In order to study the behavior and interconnection of network devices,graphs structures are used to formulate the properties in terms of mathematical models.Mesh network(meshnet)is a LAN topology in which devices are ...In order to study the behavior and interconnection of network devices,graphs structures are used to formulate the properties in terms of mathematical models.Mesh network(meshnet)is a LAN topology in which devices are connected either directly or through some intermediate devices.These terminating and intermediate devices are considered as vertices of graph whereas wired or wireless connections among these devices are shown as edges of graph.Topological indices are used to reflect structural property of graphs in form of one real number.This structural invariant has revolutionized the field of chemistry to identify molecular descriptors of chemical compounds.These indices are extensively used for establishing relationships between the structure of nanotubes and their physico-chemical properties.In this paper a representation of sodium chloride(NaCl)is studied,because structure of NaCl is same as the Cartesian product of three paths of length exactly like a mesh network.In this way the general formula obtained in this paper can be used in chemistry as well as for any degree-based topological polynomials of three-dimensional mesh networks.展开更多
Let G be a graph and di denote the degree of a vertex vi in G,and let f(x,y)be a real symmetric function.Then one can get an edge-weighted graph in such a way that for each edge vivj of G,the weight of vivj is assigne...Let G be a graph and di denote the degree of a vertex vi in G,and let f(x,y)be a real symmetric function.Then one can get an edge-weighted graph in such a way that for each edge vivj of G,the weight of vivj is assigned by the value f(d_(i),d_(j)).Hence,we have a weighted adjacency matrix Af(G)of G,in which the ij-entry is equal to f(d_(i),d_(j))if v_(i)v_(j)∈E(G)and 0 otherwise.This paper attempts to unify the study of spectral properties for the weighted adjacency matrix Af(G)of graphs with a degree-based edge-weight function f(x,y).Some lower and upper bounds of the largest weighted adjacency eigenvalueλ1 are given,and the corresponding extremal graphs are characterized.Bounds of the energy for the ε_(f)(G)weighted adjacency matrix A_(f)(G)are also obtained.By virtue of the unified method,this makes many earlier results become special cases of our results.展开更多
A class of graph invariants referred to today as topological indices are inefficient progressively acknowledged by scientific experts and others to be integral assets in the depiction of structural phenomena.The struc...A class of graph invariants referred to today as topological indices are inefficient progressively acknowledged by scientific experts and others to be integral assets in the depiction of structural phenomena.The structure of an interconnection network can be represented by a graph.In the network,vertices represent the processor nodes and edges represent the links between the processor nodes.Graph invariants play a vital feature in graph theory and distinguish the structural properties of graphs and networks.A topological descriptor is a numerical total related to a structure that portray the topology of structure and is invariant under structure automorphism.There are various uses of graph theory in the field of basic science.The main notable utilization of a topological descriptor in science was by Wiener in the investigation of paraffin breaking points.In this paper we study the topological descriptor of a newly design hexagon star network.More preciously,we have computed variation of the Randic0 R0,fourth Zagreb M4,fifth Zagreb M5,geometric-arithmetic GA;atom-bond connectivity ABC;harmonic H;symmetric division degree SDD;first redefined Zagreb,second redefined Zagreb,third redefined Zagreb,augmented Zagreb AZI,Albertson A;Irregularity measures,Reformulated Zagreb,and forgotten topological descriptors for hexagon star network.In the analysis of the quantitative structure property relationships(QSPRs)and the quantitative structure activity relationships(QSARs),graph invariants are important tools to approximate and predicate the properties of the biological and chemical compounds.We also gave the numerical and graphical representations comparisons of our different results.展开更多
文摘In order to study the behavior and interconnection of network devices,graphs structures are used to formulate the properties in terms of mathematical models.Mesh network(meshnet)is a LAN topology in which devices are connected either directly or through some intermediate devices.These terminating and intermediate devices are considered as vertices of graph whereas wired or wireless connections among these devices are shown as edges of graph.Topological indices are used to reflect structural property of graphs in form of one real number.This structural invariant has revolutionized the field of chemistry to identify molecular descriptors of chemical compounds.These indices are extensively used for establishing relationships between the structure of nanotubes and their physico-chemical properties.In this paper a representation of sodium chloride(NaCl)is studied,because structure of NaCl is same as the Cartesian product of three paths of length exactly like a mesh network.In this way the general formula obtained in this paper can be used in chemistry as well as for any degree-based topological polynomials of three-dimensional mesh networks.
基金Supported by NSFC(Grant Nos.12131013 and 12161141006)。
文摘Let G be a graph and di denote the degree of a vertex vi in G,and let f(x,y)be a real symmetric function.Then one can get an edge-weighted graph in such a way that for each edge vivj of G,the weight of vivj is assigned by the value f(d_(i),d_(j)).Hence,we have a weighted adjacency matrix Af(G)of G,in which the ij-entry is equal to f(d_(i),d_(j))if v_(i)v_(j)∈E(G)and 0 otherwise.This paper attempts to unify the study of spectral properties for the weighted adjacency matrix Af(G)of graphs with a degree-based edge-weight function f(x,y).Some lower and upper bounds of the largest weighted adjacency eigenvalueλ1 are given,and the corresponding extremal graphs are characterized.Bounds of the energy for the ε_(f)(G)weighted adjacency matrix A_(f)(G)are also obtained.By virtue of the unified method,this makes many earlier results become special cases of our results.
文摘A class of graph invariants referred to today as topological indices are inefficient progressively acknowledged by scientific experts and others to be integral assets in the depiction of structural phenomena.The structure of an interconnection network can be represented by a graph.In the network,vertices represent the processor nodes and edges represent the links between the processor nodes.Graph invariants play a vital feature in graph theory and distinguish the structural properties of graphs and networks.A topological descriptor is a numerical total related to a structure that portray the topology of structure and is invariant under structure automorphism.There are various uses of graph theory in the field of basic science.The main notable utilization of a topological descriptor in science was by Wiener in the investigation of paraffin breaking points.In this paper we study the topological descriptor of a newly design hexagon star network.More preciously,we have computed variation of the Randic0 R0,fourth Zagreb M4,fifth Zagreb M5,geometric-arithmetic GA;atom-bond connectivity ABC;harmonic H;symmetric division degree SDD;first redefined Zagreb,second redefined Zagreb,third redefined Zagreb,augmented Zagreb AZI,Albertson A;Irregularity measures,Reformulated Zagreb,and forgotten topological descriptors for hexagon star network.In the analysis of the quantitative structure property relationships(QSPRs)and the quantitative structure activity relationships(QSARs),graph invariants are important tools to approximate and predicate the properties of the biological and chemical compounds.We also gave the numerical and graphical representations comparisons of our different results.
