چكيده فارسي :
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
