A Study Of The General P-neighbor Invulnerability Parameters Of Graphs

The rapid development of the Internet has brought unprecedented convenience to people's life,study,work and so on.However,due to the complex and changeable internal and external environment,networks may be interrupted.Therefore,network invulnerab-ility has become an important research subject in graph theory.The research on network invulnerability began with the non-neighbor invulnerability represented by traditional communication networks.Afterwards,the neighbor invulnerability represented by spy networks was introduced.However,the situation of network interruption in reality is often more complicated.Most of vertices or edges failure are between neighbor and non-neighbor situations.In other words,suppose that a vertex has been destroyed,its adjacent vertices will be failed with probability p,and the failure probability can be used to unify the above situations,which is called general p-neighbor invulnerability.Inspired by the definition of toughness and neighborhood toughness,the general p-neighbor toughness is introduced in this paper.The p-neighbor toughness formula of some graphs,line graphs and combinatorial graphs are given.The algorithm for the p-neighbor toughness of a star tree is given.The monotonicity and extreme value of the general p-neighbor toughness of some graphs on vertex failure probability are discussed.Because the general p-neighbor invulnerability of some graphs is not easy to distinguish by the general p-neighbor toughness,the concept of general p-neighbor scattering number is introduced.The p-neighbor scattering number formula of some graphs,line graphs and combinatorial graphs are given.The monotonicity and extreme value of the general p-neighbor scattering of some graphs on vertex failure probability are discussed.By analyzing the relationship between the vertex failure probability p and the parameter value,it is shown that the general p-neighbor toughness and the general p-neighbor scattering number can describe the subtle differences in the invulnerability of different networks.