One of the most recent developments in the field of graph theory is the analysis of networks such as Butterfly networks,Benes networks,Interconnection networks,and David-derived networks using graph theoretic paramete...One of the most recent developments in the field of graph theory is the analysis of networks such as Butterfly networks,Benes networks,Interconnection networks,and David-derived networks using graph theoretic parameters.The topological indices(TIs)have been widely used as graph invariants among various graph theoretic tools.Quantitative structure activity relationships(QSAR)and quantitative structure property relationships(QSPR)need the use of TIs.Different structure-based parameters,such as the degree and distance of vertices in graphs,contribute to the determination of the values of TIs.Among other recently introduced novelties,the classes of ev-degree and ve-degree dependent TIs have been extensively explored for various graph families.The current research focuses on the development of formulae for different ev-degree and ve-degree dependent TIs for s−dimensional Benes network and certain networks derived from it.In the end,a comparison between the values of the TIs for these networks has been presented through graphical tools.展开更多
A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v E V(G) there is a vertex w E W such that d(u, w) ≠ d(v, w). A resolving set of minimum card...A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v E V(G) there is a vertex w E W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V(G), the distance between u and S is the number minses d(u,s). A k-partition II = {$1,$2,..., Sk} of V(G) is called a resolving partition if for every two distinct vertices u, v E V(G) there is a set Si in H such that d(u, Si) ≠ d(v, Si). The minimum k for which there is a resolving k-partition of V(G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn, an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i - j (rood n) E C, where C C Zn has the property that C = -C and 0 ¢ C. The circulant graph is denoted by Xn,△ where A = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn,3 with connection set C = {1, 3, n - 1} and prove that dim(Xn,3) is independent of choice of n by showing that dim(Xn,a) = {3 for all n ≡0 (mod 4),4 for all n≡2 (mod4).We also study the partition dimension of a family of circulant graphs Xm,4 with connection set C = {±1, ±2} and prove that pd(Xn,4) is independent of choice of n and show that pd(X5,4) = 5 and pd(Xn,a) ={ 3 for all odd n≥9,4 for all even n≥6 and n=7.展开更多
In this paper, we consider the family of generalized Petersen graphs P(n,4). We prove that the metric dimension of P(n, 4) is 3 when n = 0 (mod 4), and is 4 when n = 4k + 3 (k is even).For n = 1,2 (mod 4) a...In this paper, we consider the family of generalized Petersen graphs P(n,4). We prove that the metric dimension of P(n, 4) is 3 when n = 0 (mod 4), and is 4 when n = 4k + 3 (k is even).For n = 1,2 (mod 4) and n = 4k + 3 (k is odd), we prove that the metric dimension of P(n,4) is bounded above by 4. This shows that each graph of the family of generalized Petersen graphs P(n, 4) has constant metric dimension.展开更多
基金supported by the National Natural Science Foundation of China (Grant No.61702291)China Henan International Joint Laboratory for Multidimensional Topology and Carcinogenic Characteristics Analysis of Atmospheric Particulate Matter PM2.5.
文摘One of the most recent developments in the field of graph theory is the analysis of networks such as Butterfly networks,Benes networks,Interconnection networks,and David-derived networks using graph theoretic parameters.The topological indices(TIs)have been widely used as graph invariants among various graph theoretic tools.Quantitative structure activity relationships(QSAR)and quantitative structure property relationships(QSPR)need the use of TIs.Different structure-based parameters,such as the degree and distance of vertices in graphs,contribute to the determination of the values of TIs.Among other recently introduced novelties,the classes of ev-degree and ve-degree dependent TIs have been extensively explored for various graph families.The current research focuses on the development of formulae for different ev-degree and ve-degree dependent TIs for s−dimensional Benes network and certain networks derived from it.In the end,a comparison between the values of the TIs for these networks has been presented through graphical tools.
基金Supported by the Higher Education Commission of Pakistan (Grant No. 17-5-3(Ps3-257) HEC/Sch/2006)
文摘A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v E V(G) there is a vertex w E W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V(G), the distance between u and S is the number minses d(u,s). A k-partition II = {$1,$2,..., Sk} of V(G) is called a resolving partition if for every two distinct vertices u, v E V(G) there is a set Si in H such that d(u, Si) ≠ d(v, Si). The minimum k for which there is a resolving k-partition of V(G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn, an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i - j (rood n) E C, where C C Zn has the property that C = -C and 0 ¢ C. The circulant graph is denoted by Xn,△ where A = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn,3 with connection set C = {1, 3, n - 1} and prove that dim(Xn,3) is independent of choice of n by showing that dim(Xn,a) = {3 for all n ≡0 (mod 4),4 for all n≡2 (mod4).We also study the partition dimension of a family of circulant graphs Xm,4 with connection set C = {±1, ±2} and prove that pd(Xn,4) is independent of choice of n and show that pd(X5,4) = 5 and pd(Xn,a) ={ 3 for all odd n≥9,4 for all even n≥6 and n=7.
文摘In this paper, we consider the family of generalized Petersen graphs P(n,4). We prove that the metric dimension of P(n, 4) is 3 when n = 0 (mod 4), and is 4 when n = 4k + 3 (k is even).For n = 1,2 (mod 4) and n = 4k + 3 (k is odd), we prove that the metric dimension of P(n,4) is bounded above by 4. This shows that each graph of the family of generalized Petersen graphs P(n, 4) has constant metric dimension.