Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them exactly 0 (so sparse)? We have upper bounds on the magnitudes of all nonzero entries. We do not know whether the main diagonal terms are more or less than minus the off-diagonal nonzero terms.

Zhan's "Extremal eigenvalues of real symmetric matrices with entries in an interval" (SIAM J. Matrix Anal. Appl. Vol. 27, No. 3, pp. 851-860) provides an answer for general n*n real symmetric matrices but can we improve/simplify this given the extra conditions?