# 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?

When the entries of $$A$$ are unbounded, $$\lambda_\max(A)$$ is unbounded too and hence there isn't any constant upper bound (but of course, you may use any submultiplicative matrix norm of $$A$$ as a non-constant upper bound). E.g. when $$A=I+k\pmatrix{1&-1\\ -1&1}$$ with $$k\ge0$$, we have $$\lambda_\max(A)=2k+1$$.
When the entries of $$A$$ have moduli $$\le1$$, we have $$\lambda_\max(A)\le\rho(A)\le\|A\|_\infty\le n$$. When $$n$$ is odd, this upper bound is attainable by the checkerboard matrix $$A$$ with $$a_{ij}=(-1)^{i+j}$$. When $$n$$ is even, if we set $$A=C\oplus1$$ where $$C$$ is the checkerboard matrix of size $$n-1$$, then $$\lambda_\max(A)=n-1$$. Therefore, there always exists a symmetric generalised stochastic matrix $$A$$ with $$\lambda_\max(A)\ge n-1>1$$ for every $$n\ge3$$.