On the location of the third largest eigenvalue of graphs

‎Let $G$ be a graph with eigenvalues $\lambda_1(G)\geq\cdots\geq\lambda_n(G)$‎. ‎In this paper we study the possible value of $\lambda_3(G)$‎. ‎We prove that for every graph $G$‎, ‎$\lambda_3(G)\in\{-\sqrt{2},-1,\frac{1-\sqrt{5}}{2}\}$ or $\lambda_3(G)\in(-.59,-.5)\cup(-.496,\infty)$‎. ‎In addition‎, ‎we find that‎ ‎$\lambda_3(G)=-\sqrt{2}$ if and only if $G\cong P_3$ and $\lambda_3(G)=\frac{1-\sqrt{5}}{2}$ if and only if $G\cong P_4$‎, ‎where $P_n$ is the path on $n$ vertices‎. ‎We find some formulas for computing the characteristic polynomials of graphs $G$ such that $\lambda_3(G) 0$‎. ‎As a consequence we obtain a relation between the multiplicity of $-1$ and the sign of the third largest eigenvalue of graphs