The Generalized Connectivity Of Several Networks

Today's society is made up of a variety of networks,such as railway networks,road net-works,communication networks and wireless sensor networks and so on.In order to achieve transportation and communication purposes,the network usually meets specific parameter requirements.A network can usually be abstracted into a graph,so that the requirements for network parameters can be translated into requirements for graph parameters,such as Euler,Hamiltonian,and connectivity.Connectivity is one of the basic concepts of graph theory,which can measure the reliability of a network.The better the connectivity of the graph,the greater the possibility that the network will work in the event of a failure of the vertex and edge of the graph.In recent years,mathematicians have introduced new connectivity concepts,such as limiting connectivity,essential connectivity,and generalized connectivity,because the classical connectivity measure of network connectivity is flawed.This paper mainly studies the generalized connectivity of several networks.Given a connected graph G and S?V?G?with|S|?2,a tree T is a S-Steiner tree if S?V?T?.Two S-Steiner trees T₁and T₂are edge-disjoint if E?T₁??E?T₂?=?;Two S-Steiner trees T₁and T₂are internally disjoint if E?T₁??E?T₂?=?,and V?T₁??V?T₂?=S.Let?_G?S??resp.?_G?S??be the maximum size of a set of edge-disjoint?resp.internally disjoint?S-Steiner trees in G,and let?_k?G??resp.?_k?G??be the minimum?_G?S??resp.?_G?S??when S ranges over all k-subsets of V?G?.Clearly,?₂?G??resp.?₂?G??is the classical edge?resp.vertex?-connectivity??G??resp.??G??.In this paper,we mainly construct the edge-disjoint S-Steiner Trees through the cyclic structure of recursive circulant,Hamilton Properties,and the distribution of points in S.Finally,we can determine the ?₃ connectivity of the recursive circulant G?N,d?.Similarly,we mainly construct the internally disjoint S-Steiner Trees to determine the ?₃ connectivity of the recursive circulant G?N,d?.In addition,we can construct the edge-disjoint S-Steiner trees throng the structure and the properties of the Cartesian Product of some trees(T_n1???T_n2???···???T_nk)to determine the ?₄ connectivity.