  We consider the ensemble of N-dimensional random symmetric matrices A that
have, in average, p non-zero elements per row. We study the asymptotic behavior
of the norm of A in the limit of infinitely increasing N and p. We prove that
the value p= log N is the critical one for the norm to be either bounded or
not. The arguments are based on the calculus of the tree-type graphs.
Asymptotic properties of sparse random matrices essentially depend on the
typical degree of a tree vertex that we show to be finite.
