On The Related Topics Of Vertex Neighbor Tenacity Of Graphs

The network invulnerability analysis and design have been valued by scholars at home and abroad in recent decades.The primary study is the quantitative measure of a network invulnerability by introducing invulnerability parameters.The study of invulnerability parameters based on communication networks is in-depth and mature,but the research on the neighbor invulnerability parameters of network represented by spy networks is relatively fewer.The existing network neighbor invulnerability parameters have some deficiencies.The neighbor connectivity only considers the difficulty of breaking the network and does not consider the situation after the network was destroyed.The physical meanings and dimensions of the addition or subtraction terms in the vertex neighbor scattering number and the vertex neighbor integrity are different.The neighbor toughness does not consider the largest branch of the survived subgraphs.Based on the study of the neighbor invulnerability parameters and the idea of tenacity,this paper introduces the concept of vertex neighbor tenacity,and some related researches are done.The paper is divided into five parts,which are as follows.The first part briefly introduces the concept of network and its invulnerability,the research background,significance and research status.The second part is the basic knowledge of graph theory and the main network invulnerability parameters with research results.In the third part,we first give the definition of the vertex neighbor tenacity and the formula for calculating the vertex neighbor tenacity of several special graphs.Next,we discuss the upper and lower bounds of the vertex neighbor tenacity of general graphs,and the relationship between the vertex neighbor tenacity and some other parameters of graphs.By comparing the invulnerability of several graphs based on the main neighbor invulnerability parameters,it is concluded that the vertex neighbor tenacity is better than other neighbor invulnerability parameters and the graph invulnerability can be measured more accurately.In the fourth part,the calculation formulas of the vertex neighbor tenacity of several kinds of operation graphs(joint graphs,line graphs,complement graphs,Cartesian product graphs)are given.The fifth part is a summary of this paper,including the innovation and the issues worth studying further.