On the third largest eigenvalue of graphs

Let G   be a graph with eigenvalues λ1(G)≥⋯≥λn(G)λ1(G)≥⋯≥λn(G). In this paper we investigate the value of λ3(G)λ3(G). We show that if the multiplicity of −1 as an eigenvalue of G   is at most n−13n−13, then λ3(G)≥0λ3(G)≥0. We prove that λ3(G)∈{−2,−1,1−52} or −0.59<λ3(G)<−0.5−0.59<λ3(G)<−0.5 or λ3(G)>−0.496λ3(G)>−0.496. We find that λ3(G)=−2 if and only if G≅P3G≅P3 and λ3(G)=1−52 if and only if G≅P4G≅P4, where PnPn is the path on n vertices. In addition we characterize the graphs whose third largest eigenvalue equals −1. We find all graphs G   with −0.59<λ3(G)<−0.5−0.59<λ3(G)<−0.5. Finally we investigate the limit points of the set {λ3(G): G  is  a  graph  such  thatλ3(G)<0} and show that 0 and −0.5 are two limit points of this set.