Perturbing eigenvalues of non-negative matrices

Let $A$ be an irreducible (entrywise) nonnegative $n\times n$ matrix with eigenvalues $$\rho, b+ic,b-ic, \lambda_4,\cdots,\lambda_n,$$ where $\rho$ is the Perron eigenvalue. It is shown that for any $t \in [0, \infty)$ there is a nonnegative matrix with eigenvalues $$\rho+ \tilde t,\lambda_2+t,\lambda_3+t, \lambda_4 \cdots,\lambda_n,$$ whenever $\tilde t \ge \gamma_n t$ with $\gamma_3=1, \gamma_4 = 2, \gamma_5=\sqrt 5$ and $\gamma_n = 2.25$ for $n \ge 6$. The result improves that of Guo et al. Our proof depends on an auxiliary result in geometry asserting that the area of an $n$-sided convex polygon is bounded by $\gamma_n$ times the maximum area of the triangle lying inside the polygon.