Abstract. Let D （U, V, W） be an oriented 3-partite graph with ｜ U ｜ = p, ｜V｜ = q and ｜W ｜ = r. For any vertex x in D（U,V,W）, let dx^＋ and dui^- be the outdegree and indegree ofx respectively. Define aui （o...Abstract. Let D （U, V, W） be an oriented 3-partite graph with ｜ U ｜ = p, ｜V｜ = q and ｜W ｜ = r. For any vertex x in D（U,V,W）, let dx^＋ and dui^- be the outdegree and indegree ofx respectively. Define aui （or simply ai） = q ＋ r ＋ dui^＋ - dui^-, bvj （or simply b j） = p ＋ r ＋ d^＋vj - d^-vj and cwk （or simply ck） =p ＋ q ＋ dwk^＋ -dwk^- as the scores of ui in U,vj in V and wk in W respectively. The set A of distinct scores of the vertices of D（U, V, W） is called its score set. In this paper, we prove that if a1 is a non-negative integer, ai（2 ≤ i ≤ n - 1） are even positive integers and an is any positive integer, then for n 〉 3, there exists an oriented 3-partite graph with the score set A ={a1,Σ2i=1 ai,…,Σni=1 ai}, except when A = {0, 2, 3}. Some more results for score sets in oriented 3-partite graphs are obtained.展开更多
Given a simple graph G,the oriented graph G^(σ)is obtained from G by orienting each edge and G is called the underlying graph of G^(σ).The skew-symmetric adjacency matrix S(G^(σ))of G^(σ),where the(u,v)-entry is 1...Given a simple graph G,the oriented graph G^(σ)is obtained from G by orienting each edge and G is called the underlying graph of G^(σ).The skew-symmetric adjacency matrix S(G^(σ))of G^(σ),where the(u,v)-entry is 1 if there is an arc from u to v,and−1 if there is an arc from v to u(and 0 otherwise),has eigenvalues of 0 or pure imaginary.The k-th-skew spectral moment of Gσis the sum of power k of all eigenvalues of S(G^(σ)),where k is a non-negative integer.The skew spectral moments can be used to produce graph catalogues.In this paper,we researched the skew spectral moments of some oriented trees and oriented unicyclic graphs and produced their catalogues in lexicographical order.We determined the last 2[d/4]oriented trees with underlying graph of diameter d and the last 2[g/4]+1 oriented unicyclic graphs with underlying graph of girth g,respectively.展开更多
In this paper, the oriented matrix analysis (OMA) method of system reliability has been discussed. OMA uses a oriented fault graph instead of a traditional fault tree model. By defining the specific logic operation an...In this paper, the oriented matrix analysis (OMA) method of system reliability has been discussed. OMA uses a oriented fault graph instead of a traditional fault tree model. By defining the specific logic operation and calculating a reachability matrix, the cut sets can be formed directly and the minimum cut sets can be easily obtained.展开更多
In this paper we study the structure of k-transitive closures of directed paths and formulate several properties. Concept of k-transitive orientation generalizes the traditional concept of transitive orientation of a ...In this paper we study the structure of k-transitive closures of directed paths and formulate several properties. Concept of k-transitive orientation generalizes the traditional concept of transitive orientation of a graph.展开更多
An oriented tripartite graph is the result of assigning a direction to each edge of a simple tripartite graph. For any vertex x in an oriented tripartite graph D（U, V, W）, let dx^＋ and dx^- denote the outdegree and...An oriented tripartite graph is the result of assigning a direction to each edge of a simple tripartite graph. For any vertex x in an oriented tripartite graph D（U, V, W）, let dx^＋ and dx^- denote the outdegree and indegree respectively of x. Define aui： dui^＋ - dus^-, bvj = dvj^＋ - dvj^- and cwk = dwk^＋ - dwk^- as the imbalances of the vertices ui in U, vj in V and wk in W respectively. In this paper, we obtain criteria for sequences of integers to be the imbalances of some oriented tripartite graph. Keywords Digraph, imbalance, outdegree, indegree, oriented graph, oriented tripartite graph, arc展开更多
文摘Abstract. Let D （U, V, W） be an oriented 3-partite graph with ｜ U ｜ = p, ｜V｜ = q and ｜W ｜ = r. For any vertex x in D（U,V,W）, let dx^＋ and dui^- be the outdegree and indegree ofx respectively. Define aui （or simply ai） = q ＋ r ＋ dui^＋ - dui^-, bvj （or simply b j） = p ＋ r ＋ d^＋vj - d^-vj and cwk （or simply ck） =p ＋ q ＋ dwk^＋ -dwk^- as the scores of ui in U,vj in V and wk in W respectively. The set A of distinct scores of the vertices of D（U, V, W） is called its score set. In this paper, we prove that if a1 is a non-negative integer, ai（2 ≤ i ≤ n - 1） are even positive integers and an is any positive integer, then for n 〉 3, there exists an oriented 3-partite graph with the score set A ={a1,Σ2i=1 ai,…,Σni=1 ai}, except when A = {0, 2, 3}. Some more results for score sets in oriented 3-partite graphs are obtained.
基金Supported by the Research Project of Jianghan University(Grant No.2021yb056)the National Natural Science Foundation of China(Grant Nos.1197115812061039).
文摘Given a simple graph G,the oriented graph G^(σ)is obtained from G by orienting each edge and G is called the underlying graph of G^(σ).The skew-symmetric adjacency matrix S(G^(σ))of G^(σ),where the(u,v)-entry is 1 if there is an arc from u to v,and−1 if there is an arc from v to u(and 0 otherwise),has eigenvalues of 0 or pure imaginary.The k-th-skew spectral moment of Gσis the sum of power k of all eigenvalues of S(G^(σ)),where k is a non-negative integer.The skew spectral moments can be used to produce graph catalogues.In this paper,we researched the skew spectral moments of some oriented trees and oriented unicyclic graphs and produced their catalogues in lexicographical order.We determined the last 2[d/4]oriented trees with underlying graph of diameter d and the last 2[g/4]+1 oriented unicyclic graphs with underlying graph of girth g,respectively.
文摘In this paper, the oriented matrix analysis (OMA) method of system reliability has been discussed. OMA uses a oriented fault graph instead of a traditional fault tree model. By defining the specific logic operation and calculating a reachability matrix, the cut sets can be formed directly and the minimum cut sets can be easily obtained.
文摘In this paper we study the structure of k-transitive closures of directed paths and formulate several properties. Concept of k-transitive orientation generalizes the traditional concept of transitive orientation of a graph.
文摘An oriented tripartite graph is the result of assigning a direction to each edge of a simple tripartite graph. For any vertex x in an oriented tripartite graph D（U, V, W）, let dx^＋ and dx^- denote the outdegree and indegree respectively of x. Define aui： dui^＋ - dus^-, bvj = dvj^＋ - dvj^- and cwk = dwk^＋ - dwk^- as the imbalances of the vertices ui in U, vj in V and wk in W respectively. In this paper, we obtain criteria for sequences of integers to be the imbalances of some oriented tripartite graph. Keywords Digraph, imbalance, outdegree, indegree, oriented graph, oriented tripartite graph, arc
