  The properties of the first (largest) eigenvalue and its eigenvector (first
eigenvector) are investigated for large sparse random symmetric matrices that
are characterized by bimodal degree distributions. In principle, one should be
able to accurately calculate them by solving a functional equation concerning
auxiliary fields which come out in an analysis based on replica/cavity methods.
However, the difficulty in analytically solving this equation makes an accurate
calculation infeasible in practice. To overcome this problem, we develop
approximation schemes on the basis of two exceptionally solvable examples. The
schemes are reasonably consistent with numerical experiments when the
statistical bias of positive matrix entries is sufficiently large, and they
qualitatively explain why considerably large finite size effects of the first
eigenvalue can be observed when the bias is relatively small.
