linear algebra - Largest eigenvalue of a symmetric "generalized doubly stochastic" matrix

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix whose rows and columns sum to one. $A$ is not necessarily a doubly stochastic matrix, because negative entries are possible.

What can be said about the largest eigenvalue $\lambda$ of $A$? Is there a "good" upper bound for $\lambda$?

Additional constraint: Suppose that $|a_{ij}| \leq 1$. Does $\lambda \leq 1$ hold?