Abstract: Let G be a unicyclic graph of order n, λ₁(G) and λ₂(G) be the largest eigenvalue and second largest eigenvalue of the adjacent matrix of G, μ₁(G) and μ₂(G) be the largest eigenvalue and second largest eigenvalue of the Laplacian matrix of G, respectively. The separator of G is defined as S_A(G)=λ₁(G) -λ₂(G). The Laplacian separator of G is defined as S_L(G)=μ₁(G) -μ₂(G). In this paper, we study the (Laplacian) separator of unicyclic graphs, and give the extremal graphs which attain the second maximum separator and second maximum Laplacian separator respectively.

Key words: unicyclic graph, separator, Laplacian separator