Research On The α-spectra Problems Of Graphs

The spectral theory of graph is a very important research field in algebraic graph theory and combinatorial matrix theory.As one of the main research directions and hot spots in recent years,the spectral polar problem of graph has been widely used in computer science,chemistry,biology,physics and other fields.Its core content is to study the maximum eigenvalue of the correlation matrix of a graph and the corresponding polar graph when the maximum eigenvalue of the correlation matrix is obtained.In other words,the problem of spectral extremum of a graph is to determine the extremum of its eigenvalue and characterize its corresponding polar graph in a given graph class G.Given a graph G,we write:·Gn denotes the set of all n order graphs;·Gnk denotes the set of all graphs of order n with vertex connectivity of k:·Gnk denotes the set of all graphs of order n with edge connectivity of k;·Gn,k denotes the set of all connected graphs of order n with k cut points;·Gn,k denotes the set of all connected graphs of order n with k cut edges;·K(p,q)denotes a graph of q edges connected to Kp at a point outside Kp,where p>q;·K(p,q,r)denotes a graph with q pendant edges hanging on Kp and the maximum degree is r.The α-matrix of graph is defined as:Aα=αD(G)+(1-α)A(G),0≤α≤1.The α-spectral radius of a graph is the maximum eigenvalue of the correspondingα-matrix of the graph.In this paper,we study the problem of α-spectral extremum of graphs,depict the polar graphs with maximum α-spectral radius in graphs with given edge cut set,point cut set,cut edge and cut point,and give the upper bound of their α-spectral radius.At the same time,we give the edge condition of a pancyclic graph and the bound of its α-spectral radius:1.The graph Kn is the unique one in Gn with maximum α-spectral radius,and G=K(n-1,0)is the unique one in Gn0 or Gn0 with maximum a-spectral radius.2.For each k=1,2,…,n-2,the graph G=K(n-1.k)is the unique one with maximum α-spectral radius in Gnn and(?)3.For each k=1,2,…,n-,the graph G=K(n-1,k)is the unique one with maximum α-spectral radius in Gnk and(?)4.For each k=1,2,…,[n/2],the graph G=K(n-k,k,n-k)is the unique one with maximum α-spectral radius in Gn,k and(?)5.For each k=1,2,…,n-1,the graph G=K(n-k,k,n-1)is the unique one with maximum α-spectral radius in Gn,k and(?)6.For a connected graph G with n≥5 and δ≥2,if m≥(n-2/2)+4,then G is a pancyclic graph unless G∈NP1={K1,2 ∨ 4K1,K2∨(Kn-4+2K1),K2∨(K1+K1,3),K3 ∨(K1+K1,4),K3∨(K2+3K1),K3 ∨(K2+K1,3),K3 ∨ 4K1,K4 ∨5K1,K5 ∨ 6K1,(K2 ∨ 2K1)∨ 5K1.7.For the connected graph G with n≥5,m≥(n-2/2)+4,δ≥2,there is(?).