A digraph D is k-ordered if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a cycle C such that C encounters the vertices of S in the specified order.In particular,we say that D is k-ordered h...A digraph D is k-ordered if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a cycle C such that C encounters the vertices of S in the specified order.In particular,we say that D is k-ordered hamiltonian if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a hamiltonian cycle C such that the vertices of S are encountered on C in the specified order.In this paper,sufficient conditions for digraphs to be ordered and ordered hamiltonian have been given.展开更多
The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It ...The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on and for to be supereulerian. In particular, we show that if and are symmetrically connected or partially symmetric, then is supereulerian.展开更多
Digraph-based causal models have been widely used to model the cause and effect behavior of process systems. Signed digraphs （SDG） capture the direction of the effect. It should be mentioned that there are loops in ...Digraph-based causal models have been widely used to model the cause and effect behavior of process systems. Signed digraphs （SDG） capture the direction of the effect. It should be mentioned that there are loops in SDG generated from chemical process. From the point of the inherent operability, the worst unsafe factor is the SDG having positive loops that means any disturbance occurring within the loop will propagate through the nodes one by one and are amplified gradually, so the system may lose control, which may lead to an accident. So finding the positive loops in a SDG and treating these unsafe factors in a proper manner can improve the inherent safety of a chemical process. This article proposed a method that can detect the above-mentioned unsafe factors in the proc- ess conceptual design stage automatically through the analysis of the SDG generated from the chemical process. A case study is illustrated to show the working of the algorithm, and then a complicated case from industry is studied to depict the effectiveness of the proposed algorithm.展开更多
This paper presents an efficient parallel algorithm for the shortest path problem in planar layered digraphs that runs in O(log^3n) time with n processors. The algorithms uses a divide and conquer approach and is base...This paper presents an efficient parallel algorithm for the shortest path problem in planar layered digraphs that runs in O(log^3n) time with n processors. The algorithms uses a divide and conquer approach and is based on the novel idea of a one-way separator, which has the property that any directed path can be crossed only once.展开更多
Recently,scrambling index and competition index are widely applied to stochastic matrices and food webs. By analyzing the relationship of scrambling index and 2-competition index,n-「d/2」+ 1 was proved to be an upper...Recently,scrambling index and competition index are widely applied to stochastic matrices and food webs. By analyzing the relationship of scrambling index and 2-competition index,n-「d/2」+ 1 was proved to be an upper bound of the 2-competition2 index of a primitive digraph with exact d loops in this article.Moreover,the maximum index problem and the index set problem for the 2-competition index of primitive digraphs with minimally strong digraphs were settled.展开更多
Denote by C n(S) the circulant digraph with vertex set Z n={0,1,2,...,n-1} and symbol set S(≠-S)Z n }. Let X be the automorphism group of C n(S) and X 0 the stabilizer of 0 in X. Then C n(S) is arc ...Denote by C n(S) the circulant digraph with vertex set Z n={0,1,2,...,n-1} and symbol set S(≠-S)Z n }. Let X be the automorphism group of C n(S) and X 0 the stabilizer of 0 in X. Then C n(S) is arc transitive if and only if X 0 acts transitively on S. In this paper, C n(S) with X 0| S being the symmetric group is characterized by its symbol set. By the way all the arc transitive circulant digraphs of degree 2 and 3 are given.展开更多
The strong product digraph G1■G2 is constructed by the known digraph G1 and G2 of small order. The digraph G1■G2 constructed by the strong product method contain G1 and G2 as its sub-graphs. Therefore, the topologic...The strong product digraph G1■G2 is constructed by the known digraph G1 and G2 of small order. The digraph G1■G2 constructed by the strong product method contain G1 and G2 as its sub-graphs. Therefore, the topological structure and properties of these small digraphs G1 and G2 must affect the topological structure and properties of the large digraph. By using group theory, we prove some algebraic properties of strong product of digraphs, such as commutative law, associative law and so on.展开更多
Abstract In this paper, the problem of fault tolerant routings in fault tolerant networks is considered. A routing in a network assigns to each ordered pair of nodes a fixed path. All communication among nodes must ...Abstract In this paper, the problem of fault tolerant routings in fault tolerant networks is considered. A routing in a network assigns to each ordered pair of nodes a fixed path. All communication among nodes must go on this routing. When either a node or a link in a fault tolerant network fails, the communication from one node to another using this faulty element must be sent via one or more intermediate nodes along a sequence of paths determined by this routing. An important and practical problem is how to choose a routing in the network such that intermediate nodes to ensure communication are small for any fault set. Let C d be a directed cycle of order d . In this paper. The author first discusses connectivity of Cartesian product digraphs, then proves that the Cartesian product digraph C d 1 ×C d 2 ×...×C d n (d i≥2,1≤i≤n) has a routing such that at most one intermediate node is needed to ensure transmission of messages among all non faulty nodes so long as the number of faults is less than n . This is a generalization of Dolev et al's result for the n dimensional cube.展开更多
The generalized Kautz digraphs have many good properties as interconnection network topologies. In this note, the bounds of the absorbant number for the generalized Kautz digraph are given, and some sufficient conditi...The generalized Kautz digraphs have many good properties as interconnection network topologies. In this note, the bounds of the absorbant number for the generalized Kautz digraph are given, and some sufficient conditions for the absorbant number of the generalized Kautz digraph attaining the bounds are presented.展开更多
A digraph D is supereulerian if D has a spanning eulerian subdigraph. Bang- Jensen and Thomasse conjectured that if the arc-strong connectivity ）,（D） of α digraph D is not less than the independence number α（D）...A digraph D is supereulerian if D has a spanning eulerian subdigraph. Bang- Jensen and Thomasse conjectured that if the arc-strong connectivity ）,（D） of α digraph D is not less than the independence number α（D）, then D is supereulerian. In this paper, we prove that if D is an extended cycle, an extended hamiltonian digraph, an arc-locally semicomplete digraph, an extended arc-locally semicomplete digraph, an extension of two kinds of eulerian digraph, a hypo-semicomplete digraph or an extended hypo-semicomplete digraph satisfying λ（D） ≥α（D）, then D is supereulerian.展开更多
we prove that the Connectivities of Minimal Cayley Coset Digraphs are their regular degrees. Connectivity of transitive digraphs and a combinatorial propertyof finite groups Ann., Discrete Math., 8 1980 61--64 ...we prove that the Connectivities of Minimal Cayley Coset Digraphs are their regular degrees. Connectivity of transitive digraphs and a combinatorial propertyof finite groups Ann., Discrete Math., 8 1980 61--64 Meng Jixiang and Huang Qiongxiang On the connectivity of Cayley digraphs, to appear Sabidussi, G. Vertex transitive graphs Monatsh. Math., 68 1969 426--438 Watkins, M. E. Connectivity of transitive graphs J. Combin. Theory, 8 1970 23--29 Zemor, G. On positive and negative atoms of Cayley digraphs Discrete Applied Math., 23 1989 193--195 Department of Mathematics,Xinjiang University,Urumpi 830046.APPLIED MATHEMATICS 3. Statement of Inexact Method Here we assume F to be continuousely differentiable. Inexact Newton method was first studied in the solution of smooth equations (see ). Now, such a technique has been widely used in optimizations, nonlinear complementarity problems and nonsmooth equations (see, and , etc.) In order to establish the related inexact methods,we introduce a nonlinear operator T(x): R n R n . Its components are defined as follows: (T(x)p) i=[HL(2:1,Z;2,Z] (x k+p k) i, if i∈(x k), H i(x k)+ min {(p k) i,F i(x k) Tp k}, if i∈(x k), F i(x k)+F i(x k) Tp k, i∈(x k).(3.1) Then, it is clear that the subproblem (2.5) turns to T(x k)p k=0.(3.2) In inexact algorithm, we determine p k in the followinginexact way ( see ). ‖T(x k)p k‖ υ k‖H(x k)‖,(3.3) where υ k is a given positive sequence. It is then obviously that (3.2),or equivalently (2.5), is a special case of (3.3) corresponding to υ k=0 . In particular, (3.3) can be used as a termination rule of the iterative process for solving (2.5). The following proposition shows the existence of λ k satisfying (2.4). Proposition 3.1. Let F be continuously differe ntiable. υ k is chosen so that υ k for some constant ∈(0,1). Then p k generated by (3.3) is a descent direction of θ at x k, and for some constant σ∈(0, min (1/2,1- holds θ(x k)-θ(x k+λ kp k) 2σλ kθ(x k)(3.4) for all sufficiently small λ k>0. Proof For simplification, we omit the lower subscripts k and denote (x k) i , H i(x k) , (BH(x k)p k) i , etc.by x i , H i , (BHp) i , etc. respectively. To estimate the directional derivative of θ at x k along p k , we divide it into three parts: D p k θ(x k)=H T(x k)BH(x k)p k=T 1+T 2+T 3,(3.5) where T 1=Σ i∈α k H i(BHp) i , T 2=Σ i∈β k H i(BHp) i , T 3=Σ i∈γ k H i(BHp) i . Consider i∈α k= k∪α -(x k) . In this case, we always have H i(BH(x)p) i=H i 2+H i(x i+p i) . If i∈ k , then H i(BHp) i -H i 2+|H i‖(T(x)p) i|. If i∈α -(x k) , then x i<0 . We have either x i+p i 0 , or x i+p i<0 . When x i+p i 0 , we get H i(BH(x)p) i -H i 2 .In the later case, x i+p i<0 , so H i(BH(x)p) i=-H i 2+|H i‖x i+p i|. Then, by elementary computation, we deduce that T 1 -Σi∈α kH i 2+Σ i∈α k|H i‖(T(x)p) i|.(3.6) Received March 1, 1995. 1991 MR Subject Classification: 05C25展开更多
In this paper, we study the bases and base sets of primitive symmetric loop-free （generalized） signed digraphs on n vertices. We obtain sharp upper bounds of the bases, and show that the base sets of the classes of ...In this paper, we study the bases and base sets of primitive symmetric loop-free （generalized） signed digraphs on n vertices. We obtain sharp upper bounds of the bases, and show that the base sets of the classes of such digraphs are （2, 3,..., 2n - 1}. We also give a new proof of an important result obtained by Cheng and Liu.展开更多
As far as the minimal spanning tree problem for the digraph with asymmetric weightsis concerned, an explicit integer programming model is proposed, which could be solved successfullyusing the integer programming packa...As far as the minimal spanning tree problem for the digraph with asymmetric weightsis concerned, an explicit integer programming model is proposed, which could be solved successfullyusing the integer programming packages such as LINDO, and furthermore this model is extendedinto the stochastic version, that is, the minimal spanning tree problem for the digraph with theweights is not constant but random variables. Several algorithms are also developed to solve themodels. Finally, a numerical demonstration is given.展开更多
Let G = （V,A） be a digraph.A set T of vertices of G is a twin dominating set of G if for every vertex v ∈ V ／ T.There exist u,w ∈ T （possibly u = w） such that （u,v）,（v,w） ∈ A.The twin domination number γ...Let G = （V,A） be a digraph.A set T of vertices of G is a twin dominating set of G if for every vertex v ∈ V ／ T.There exist u,w ∈ T （possibly u = w） such that （u,v）,（v,w） ∈ A.The twin domination number γ＊（G） of G is the cardinality of a minimum twin dominating set of G.In this paper we consider the twin domination number in generalized Kautz digraphs GK（n,d）.In these digraphs,we establish bounds on the twin domination number and give a sufficient condition for the twin domination number attaining the lower bound.We give the exact values of the twin domination numbers by constructing minimum twin dominating sets for some special generalized Kautz digraphs.展开更多
A jump digraph J(D)of a directed multigraph D has as its vertex set being A(D),the set of arcs of D;where(a,b)is an arc of J(D)if and only if there are vertices u1,v1,u2,v2in D such that a=(u1,v1),b=(u2,v2)and v1≠u2....A jump digraph J(D)of a directed multigraph D has as its vertex set being A(D),the set of arcs of D;where(a,b)is an arc of J(D)if and only if there are vertices u1,v1,u2,v2in D such that a=(u1,v1),b=(u2,v2)and v1≠u2.In this paper,we give a well characterized directed multigraph families H1and H2,and prove that a jump digraph J(D)of a directed multigraph D is strongly connected if and only if D?H1.Specially,J(D)is weakly connected if and only if D?H2.The following results are obtained:(ⅰ)There exists a family D of wellcharacterized directed multigraphs such that strongly connected jump digraph J(D)of directed multigraph is strongly trail-connected if and only if D?D.(ⅱ)Every strongly connected jump digraph J(D)of directed multigraph D is weakly trail-connected,and so is supereulerian.(ⅲ)Every weakly connected jump digraph J(D)of directed multigraph D has a spanning trail.展开更多
For given two digraphs,we can construct a larger digraph through join.The two digraphs that make up the join are called the factors of the join.In this paper,we give a necessary and sufficient condition that the funct...For given two digraphs,we can construct a larger digraph through join.The two digraphs that make up the join are called the factors of the join.In this paper,we give a necessary and sufficient condition that the function on the join determined by the discrete Morse functions on factors is a discrete Morse function.Moreover,we further prove the discrete Morse theory on join when the factors satisfy certain conditions.展开更多
Recently, the primitive symmetric signed digraphs on $n$ vertices with the maximum base 2n and the primitive symmetric loop-free signed digraphs on n vertices with the maximum base 2n-1 are characterized, respectively...Recently, the primitive symmetric signed digraphs on $n$ vertices with the maximum base 2n and the primitive symmetric loop-free signed digraphs on n vertices with the maximum base 2n-1 are characterized, respectively. In this paper, the primitive symmetric signed digraphs with loops on n vertices with the base 2n-1 are characterized, and then the primitive symmetric signed digraphs on n vertices with the second maximum base 2n-1 are characterized.展开更多
Let S be a primitive non-powerful symmetric loop-free signed digraph on even n vertices with base 3 and minimum number of arcs. In [Lihua YOU, Yuhan WU. Primitive non- powerful symmetric loop-free signed digraphs with...Let S be a primitive non-powerful symmetric loop-free signed digraph on even n vertices with base 3 and minimum number of arcs. In [Lihua YOU, Yuhan WU. Primitive non- powerful symmetric loop-free signed digraphs with given base and minimum number of arcs. Linear Algebra Appl., 2011, 434（5）, 1215-1227], authors conjectured that D is the underlying digraph of S with exp（D）= 3 if and only if D is isomorphic to EDn,3,3, where EDn,3,3 = （V, A） is a digraph with V = {1, 2,..., n}, A = {（1, i）, （i, 1） [ 3 〈： i 〈 n} U {（2i - 1, 2i）, （2i, 2i - 1） [ 2 〈 〈 2} U {（2, 3）, （3, 2）, （2, 4）, （4, 2）}）. In this paper, we show the conjecture is true and completely characterize the underlying digraphs which have base 3 and the minimum number of arcs.展开更多
基金Foundation item： Supported by the National Natural Science Foundation of China（61070229） Supported by the Natural Science Foundation of Shanxi Province（2008011010）
文摘A digraph D is k-ordered if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a cycle C such that C encounters the vertices of S in the specified order.In particular,we say that D is k-ordered hamiltonian if for every sequence S：v 1,v 2,…,v k of k distinct vertices,there exists a hamiltonian cycle C such that the vertices of S are encountered on C in the specified order.In this paper,sufficient conditions for digraphs to be ordered and ordered hamiltonian have been given.
文摘The 2-sum of two digraphs and , denoted , is the digraph obtained from the disjoint union of and by identifying an arc in with an arc in . A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on and for to be supereulerian. In particular, we show that if and are symmetrically connected or partially symmetric, then is supereulerian.
文摘Digraph-based causal models have been widely used to model the cause and effect behavior of process systems. Signed digraphs （SDG） capture the direction of the effect. It should be mentioned that there are loops in SDG generated from chemical process. From the point of the inherent operability, the worst unsafe factor is the SDG having positive loops that means any disturbance occurring within the loop will propagate through the nodes one by one and are amplified gradually, so the system may lose control, which may lead to an accident. So finding the positive loops in a SDG and treating these unsafe factors in a proper manner can improve the inherent safety of a chemical process. This article proposed a method that can detect the above-mentioned unsafe factors in the proc- ess conceptual design stage automatically through the analysis of the SDG generated from the chemical process. A case study is illustrated to show the working of the algorithm, and then a complicated case from industry is studied to depict the effectiveness of the proposed algorithm.
文摘This paper presents an efficient parallel algorithm for the shortest path problem in planar layered digraphs that runs in O(log^3n) time with n processors. The algorithms uses a divide and conquer approach and is based on the novel idea of a one-way separator, which has the property that any directed path can be crossed only once.
基金National Natural Science Foundations of China(No.11272100,No.50865001)
文摘Recently,scrambling index and competition index are widely applied to stochastic matrices and food webs. By analyzing the relationship of scrambling index and 2-competition index,n-「d/2」+ 1 was proved to be an upper bound of the 2-competition2 index of a primitive digraph with exact d loops in this article.Moreover,the maximum index problem and the index set problem for the 2-competition index of primitive digraphs with minimally strong digraphs were settled.
文摘Denote by C n(S) the circulant digraph with vertex set Z n={0,1,2,...,n-1} and symbol set S(≠-S)Z n }. Let X be the automorphism group of C n(S) and X 0 the stabilizer of 0 in X. Then C n(S) is arc transitive if and only if X 0 acts transitively on S. In this paper, C n(S) with X 0| S being the symmetric group is characterized by its symbol set. By the way all the arc transitive circulant digraphs of degree 2 and 3 are given.
基金Supported by National Natural Science Foundation of China(Grant No. 11551002)Natural Science Foundation of Qinghai Province (Grant No. 2019-ZJ-7093)。
文摘The strong product digraph G1■G2 is constructed by the known digraph G1 and G2 of small order. The digraph G1■G2 constructed by the strong product method contain G1 and G2 as its sub-graphs. Therefore, the topological structure and properties of these small digraphs G1 and G2 must affect the topological structure and properties of the large digraph. By using group theory, we prove some algebraic properties of strong product of digraphs, such as commutative law, associative law and so on.
文摘Abstract In this paper, the problem of fault tolerant routings in fault tolerant networks is considered. A routing in a network assigns to each ordered pair of nodes a fixed path. All communication among nodes must go on this routing. When either a node or a link in a fault tolerant network fails, the communication from one node to another using this faulty element must be sent via one or more intermediate nodes along a sequence of paths determined by this routing. An important and practical problem is how to choose a routing in the network such that intermediate nodes to ensure communication are small for any fault set. Let C d be a directed cycle of order d . In this paper. The author first discusses connectivity of Cartesian product digraphs, then proves that the Cartesian product digraph C d 1 ×C d 2 ×...×C d n (d i≥2,1≤i≤n) has a routing such that at most one intermediate node is needed to ensure transmission of messages among all non faulty nodes so long as the number of faults is less than n . This is a generalization of Dolev et al's result for the n dimensional cube.
基金supported by the National Natural Science Foundation of China (Grant Nos.10571117,60773078)Shu Guang Plan of Shanghai Education Development Foundation (Grant No.06SG42)the Shanghai Leading Academic Discipline Project(Grant No.J50101)
文摘The generalized Kautz digraphs have many good properties as interconnection network topologies. In this note, the bounds of the absorbant number for the generalized Kautz digraph are given, and some sufficient conditions for the absorbant number of the generalized Kautz digraph attaining the bounds are presented.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1176107161363020)+1 种基金Science and Technology Innovation Project of Xinjiang Normal University(Grant No.XSY201602013)the"13th Five-Year"Plan for Key Discipline Mathematics of Xinjiang Normal University(Grant No.17SDKD1107)
文摘A digraph D is supereulerian if D has a spanning eulerian subdigraph. Bang- Jensen and Thomasse conjectured that if the arc-strong connectivity ）,（D） of α digraph D is not less than the independence number α（D）, then D is supereulerian. In this paper, we prove that if D is an extended cycle, an extended hamiltonian digraph, an arc-locally semicomplete digraph, an extended arc-locally semicomplete digraph, an extension of two kinds of eulerian digraph, a hypo-semicomplete digraph or an extended hypo-semicomplete digraph satisfying λ（D） ≥α（D）, then D is supereulerian.
文摘we prove that the Connectivities of Minimal Cayley Coset Digraphs are their regular degrees. Connectivity of transitive digraphs and a combinatorial propertyof finite groups Ann., Discrete Math., 8 1980 61--64 Meng Jixiang and Huang Qiongxiang On the connectivity of Cayley digraphs, to appear Sabidussi, G. Vertex transitive graphs Monatsh. Math., 68 1969 426--438 Watkins, M. E. Connectivity of transitive graphs J. Combin. Theory, 8 1970 23--29 Zemor, G. On positive and negative atoms of Cayley digraphs Discrete Applied Math., 23 1989 193--195 Department of Mathematics,Xinjiang University,Urumpi 830046.APPLIED MATHEMATICS 3. Statement of Inexact Method Here we assume F to be continuousely differentiable. Inexact Newton method was first studied in the solution of smooth equations (see ). Now, such a technique has been widely used in optimizations, nonlinear complementarity problems and nonsmooth equations (see, and , etc.) In order to establish the related inexact methods,we introduce a nonlinear operator T(x): R n R n . Its components are defined as follows: (T(x)p) i=[HL(2:1,Z;2,Z] (x k+p k) i, if i∈(x k), H i(x k)+ min {(p k) i,F i(x k) Tp k}, if i∈(x k), F i(x k)+F i(x k) Tp k, i∈(x k).(3.1) Then, it is clear that the subproblem (2.5) turns to T(x k)p k=0.(3.2) In inexact algorithm, we determine p k in the followinginexact way ( see ). ‖T(x k)p k‖ υ k‖H(x k)‖,(3.3) where υ k is a given positive sequence. It is then obviously that (3.2),or equivalently (2.5), is a special case of (3.3) corresponding to υ k=0 . In particular, (3.3) can be used as a termination rule of the iterative process for solving (2.5). The following proposition shows the existence of λ k satisfying (2.4). Proposition 3.1. Let F be continuously differe ntiable. υ k is chosen so that υ k for some constant ∈(0,1). Then p k generated by (3.3) is a descent direction of θ at x k, and for some constant σ∈(0, min (1/2,1- holds θ(x k)-θ(x k+λ kp k) 2σλ kθ(x k)(3.4) for all sufficiently small λ k>0. Proof For simplification, we omit the lower subscripts k and denote (x k) i , H i(x k) , (BH(x k)p k) i , etc.by x i , H i , (BHp) i , etc. respectively. To estimate the directional derivative of θ at x k along p k , we divide it into three parts: D p k θ(x k)=H T(x k)BH(x k)p k=T 1+T 2+T 3,(3.5) where T 1=Σ i∈α k H i(BHp) i , T 2=Σ i∈β k H i(BHp) i , T 3=Σ i∈γ k H i(BHp) i . Consider i∈α k= k∪α -(x k) . In this case, we always have H i(BH(x)p) i=H i 2+H i(x i+p i) . If i∈ k , then H i(BHp) i -H i 2+|H i‖(T(x)p) i|. If i∈α -(x k) , then x i<0 . We have either x i+p i 0 , or x i+p i<0 . When x i+p i 0 , we get H i(BH(x)p) i -H i 2 .In the later case, x i+p i<0 , so H i(BH(x)p) i=-H i 2+|H i‖x i+p i|. Then, by elementary computation, we deduce that T 1 -Σi∈α kH i 2+Σ i∈α k|H i‖(T(x)p) i|.(3.6) Received March 1, 1995. 1991 MR Subject Classification: 05C25
基金Supported by the National Natural Science Foundation of China(Grant Nos.1090106111071088)the Zhujiang Technology New Star Foundation of Guangzhou(Grant No.2011J2200090)
文摘In this paper, we study the bases and base sets of primitive symmetric loop-free （generalized） signed digraphs on n vertices. We obtain sharp upper bounds of the bases, and show that the base sets of the classes of such digraphs are （2, 3,..., 2n - 1}. We also give a new proof of an important result obtained by Cheng and Liu.
文摘As far as the minimal spanning tree problem for the digraph with asymmetric weightsis concerned, an explicit integer programming model is proposed, which could be solved successfullyusing the integer programming packages such as LINDO, and furthermore this model is extendedinto the stochastic version, that is, the minimal spanning tree problem for the digraph with theweights is not constant but random variables. Several algorithms are also developed to solve themodels. Finally, a numerical demonstration is given.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.10571117, 60773078)the Shuguang Plan of Shanghai Education Development Foundation (Grant No.06SG42)the Shanghai Leading Academic Discipline Project(Grant No.J50101)
文摘Let G = （V,A） be a digraph.A set T of vertices of G is a twin dominating set of G if for every vertex v ∈ V ／ T.There exist u,w ∈ T （possibly u = w） such that （u,v）,（v,w） ∈ A.The twin domination number γ＊（G） of G is the cardinality of a minimum twin dominating set of G.In this paper we consider the twin domination number in generalized Kautz digraphs GK（n,d）.In these digraphs,we establish bounds on the twin domination number and give a sufficient condition for the twin domination number attaining the lower bound.We give the exact values of the twin domination numbers by constructing minimum twin dominating sets for some special generalized Kautz digraphs.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11761071,11861068)Guizhou Key Laboratory of Big Data Statistical Analysis,China(Grant No.[2019]5103)the Natural Science Foundation of Xinjiang Uygur Autonomous Region(Grant No.2022D01E13)。
文摘A jump digraph J(D)of a directed multigraph D has as its vertex set being A(D),the set of arcs of D;where(a,b)is an arc of J(D)if and only if there are vertices u1,v1,u2,v2in D such that a=(u1,v1),b=(u2,v2)and v1≠u2.In this paper,we give a well characterized directed multigraph families H1and H2,and prove that a jump digraph J(D)of a directed multigraph D is strongly connected if and only if D?H1.Specially,J(D)is weakly connected if and only if D?H2.The following results are obtained:(ⅰ)There exists a family D of wellcharacterized directed multigraphs such that strongly connected jump digraph J(D)of directed multigraph is strongly trail-connected if and only if D?D.(ⅱ)Every strongly connected jump digraph J(D)of directed multigraph D is weakly trail-connected,and so is supereulerian.(ⅲ)Every weakly connected jump digraph J(D)of directed multigraph D has a spanning trail.
基金Supported by Science and Technology Project of Hebei Education Department(ZD2022168)Project of Cangzhou Normal University(XNJJLYB2021006)Science and Technology Project of Hebei Education Department(ZD2020410)
文摘For given two digraphs,we can construct a larger digraph through join.The two digraphs that make up the join are called the factors of the join.In this paper,we give a necessary and sufficient condition that the function on the join determined by the discrete Morse functions on factors is a discrete Morse function.Moreover,we further prove the discrete Morse theory on join when the factors satisfy certain conditions.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1090106111071088)+1 种基金the Zhujiang Technology New Star Foundation of Guangzhou(Grant No.2011J2200090)Program on International Cooperation and Innovation of Guangdong Province Education Department(Grant No.2012gjhz0007)
文摘Recently, the primitive symmetric signed digraphs on $n$ vertices with the maximum base 2n and the primitive symmetric loop-free signed digraphs on n vertices with the maximum base 2n-1 are characterized, respectively. In this paper, the primitive symmetric signed digraphs with loops on n vertices with the base 2n-1 are characterized, and then the primitive symmetric signed digraphs on n vertices with the second maximum base 2n-1 are characterized.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1090106111071088)+1 种基金Programon International Cooperation and Innovation,Department of Education,Guangdong Province(Grant No.2012gjhz0007)the Zhujiang Technology New Star Foundation of Guangzhou City(Grant No.2011J2200090)
文摘Let S be a primitive non-powerful symmetric loop-free signed digraph on even n vertices with base 3 and minimum number of arcs. In [Lihua YOU, Yuhan WU. Primitive non- powerful symmetric loop-free signed digraphs with given base and minimum number of arcs. Linear Algebra Appl., 2011, 434（5）, 1215-1227], authors conjectured that D is the underlying digraph of S with exp（D）= 3 if and only if D is isomorphic to EDn,3,3, where EDn,3,3 = （V, A） is a digraph with V = {1, 2,..., n}, A = {（1, i）, （i, 1） [ 3 〈： i 〈 n} U {（2i - 1, 2i）, （2i, 2i - 1） [ 2 〈 〈 2} U {（2, 3）, （3, 2）, （2, 4）, （4, 2）}）. In this paper, we show the conjecture is true and completely characterize the underlying digraphs which have base 3 and the minimum number of arcs.
