Nordhaus-Gaddum type inequalities of the second Aα-eigenvalue of a graph

Abstract

Let G be a graph with adjacency matrix A(G) and the degree diagonal matrix D(G). For any real α∈[0,1], the matrix Aα(G) of a graph is defined as Aα(G)=αD(G)+(1−α)A(G). Denote its eigenvalues by λ1(Aα(G))≥λ2(Aα(G))≥⋯≥λn(Aα(G)). In this paper, for 1/2<α<1, the Nordhaus-Gaddum type bound for the second largest Aα-eigenvalue of a graph is considered. We show that λ2(Aα(G))+λ2(Aα(G‾))≥nα−1, and the extremal graphs for which the equality holds are Kn and Sn. If G∉{Kn,Sn,Kn−e}, then λ2(Aα(G))+λ2(Aα(G‾))≥(n−1)α. Moreover, we give two upper Nordhaus-Gaddum type bounds for the second largest Aα-eigenvalue of a graph.