On Unicyclic Graphs Whose Second Largest Eigenvalue Is In [1,√2]

Let G be a simple and undirected graph with n vertices,and let A be the adjacency matrix of G.Its eigenvalues denoted by λ1,λ2,…,λn,and we assume that λ1≥λ2≥…≥λn.The eigenvalues of A also are the eigenvalues of G,and λ2is the second largest eigenvalue of G.Connected graphs in which the number of edges equals the number of vertices are called the unicyclic graphs.Denote P(G；λ)=P(G) be the characteristic polynomial of the adjacency matrix A of G.If P(G；λ)=P(H；λ),then G and H are cospectral for A, denoted by G～H.If H(?)G whenever H～G.If G(?)H,λ1=λ1’,λ2=λ2’,…, λn=λn’.λ1’≥λ2’≥…≥λn’ are the eigenvalues of H.In the first chapter of this paper we introduce the research progress of the second largest eigenvalues as well as some basic concepts；in chapter2,we study the unicyclic graphs whose second largest eigenvalue is in[1,√2),and obtain some basic characteristics of the kind of the graphs；in chapter3,we determine all unicyclic graphs with λ2=√2,and obtain all minimal forbidden subgraphs with λ2(Un)<√2.