges of carbon knowledge graph technology are summarized in achieving carbon traceability,and several research ideas and suggestions are proposed.展开更多
Xiong and Liu[21]gave a characterization of the graphs G for which the n-iterated line graph L^(n)(G)is hamiltonian,for n≥2.In this paper,we study the existence of a hamiltonian path in L^(n)(G),and give a characteri...Xiong and Liu[21]gave a characterization of the graphs G for which the n-iterated line graph L^(n)(G)is hamiltonian,for n≥2.In this paper,we study the existence of a hamiltonian path in L^(n)(G),and give a characterization of G for which L^(n)(G)has a hamiltonian path.As applications,we use this characterization to give several upper bounds on the hamiltonian path index of a graph.展开更多
The exponential Randić index has important applications in the fields of biology and chemistry. The exponential Randić index of a graph G is defined as the sum of the weights e 1 d( u )d( v ) of all edges uv of G, whe...The exponential Randić index has important applications in the fields of biology and chemistry. The exponential Randić index of a graph G is defined as the sum of the weights e 1 d( u )d( v ) of all edges uv of G, where d( u ) denotes the degree of a vertex u in G. The paper mainly provides the upper and lower bounds of the exponential Randić index in quasi-tree graphs, and characterizes the extremal graphs when the bounds are achieved.展开更多
In this paper,we consider the graph of the product of continuous functions in terms of Hausdorff and packing dimensions.More precisely,we show that,given a real number 1≤β≤2,any real-valued continuous function in C...In this paper,we consider the graph of the product of continuous functions in terms of Hausdorff and packing dimensions.More precisely,we show that,given a real number 1≤β≤2,any real-valued continuous function in C([0,1])can be decomposed into a product of two real-valued continuous functions,each having a graph of Hausdorff dimensionβ.In addition,a product decomposition result for the packing dimension is obtained.This work answers affirmatively two questions raised by Verma and Priyadarshi[14].展开更多
The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. ...The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.展开更多
The Estrada index of a graph G on n vertices is defined by EE(G)=∑^(n)_(i=1)^(eλ_(i)),whereλ_(1),λ_(2),···,λ_(n)are the adjacency eigenvalues of G.We define two general types of dynamic graphs evol...The Estrada index of a graph G on n vertices is defined by EE(G)=∑^(n)_(i=1)^(eλ_(i)),whereλ_(1),λ_(2),···,λ_(n)are the adjacency eigenvalues of G.We define two general types of dynamic graphs evolving according to continuous-time Markov processes with their stationary distributions matching the Erd¨os-R´enyi random graph and the random graph with given expected degrees,respectively.We formulate some new estimates and upper and lower bounds for the Estrada indices of these dynamic graphs.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
Over the past era,subgraph mining from a large collection of graph database is a crucial problem.In addition,scalability is another big problem due to insufficient storage.There are several security challenges associa...Over the past era,subgraph mining from a large collection of graph database is a crucial problem.In addition,scalability is another big problem due to insufficient storage.There are several security challenges associated with subgraph mining in today’s on-demand system.To address this downside,our proposed work introduces a Blockchain-based Consensus algorithm for Authenticated query search in the Large-Scale Dynamic Graphs(BCCA-LSDG).The two-fold process is handled in the proposed BCCA-LSDG:graph indexing and authenticated query search(query processing).A blockchain-based reputation system is meant to maintain the trust blockchain and cloud server of the proposed architecture.To resolve the issues and provide safe big data transmission,the proposed technique also combines blockchain with a consensus algorithm architecture.Security of the big data is ensured by dividing the BC network into distinct networks,each with a restricted number of allowed entities,data kept in the cloud gate server,and data analysis in the blockchain.The consensus algorithm is crucial for maintaining the speed,performance and security of the blockchain.Then Dual Similarity based MapReduce helps in mapping and reducing the relevant subgraphs with the use of optimal feature sets.Finally,the graph index refinement process is undertaken to improve the query results.Concerning query error,fuzzy logic is used to refine the index of the graph dynamically.The proposed technique outperforms advanced methodologies in both blockchain and non-blockchain systems,and the combination of blockchain and subgraph provides a secure communication platform,according to the findings.展开更多
Let be a graph with n vertices and m edges. The sum of absolute value of all coefficients of matching polynomial is called Hosoya index. In this paper, we determine 2<sup>nd</sup> to 4<sup>th</sup...Let be a graph with n vertices and m edges. The sum of absolute value of all coefficients of matching polynomial is called Hosoya index. In this paper, we determine 2<sup>nd</sup> to 4<sup>th</sup> minimum Hosoya index of a kind of tetracyclic graph, with m = n +3.展开更多
A lot of combinatorial objects have a natural bialgebra structure. In this paper, we prove that the vector space spanned by labeled simple graphs is a bialgebra with the conjunction product and the unshuffle coproduct...A lot of combinatorial objects have a natural bialgebra structure. In this paper, we prove that the vector space spanned by labeled simple graphs is a bialgebra with the conjunction product and the unshuffle coproduct. In fact, it is a Hopf algebra since it is graded connected. The main conclusions are that the vector space spanned by labeled simple graphs arising from the unshuffle coproduct is a Hopf algebra and that there is a Hopf homomorphism from permutations to label simple graphs.展开更多
A graph G is said to be one modulo N-difference mean graph if there is an injective function f from the vertex set of G to the set , where N is the natural number and q is the number of edges of G and f induces a bije...A graph G is said to be one modulo N-difference mean graph if there is an injective function f from the vertex set of G to the set , where N is the natural number and q is the number of edges of G and f induces a bijection from the edge set of G to given by and the function f is called a one modulo N-difference mean labeling of G. In this paper, we show that the graphs such as arbitrary union of paths, , ladder, slanting ladder, diamond snake, quadrilateral snake, alternately quadrilateral snake, , , , , friendship graph and admit one modulo N-difference mean labeling.展开更多
Since many large graphs are composed from some existing smaller graphs by using graph operations, say, the Cartesian product, the Lexicographic product and the Strong product. Many properties of such large graphs are ...Since many large graphs are composed from some existing smaller graphs by using graph operations, say, the Cartesian product, the Lexicographic product and the Strong product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the Strong product of vertex-transitive graphs. In particular, we show that the Strong product of Cayley graphs is still a Cayley graph.展开更多
The present paper deals with the gracefulness of unconnected graph (jC_(4n))∪P_m,and proves the following result:for positive integers n,j and m with n≥1,j≥2,the unconnected graph(jC_(4n))∪P_m is a gracef...The present paper deals with the gracefulness of unconnected graph (jC_(4n))∪P_m,and proves the following result:for positive integers n,j and m with n≥1,j≥2,the unconnected graph(jC_(4n))∪P_m is a graceful graph for m=j-1 or m≥n+j,where C_(4n) is a cycle with 4n vertexes,P_m is a path with m+1 vertexes,and(jC_(4n))∪P_m denotes the disjoint union of j-C_(4n) and P_m.展开更多
This paper considers the edge-connectivity and the restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and ...This paper considers the edge-connectivity and the restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, we construct a Cayley graph with degree d whose restricted edge-connectivity is equal to d + s for given odd integer d and integer s with d 5 and 1 s d- 3, which answers a problem proposed ten years ago.展开更多
The crossing number of cartesian products of paths and cycles with 5-vertex graphs mostly are known, but only few cartesian products of 5-vertex graphs with star K 1,n are known. In this paper, we will extent those re...The crossing number of cartesian products of paths and cycles with 5-vertex graphs mostly are known, but only few cartesian products of 5-vertex graphs with star K 1,n are known. In this paper, we will extent those results, and determine the crossing numbers of cartesian products of two 5-vertex graphs with star K 1,n .展开更多
In this paper we define direct product of graphs and give a recipe for obtaining probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability o...In this paper we define direct product of graphs and give a recipe for obtaining probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph is obtained by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determining probability of walk on compficated graphs. Using this method, we calculate the probability of Continuous-time classical and quantum random walks on many of finite direct product Cayley graphs (complete cycle, complete Kn, charter and n-cube). Also, we inquire that the classical state the stationary uniform distribution is reached as t→∞ but for quantum state is not always satisfied.展开更多
Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n...Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n×P m (grid graph) with small n and m . But for large n and m , it is difficult to decide the exact fractional domination number. Motivated by this, nearly sharp upper and lower bounds are given to the fractional domination number of grid graphs. Furthermore, upper and lower bounds on the fractional total domination number of strong direct product of graphs are given.展开更多
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The induced matching partition number of graph G is the minimum integer k such that there exists a k-partition(V1,V2,…,Vk) of V(G)such that,for each i(1≤i≤k),G[Vi] is 1-regular.In this paper,we study the induced matching partition number of product graphs.We provide a lower bound and an upper bound for the induced matching partition number of product graphs,and exact results are given for some special product graphs.
基金This research is supported by the Chinese Special Projects of the National Key Research and Development Plan(2019YFB1405702).
文摘The acquisition of valuable design knowledge from massive fragmentary data is challenging for designers in conceptual product design.This study proposes a novel method for acquiring design knowledge by combining deep learning with knowledge graph.Specifically,the design knowledge acquisition method utilises the knowledge extraction model to extract design-related entities and relations from fragmentary data,and further constructs the knowledge graph to support design knowledge acquisition for conceptual product design.Moreover,the knowledge extraction model introduces ALBERT to solve memory limitation and communication overhead in the entity extraction module,and uses multi-granularity information to overcome segmentation errors and polysemy ambiguity in the relation extraction module.Experimental comparison verified the effectiveness and accuracy of the proposed knowledge extraction model.The case study demonstrated the feasibility of the knowledge graph construction with real fragmentary porcelain data and showed the capability to provide designers with interconnected and visualised design knowledge.
基金Natural Science Foundation of Shanghai,China (No. 21ZR1400800)。
文摘Textile production has received considerable attention owing to its significance in production value,the complexity of its manufacturing processes and the extensive reach of its supply chains.However,textile industry consumes substantial energy and materials and emits greenhouse gases that severely harm the environment.In addressing this challenge,the concept of sustainable production offers crucial guidance for the sustainable development of the textile industry.Low-carbon manufacturing technologies provide robust technical support for the textile industry to transition to a low-carbon model by optimizing production processes,enhancing energy efficiency and minimizing material waste.Consequently,low-carbon manufacturing technologies have gradually been implemented in sustainable textile production scenarios.However,while research on low-carbon manufacturing technologies for textile production has advanced,these studies predominantly concentrate on theoretical methods,with relatively limited exploration of practical applications.To address this gap,a thorough overview of carbon emission management methods and tools in textile production,as well as the characteristics and influencing factors of carbon emissions in key textile manufacturing processes is presented to identify common issues.Additionally,two new concepts,carbon knowledge graph and carbon traceability,are introduced,offering strategic recommendations and application directions for the low-carbon development of sustainable textile production.Beginning with seven key aspects of sustainable textile production,the characteristics of carbon emissions and their influencing factors in key textile manufacturing process are systematically summarized.The aim is to provide guidance and optimization strategies for future emission reduction efforts by exploring the carbon emission situations and influencing factors at each stage.Furthermore,the potential and challenges of carbon knowledge graph technology are summarized in achieving carbon traceability,and several research ideas and suggestions are proposed.
基金Supported by the Natural Science Foundation of China(12131013,12371356)the special fund for Science and Technology Innovation Teams of Shanxi Province(202204051002015)the Fundamental Research Program of Shanxi Province(202303021221064).
文摘Xiong and Liu[21]gave a characterization of the graphs G for which the n-iterated line graph L^(n)(G)is hamiltonian,for n≥2.In this paper,we study the existence of a hamiltonian path in L^(n)(G),and give a characterization of G for which L^(n)(G)has a hamiltonian path.As applications,we use this characterization to give several upper bounds on the hamiltonian path index of a graph.
文摘The exponential Randić index has important applications in the fields of biology and chemistry. The exponential Randić index of a graph G is defined as the sum of the weights e 1 d( u )d( v ) of all edges uv of G, where d( u ) denotes the degree of a vertex u in G. The paper mainly provides the upper and lower bounds of the exponential Randić index in quasi-tree graphs, and characterizes the extremal graphs when the bounds are achieved.
基金supported by the NSFC (11701001,11626030)the Support Plan for Outstanding Young Talents in Colleges in Anhui Province (Key project) (gxyqzD2020021)the Scientific Research Project of Colleges and Universities in Anhui Province,2023。
文摘In this paper,we consider the graph of the product of continuous functions in terms of Hausdorff and packing dimensions.More precisely,we show that,given a real number 1≤β≤2,any real-valued continuous function in C([0,1])can be decomposed into a product of two real-valued continuous functions,each having a graph of Hausdorff dimensionβ.In addition,a product decomposition result for the packing dimension is obtained.This work answers affirmatively two questions raised by Verma and Priyadarshi[14].
文摘The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.
基金Supported by a starting grant of Northumbria University.
文摘The Estrada index of a graph G on n vertices is defined by EE(G)=∑^(n)_(i=1)^(eλ_(i)),whereλ_(1),λ_(2),···,λ_(n)are the adjacency eigenvalues of G.We define two general types of dynamic graphs evolving according to continuous-time Markov processes with their stationary distributions matching the Erd¨os-R´enyi random graph and the random graph with given expected degrees,respectively.We formulate some new estimates and upper and lower bounds for the Estrada indices of these dynamic graphs.
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Over the past era,subgraph mining from a large collection of graph database is a crucial problem.In addition,scalability is another big problem due to insufficient storage.There are several security challenges associated with subgraph mining in today’s on-demand system.To address this downside,our proposed work introduces a Blockchain-based Consensus algorithm for Authenticated query search in the Large-Scale Dynamic Graphs(BCCA-LSDG).The two-fold process is handled in the proposed BCCA-LSDG:graph indexing and authenticated query search(query processing).A blockchain-based reputation system is meant to maintain the trust blockchain and cloud server of the proposed architecture.To resolve the issues and provide safe big data transmission,the proposed technique also combines blockchain with a consensus algorithm architecture.Security of the big data is ensured by dividing the BC network into distinct networks,each with a restricted number of allowed entities,data kept in the cloud gate server,and data analysis in the blockchain.The consensus algorithm is crucial for maintaining the speed,performance and security of the blockchain.Then Dual Similarity based MapReduce helps in mapping and reducing the relevant subgraphs with the use of optimal feature sets.Finally,the graph index refinement process is undertaken to improve the query results.Concerning query error,fuzzy logic is used to refine the index of the graph dynamically.The proposed technique outperforms advanced methodologies in both blockchain and non-blockchain systems,and the combination of blockchain and subgraph provides a secure communication platform,according to the findings.
文摘Let be a graph with n vertices and m edges. The sum of absolute value of all coefficients of matching polynomial is called Hosoya index. In this paper, we determine 2<sup>nd</sup> to 4<sup>th</sup> minimum Hosoya index of a kind of tetracyclic graph, with m = n +3.
文摘A lot of combinatorial objects have a natural bialgebra structure. In this paper, we prove that the vector space spanned by labeled simple graphs is a bialgebra with the conjunction product and the unshuffle coproduct. In fact, it is a Hopf algebra since it is graded connected. The main conclusions are that the vector space spanned by labeled simple graphs arising from the unshuffle coproduct is a Hopf algebra and that there is a Hopf homomorphism from permutations to label simple graphs.
文摘A graph G is said to be one modulo N-difference mean graph if there is an injective function f from the vertex set of G to the set , where N is the natural number and q is the number of edges of G and f induces a bijection from the edge set of G to given by and the function f is called a one modulo N-difference mean labeling of G. In this paper, we show that the graphs such as arbitrary union of paths, , ladder, slanting ladder, diamond snake, quadrilateral snake, alternately quadrilateral snake, , , , , friendship graph and admit one modulo N-difference mean labeling.
基金Supported by the National Natural Science Foundation of China(61164005,11161037,11101232,61440005,11461054)Supported by the Program for Changjiang Scholars and Innovative Research Team in Universities(IRT1068)+1 种基金Supported by the Research Fund for the Chunhui Program of Ministry of Education of China(Z2014022)Supported by the Nature Science Foundation from Qinghai Province(2014-ZJ-721,2014-ZJ-907,2015-ZJ-905)
文摘Since many large graphs are composed from some existing smaller graphs by using graph operations, say, the Cartesian product, the Lexicographic product and the Strong product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the Strong product of vertex-transitive graphs. In particular, we show that the Strong product of Cayley graphs is still a Cayley graph.
文摘The present paper deals with the gracefulness of unconnected graph (jC_(4n))∪P_m,and proves the following result:for positive integers n,j and m with n≥1,j≥2,the unconnected graph(jC_(4n))∪P_m is a graceful graph for m=j-1 or m≥n+j,where C_(4n) is a cycle with 4n vertexes,P_m is a path with m+1 vertexes,and(jC_(4n))∪P_m denotes the disjoint union of j-C_(4n) and P_m.
基金supported by National Natural Science Foundation of China (Grant Nos. 61272008 and 11571044)University Natural Science Research Project of Anhui Province (Grant No. KJ2016A003)Scientific Research Fund of Anhui University of Finance & Economics (Grant No. ACKY1532)
文摘This paper considers the edge-connectivity and the restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, we construct a Cayley graph with degree d whose restricted edge-connectivity is equal to d + s for given odd integer d and integer s with d 5 and 1 s d- 3, which answers a problem proposed ten years ago.
基金Supported by the Scientific Research Fund of Education Department of Hunan Province(08C518)
文摘The crossing number of cartesian products of paths and cycles with 5-vertex graphs mostly are known, but only few cartesian products of 5-vertex graphs with star K 1,n are known. In this paper, we will extent those results, and determine the crossing numbers of cartesian products of two 5-vertex graphs with star K 1,n .
文摘In this paper we define direct product of graphs and give a recipe for obtaining probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph is obtained by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determining probability of walk on compficated graphs. Using this method, we calculate the probability of Continuous-time classical and quantum random walks on many of finite direct product Cayley graphs (complete cycle, complete Kn, charter and n-cube). Also, we inquire that the classical state the stationary uniform distribution is reached as t→∞ but for quantum state is not always satisfied.
文摘Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n×P m (grid graph) with small n and m . But for large n and m , it is difficult to decide the exact fractional domination number. Motivated by this, nearly sharp upper and lower bounds are given to the fractional domination number of grid graphs. Furthermore, upper and lower bounds on the fractional total domination number of strong direct product of graphs are given.