Outer-independent Roman domination on graphs originates from the defensive strategy of Ancient Rome,which is that if any city without an army is attacked,a neighboring city with two armies could mobilize an army to su...Outer-independent Roman domination on graphs originates from the defensive strategy of Ancient Rome,which is that if any city without an army is attacked,a neighboring city with two armies could mobilize an army to support it and any two cities that have no army cannot be adjacent.The outer-independent Roman domination on graphs is an attractive topic in graph theory,and the definition is described as follows.Given a graph G=(V,E),a function f:V(G)→{0,1,2}is an outer-independent Roman dominating function(OIRDF)if f satisfies that every vertex v∈V with f(v)=0 has at least one adjacent vertex u∈N(v)with f(u)=2,where N(v)is the open neighborhood of v,and the set V0={v|f(v)=0}is an independent set.The weight of an OIRDF f is w(f)=∑_(v∈V)f(v).The value of minf w(f)is the outerindependent Roman domination number of G,denoted asγoiR(G).This paper is devoted to the study of the outer-independent Roman domination number of the Cartesian product of paths P_(n)□P_(m).With the help of computer,we find some recursive OIRDFs and then we present an upper bound ofγoiR(P_(n)□P_(m)).Furthermore,we prove the lower bound ofγoiR(P_(n)□P_(m))(n≤3)is equal to the upper bound.Hence,we achieve the exact value ofγoiR(P_(n)□P_(m))for n≤3 and the upper bound ofγoiR(P_(n)□P_(m))for n≥4.展开更多
Let G = （V,E） be a simple graph. For any real function g ： V →R and a subset S 包涵于V, we write g（S） = ∑v∈sg（v）. A function f ： V → [0,1] is said to be a fractional dominating function （FDF） of G if f（...Let G = （V,E） be a simple graph. For any real function g ： V →R and a subset S 包涵于V, we write g（S） = ∑v∈sg（v）. A function f ： V → [0,1] is said to be a fractional dominating function （FDF） of G if f（N[v]） ≥ 1 holds for every vertex v ∈ V（G）. The fractional domination number γf（G） of G is defined as γf（G） = min{f（V）｜f is an FDF of G }. The fractional total dominating function f is defined just as the fractional dominating function, the difference being that f（N（v）） ≥ 1 instead of f（N[v]）≥ 1. The fractional total domination number γ^0f（G） of G is analogous. In this note we give the exact values of γf（Cm× Pn） and γ^0f（Cm × Pn） for all integers m ≥ 3 and n ≥ 2.展开更多
The intuitionistic fuzzy set(IFS) based on fuzzy theory,which is of high efficiency to solve the fuzzy problem, has been introduced by Atanassov. Subsequently, he pushed the research one step further from the IFS to t...The intuitionistic fuzzy set(IFS) based on fuzzy theory,which is of high efficiency to solve the fuzzy problem, has been introduced by Atanassov. Subsequently, he pushed the research one step further from the IFS to the interval valued intuitionistic fuzzy set(IVIFS). On the basis of fuzzy set(FS), the IFS is a generalization concept. And the IFS is generalized to the IVIFS.In this paper, the definition of the sixth Cartesian product over IVIFSs is first introduced and its some properties are explored.We prove some equalities based on the operation and the relation over IVIFSs. Finally, we present one geometric interpretation and a numerical example of the sixth Cartesian product over IVIFSs.展开更多
A linear forest is a forest whose components are paths. The linear arboricity la （G） of a graph G is the minimum number of linear forests which partition the edge set E（G） of G. The Cartesian product G□H of two g...A linear forest is a forest whose components are paths. The linear arboricity la （G） of a graph G is the minimum number of linear forests which partition the edge set E（G） of G. The Cartesian product G□H of two graphs G and H is defined as the graph with vertex set V（G□H） = {（u, v）｜ u ∈V（G）, v∈V（H） } and edge set E（G□H） = { （ u, x） （ v, Y）｜u=v and xy∈E（H）, or uv∈E（G） and x=y}. Let Pm and Cm,, respectively, denote the path and cycle on m vertices and K, denote the complete graph on n vertices. It is proved that （Km□Pm）=[n＋1/2]for m≥2,la（Km□Cm）=[n＋2/2],and la（Km□Km）=[n＋m-1/2]. The methods to decompose these graphs into linear forests are given in the proofs. Furthermore, the linear arboricity conjecture is true for these classes of graphs.展开更多
Knowledge-based transfer learning techniques have shown good performance for brain tumor classification,especially with small datasets.However,to obtain an optimized model for targeted brain tumor classification,it is...Knowledge-based transfer learning techniques have shown good performance for brain tumor classification,especially with small datasets.However,to obtain an optimized model for targeted brain tumor classification,it is challenging to select a pre-trained deep learning(DL)model,optimal values of hyperparameters,and optimization algorithm(solver).This paper first presents a brief review of recent literature related to brain tumor classification.Secondly,a robust framework for implementing the transfer learning technique is proposed.In the proposed framework,a Cartesian product matrix is generated to determine the optimal values of the two important hyperparameters:batch size and learning rate.An extensive exercise consisting of 435 simulations for 11 state-of-the-art pre-trained DL models was performed using 16 paired hyperparameters from the Cartesian product matrix to input the model with the three most popular solvers(stochastic gradient descent with momentum(SGDM),adaptive moment estimation(ADAM),and root mean squared propagation(RMSProp)).The 16 pairs were formed using individual hyperparameter values taken from literature,which generally addressed only one hyperparameter for optimization,rather than making a grid for a particular range.The proposed framework was assessed using a multi-class publicly available dataset consisting of glioma,meningioma,and pituitary tumors.Performance assessment shows that ResNet18 outperforms all other models in terms of accuracy,precision,specificity,and recall(sensitivity).The results are also compared with existing state-of-the-art research work that used the same dataset.The comparison was mainly based on performance metric“accuracy”with support of three other parameters“precision,”“recall,”and“specificity.”The comparison shows that the transfer learning technique,implemented through our proposed framework for brain tumor classification,outperformed all existing approaches.To the best of our knowledge,the proposed framework is an efficient framework that helped reduce the computational complexity and the time to attain optimal values of two important hyperparameters and consequently the optimized model with an accuracy of 99.56%.展开更多
Let γ＊（D） denote the twin domination number of digraph D and let Cm Cn denote the Cartesian product of C_m and C_n, the directed cycles of length m, n ≥ 2. In this paper, we determine the exact values： γ＊（C_2...Let γ＊（D） denote the twin domination number of digraph D and let Cm Cn denote the Cartesian product of C_m and C_n, the directed cycles of length m, n ≥ 2. In this paper, we determine the exact values： γ＊（C_2□C_n） = n; γ＊（C_3 □C_n） = n if n ≡ 0（mod 3）,otherwise, γ＊（C_3□C_n） = n ＋ 1; γ＊（C_4□C_n） = n ＋ n/2 if n ≡ 0, 3, 5（mod 8）, otherwise,γ＊（C_4□C_n） = n ＋ n/2 ＋ 1; γ＊（C_5□C_n） = 2n; γ＊（C_6□C_n） = 2n if n ≡ 0（mod 3）, otherwise,γ＊（C_6□C_n） = 2n ＋ 2.展开更多
The g-good-neighbor connectivity of G is a generalization of the concept of connectivity, which is just for, and an important parameter in measuring the fault tolerance and reliability of interconnection network. Many...The g-good-neighbor connectivity of G is a generalization of the concept of connectivity, which is just for, and an important parameter in measuring the fault tolerance and reliability of interconnection network. Many well-known networks can be constructed by the Cartesian products of some simple graphs. In this paper, we determine the g-good-neighbor connectivity of some Cartesian product graphs. We give the exact value of g-good-neighbor connectivity of the Cartesian product of two complete graphs and for , mesh for , cylindrical grid and torus for .展开更多
The book embedding of a graph G consists of placing the vertices of G in a line called spine and assigning edges of the graph to pages so that the edges assigned to the same page do not intersect. The number of pages ...The book embedding of a graph G consists of placing the vertices of G in a line called spine and assigning edges of the graph to pages so that the edges assigned to the same page do not intersect. The number of pages is the minimum number in which the graph can be embedded. In this paper, we study the book embedding of the Cartesian product Pm × Sn, Pm × Wn, Cn × Sm, Cn × Wm, and get an upper bound of their pagenumber.展开更多
For the subset S■V(G), if every path with k vertices in a graph G contains at least one vertex from S, we call that S is a k-path vertex cover set of the graph G. Obviously, the subset is not unique. The cardinality ...For the subset S■V(G), if every path with k vertices in a graph G contains at least one vertex from S, we call that S is a k-path vertex cover set of the graph G. Obviously, the subset is not unique. The cardinality of the minimum k-path vertex cover set of a graph G is called the k-path vertex cover number, we denote it by ψk(G). In this paper, a lower or upper bound of ψk for some Cartesian product graphs is presented.展开更多
A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality...A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by <img src="Edit_82259359-0135-4a65-9378-b767f0405b48.png" alt="" />. A set <em>C ⊆ V (G)</em> is called a chromatic set if <em>C</em> contains all vertices of different colors in<em> G</em>, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by <img src="Edit_d849148d-5778-459b-abbb-ff25b5cd659b.png" alt="" />. A geo-chromatic set<em> S</em><sub><em>c</em></sub><em> ⊆ V (G</em><em>)</em> is both a geodetic set and a chromatic set. The geo-chromatic number <img src="Edit_505e203c-888c-471c-852d-4b9c2dd1a31c.png" alt="" /><em> </em>of<em> G</em> is the minimum cardinality among all geo-chromatic sets of<em> G</em>. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.展开更多
The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian...The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian product graphs.展开更多
Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more verti...Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more vertices with function values 1 than with −1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate The signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 3, 4, 5 and arbitrary n.展开更多
A set ?is a dominating set of G if every vertex of ?is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obta...A set ?is a dominating set of G if every vertex of ?is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obtained from G by adding new edges between every two vertices having distance 2 in G. In this paper we study the domination number of square of graphs, find a bound for domination number of square of Cartesian product of cycles, and find the exact value for some of them.展开更多
Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination numb...Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, ?(mod 3) and bounds for otherwise.展开更多
Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i...Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i） = ｜f＊^-1（i）｜. A labeling f is called friendly if ｜vf（1） - vf（0）｜ ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if（G） = e（1） - el（0）. The set [if（G） ｜ f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI（G）. In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.展开更多
Most results on crossing numbers of graphs focus on some special graphs, such as the Cartesian products of small graphs with path, star and cycle. In this paper, we obtain the crossing number formula of Cartesian prod...Most results on crossing numbers of graphs focus on some special graphs, such as the Cartesian products of small graphs with path, star and cycle. In this paper, we obtain the crossing number formula of Cartesian products of wheel Wm with path Pn for arbitrary m ≥ 3 and n ≥ 1.展开更多
M（J, {ms ＊ ns}, {Cs}） be the collection of Cartesian products of two homogenous Moran sets with the same ratios {cs} Where J = [0, 1] × [0, 1]. Then the maximal and minimal values of the Hausdorff dimensions f...M（J, {ms ＊ ns}, {Cs}） be the collection of Cartesian products of two homogenous Moran sets with the same ratios {cs} Where J = [0, 1] × [0, 1]. Then the maximal and minimal values of the Hausdorff dimensions for the elements in M are obtained without any restriction on {msns} or {cs}.展开更多
For positive integers j and k with j ≥ k, an L（j, k）-labeling of a graph G is an assignment of nonnegative integers to V（G） such that the difference between labels of adjacent vertices is at least j, and the diff...For positive integers j and k with j ≥ k, an L（j, k）-labeling of a graph G is an assignment of nonnegative integers to V（G） such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L（j, k）-labeling of a graph G is the difference between the maximum and minimum integers it uses. The λj, k-number of G is the minimum span taken over all L（j, k）-labelings of G. An m-（j, k）-circular labeling of a graph G is a function f ： V（G） →{0, 1, 2,..., m - 1} such that ｜f（u） - f（v）｜m ≥ j if u and v are adjacent; and ｜f（u） - f（v）｜m 〉 k ifu and v are at distance two, where ｜x｜m = min{｜xl｜, m-｜x｜}. The minimum integer m such that there exists an m-（j, k）-circular labeling of G is called the σj,k-number of G and is denoted by σj,k（G）. This paper determines the σ2,1-number of the Cartesian product of any three complete graphs.展开更多
An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence ...An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence chromatic number of the Cartesian product of a path and a path,a path and a wheel,a path and a fan,and a path and a star.展开更多
文摘Outer-independent Roman domination on graphs originates from the defensive strategy of Ancient Rome,which is that if any city without an army is attacked,a neighboring city with two armies could mobilize an army to support it and any two cities that have no army cannot be adjacent.The outer-independent Roman domination on graphs is an attractive topic in graph theory,and the definition is described as follows.Given a graph G=(V,E),a function f:V(G)→{0,1,2}is an outer-independent Roman dominating function(OIRDF)if f satisfies that every vertex v∈V with f(v)=0 has at least one adjacent vertex u∈N(v)with f(u)=2,where N(v)is the open neighborhood of v,and the set V0={v|f(v)=0}is an independent set.The weight of an OIRDF f is w(f)=∑_(v∈V)f(v).The value of minf w(f)is the outerindependent Roman domination number of G,denoted asγoiR(G).This paper is devoted to the study of the outer-independent Roman domination number of the Cartesian product of paths P_(n)□P_(m).With the help of computer,we find some recursive OIRDFs and then we present an upper bound ofγoiR(P_(n)□P_(m)).Furthermore,we prove the lower bound ofγoiR(P_(n)□P_(m))(n≤3)is equal to the upper bound.Hence,we achieve the exact value ofγoiR(P_(n)□P_(m))for n≤3 and the upper bound ofγoiR(P_(n)□P_(m))for n≥4.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1136102411061014)the Jiangxi Provincial Science and Technology Project(Grant No.KJLD12067)
文摘Let G = （V,E） be a simple graph. For any real function g ： V →R and a subset S 包涵于V, we write g（S） = ∑v∈sg（v）. A function f ： V → [0,1] is said to be a fractional dominating function （FDF） of G if f（N[v]） ≥ 1 holds for every vertex v ∈ V（G）. The fractional domination number γf（G） of G is defined as γf（G） = min{f（V）｜f is an FDF of G }. The fractional total dominating function f is defined just as the fractional dominating function, the difference being that f（N（v）） ≥ 1 instead of f（N[v]）≥ 1. The fractional total domination number γ^0f（G） of G is analogous. In this note we give the exact values of γf（Cm× Pn） and γ^0f（Cm × Pn） for all integers m ≥ 3 and n ≥ 2.
基金supported by the National Natural Science Foundation of China(61373174)
文摘The intuitionistic fuzzy set(IFS) based on fuzzy theory,which is of high efficiency to solve the fuzzy problem, has been introduced by Atanassov. Subsequently, he pushed the research one step further from the IFS to the interval valued intuitionistic fuzzy set(IVIFS). On the basis of fuzzy set(FS), the IFS is a generalization concept. And the IFS is generalized to the IVIFS.In this paper, the definition of the sixth Cartesian product over IVIFSs is first introduced and its some properties are explored.We prove some equalities based on the operation and the relation over IVIFSs. Finally, we present one geometric interpretation and a numerical example of the sixth Cartesian product over IVIFSs.
基金The National Natural Science Foundation of China(No.10971025)
文摘A linear forest is a forest whose components are paths. The linear arboricity la （G） of a graph G is the minimum number of linear forests which partition the edge set E（G） of G. The Cartesian product G□H of two graphs G and H is defined as the graph with vertex set V（G□H） = {（u, v）｜ u ∈V（G）, v∈V（H） } and edge set E（G□H） = { （ u, x） （ v, Y）｜u=v and xy∈E（H）, or uv∈E（G） and x=y}. Let Pm and Cm,, respectively, denote the path and cycle on m vertices and K, denote the complete graph on n vertices. It is proved that （Km□Pm）=[n＋1/2]for m≥2,la（Km□Cm）=[n＋2/2],and la（Km□Km）=[n＋m-1/2]. The methods to decompose these graphs into linear forests are given in the proofs. Furthermore, the linear arboricity conjecture is true for these classes of graphs.
文摘Knowledge-based transfer learning techniques have shown good performance for brain tumor classification,especially with small datasets.However,to obtain an optimized model for targeted brain tumor classification,it is challenging to select a pre-trained deep learning(DL)model,optimal values of hyperparameters,and optimization algorithm(solver).This paper first presents a brief review of recent literature related to brain tumor classification.Secondly,a robust framework for implementing the transfer learning technique is proposed.In the proposed framework,a Cartesian product matrix is generated to determine the optimal values of the two important hyperparameters:batch size and learning rate.An extensive exercise consisting of 435 simulations for 11 state-of-the-art pre-trained DL models was performed using 16 paired hyperparameters from the Cartesian product matrix to input the model with the three most popular solvers(stochastic gradient descent with momentum(SGDM),adaptive moment estimation(ADAM),and root mean squared propagation(RMSProp)).The 16 pairs were formed using individual hyperparameter values taken from literature,which generally addressed only one hyperparameter for optimization,rather than making a grid for a particular range.The proposed framework was assessed using a multi-class publicly available dataset consisting of glioma,meningioma,and pituitary tumors.Performance assessment shows that ResNet18 outperforms all other models in terms of accuracy,precision,specificity,and recall(sensitivity).The results are also compared with existing state-of-the-art research work that used the same dataset.The comparison was mainly based on performance metric“accuracy”with support of three other parameters“precision,”“recall,”and“specificity.”The comparison shows that the transfer learning technique,implemented through our proposed framework for brain tumor classification,outperformed all existing approaches.To the best of our knowledge,the proposed framework is an efficient framework that helped reduce the computational complexity and the time to attain optimal values of two important hyperparameters and consequently the optimized model with an accuracy of 99.56%.
基金Supported by the National Natural Science Foundation of China(Grant Nos.6136302011301450+2 种基金11226294)the Youth Science and Technology Education Project of Xinjiang(Grant No.2013731011)China Scholarship Council
文摘Let γ＊（D） denote the twin domination number of digraph D and let Cm Cn denote the Cartesian product of C_m and C_n, the directed cycles of length m, n ≥ 2. In this paper, we determine the exact values： γ＊（C_2□C_n） = n; γ＊（C_3 □C_n） = n if n ≡ 0（mod 3）,otherwise, γ＊（C_3□C_n） = n ＋ 1; γ＊（C_4□C_n） = n ＋ n/2 if n ≡ 0, 3, 5（mod 8）, otherwise,γ＊（C_4□C_n） = n ＋ n/2 ＋ 1; γ＊（C_5□C_n） = 2n; γ＊（C_6□C_n） = 2n if n ≡ 0（mod 3）, otherwise,γ＊（C_6□C_n） = 2n ＋ 2.
文摘The g-good-neighbor connectivity of G is a generalization of the concept of connectivity, which is just for, and an important parameter in measuring the fault tolerance and reliability of interconnection network. Many well-known networks can be constructed by the Cartesian products of some simple graphs. In this paper, we determine the g-good-neighbor connectivity of some Cartesian product graphs. We give the exact value of g-good-neighbor connectivity of the Cartesian product of two complete graphs and for , mesh for , cylindrical grid and torus for .
基金The NSF(A2015202301)HUSTP(ZD201506)RFHED(QN2016044)of Hebei Province
文摘The book embedding of a graph G consists of placing the vertices of G in a line called spine and assigning edges of the graph to pages so that the edges assigned to the same page do not intersect. The number of pages is the minimum number in which the graph can be embedded. In this paper, we study the book embedding of the Cartesian product Pm × Sn, Pm × Wn, Cn × Sm, Cn × Wm, and get an upper bound of their pagenumber.
基金Supported by the National Natural Science Foundation of China(Grant Nos.61463026,61463027).
文摘For the subset S■V(G), if every path with k vertices in a graph G contains at least one vertex from S, we call that S is a k-path vertex cover set of the graph G. Obviously, the subset is not unique. The cardinality of the minimum k-path vertex cover set of a graph G is called the k-path vertex cover number, we denote it by ψk(G). In this paper, a lower or upper bound of ψk for some Cartesian product graphs is presented.
文摘A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by <img src="Edit_82259359-0135-4a65-9378-b767f0405b48.png" alt="" />. A set <em>C ⊆ V (G)</em> is called a chromatic set if <em>C</em> contains all vertices of different colors in<em> G</em>, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by <img src="Edit_d849148d-5778-459b-abbb-ff25b5cd659b.png" alt="" />. A geo-chromatic set<em> S</em><sub><em>c</em></sub><em> ⊆ V (G</em><em>)</em> is both a geodetic set and a chromatic set. The geo-chromatic number <img src="Edit_505e203c-888c-471c-852d-4b9c2dd1a31c.png" alt="" /><em> </em>of<em> G</em> is the minimum cardinality among all geo-chromatic sets of<em> G</em>. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.
文摘The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian product graphs.
文摘Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more vertices with function values 1 than with −1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate The signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 3, 4, 5 and arbitrary n.
文摘A set ?is a dominating set of G if every vertex of ?is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obtained from G by adding new edges between every two vertices having distance 2 in G. In this paper we study the domination number of square of graphs, find a bound for domination number of square of Cartesian product of cycles, and find the exact value for some of them.
文摘Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, ?(mod 3) and bounds for otherwise.
基金Supported by FRG/07-08/II-08 Hong Kong Baptist University
文摘Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i） = ｜f＊^-1（i）｜. A labeling f is called friendly if ｜vf（1） - vf（0）｜ ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if（G） = e（1） - el（0）. The set [if（G） ｜ f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI（G）. In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.
基金the National Natural Science Foundation of China （No. 10771062） and New Century Excellent Talents in University （No. 07-0276）.
文摘Most results on crossing numbers of graphs focus on some special graphs, such as the Cartesian products of small graphs with path, star and cycle. In this paper, we obtain the crossing number formula of Cartesian products of wheel Wm with path Pn for arbitrary m ≥ 3 and n ≥ 1.
基金Supported by the National Natural Science Foundation of China (No.10771082 and 10871180)
文摘M（J, {ms ＊ ns}, {Cs}） be the collection of Cartesian products of two homogenous Moran sets with the same ratios {cs} Where J = [0, 1] × [0, 1]. Then the maximal and minimal values of the Hausdorff dimensions for the elements in M are obtained without any restriction on {msns} or {cs}.
基金Foundation item： the National Natural Science Foundation of China （No. 10671033） the Science Foundation of Southeast University （No. XJ0607230） the Natural Science Foundation of Nantong University （No. 08Z003）.
文摘For positive integers j and k with j ≥ k, an L（j, k）-labeling of a graph G is an assignment of nonnegative integers to V（G） such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L（j, k）-labeling of a graph G is the difference between the maximum and minimum integers it uses. The λj, k-number of G is the minimum span taken over all L（j, k）-labelings of G. An m-（j, k）-circular labeling of a graph G is a function f ： V（G） →{0, 1, 2,..., m - 1} such that ｜f（u） - f（v）｜m ≥ j if u and v are adjacent; and ｜f（u） - f（v）｜m 〉 k ifu and v are at distance two, where ｜x｜m = min{｜xl｜, m-｜x｜}. The minimum integer m such that there exists an m-（j, k）-circular labeling of G is called the σj,k-number of G and is denoted by σj,k（G）. This paper determines the σ2,1-number of the Cartesian product of any three complete graphs.
基金Supported by the State Ethnic Affairs Commission of China (Grant No. 08XB07)
文摘An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence chromatic number of the Cartesian product of a path and a path,a path and a wheel,a path and a fan,and a path and a star.
