  In this paper we return to the classical work by Br\'ezin, Itzykson, Parisi
and Zuber, in which, among other things, the authors explicitly calculated the
coefficients of the topological expansion in the cubic random matrix model in
genus 0. Our main goal will be to rigorously prove the results of Br\'ezin,
Itzykson, Parisi and Zuber and to obtain an explicit formula for the
coefficients of the topological expansion in genus 1. We will also prove some
formulae and asymptotic results for the coefficients of the topological
expansion in higher genera.
