On the third largest eigenvalue of a graph Original Research Article

Let λ1(G)greater-or-equal, slantedcdots, three dots, centeredgreater-or-equal, slantedλn(G) be the eigenvalues of a graph G. We explore the distribution of eigenvalues of a graph G with λ3(G)<0 and prove thatandwhere n,m are the number of vertices and edges of G, respectively. According to above property, let λk(G)=−1+α,0less-than-or-equals, slantαless-than-or-equals, slant1,3less-than-or-equals, slantkless-than-or-equals, slant(n+1)/2. Then we show that−1−λj(G)less-than-or-equals, slantα, kless-than-or-equals, slantjless-than-or-equals, slantn−k+1.If Gc, the complement of G, is bipartite, then λ2(Gc)<1 implies λ3(G)<0,λ2(Gc)=λ3(Gc)=λ implies λ3(G)=−1+λ. If Gc is not bipartite, then λ3(G)greater-or-equal, slantedλ3(P7c). Finally for a tree T that is not in two special classes, we prove that image with equality if and only if T congruent with P6.