Spectral Characterization Of Some Bounded Graphs

The main results of this paper can be divided into two parts. The first part characterizesgeneralized θ-graphs whose second largest eigenvalue between1and1.5. The main resultis that a generalized θ-graph G satisfies1<λ₂<1.5if and only if G is isomorphic toΓ₁, Γ₂,···, Γ₁₇depicted in Fig.1.In the second part we consider the eigenvalues of Laplacians of graphs and obtain thefollowing two results。（1） Let G-v be the graph obtained from graph G by deleting its vertex v. Thenμ_i+1（G）-1≤μ_i（G-v）≤μ_i（G）.where μ-i（G） is the ith largest eigenvalue of the Laplacian of G.（2） Let d3（G） be the third maximum degree of a graph G. Then μ₃（G）≥d₃（G）-1.