  A methodology to analyze the properties of the first (largest) eigenvalue and
its eigenvector is developed for large symmetric random sparse matrices
utilizing the cavity method of statistical mechanics. Under a tree
approximation, which is plausible for infinitely large systems, in conjunction
with the introduction of a Lagrange multiplier for constraining the length of
the eigenvector, the eigenvalue problem is reduced to a bunch of optimization
problems of a quadratic function of a single variable, and the coefficients of
the first and the second order terms of the functions act as cavity fields that
are handled in cavity analysis. We show that the first eigenvalue is determined
in such a way that the distribution of the cavity fields has a finite value for
the second order moment with respect to the cavity fields of the first order
coefficient. The validity and utility of the developed methodology are examined
by applying it to two analytically solvable and one simple but non-trivial
examples in conjunction with numerical justification.
