  The Leray number of an abstract simplicial complex is the minimal integer $d$
where its induced subcomplexes have trivial homology groups in dimension $d$ or
greater. We give an upper bound on the Leray number of a complex in terms of
how the facets are attached to each other. We also describe the structure of
complexes for the equality of the bound that we found. Through the
Stanley-Reisner correspondence, our results give an Eisenbud-Goto type
inequality for any square-free monomial ideals. This generalizes Terai's
result.
