This paper studies the undirected partial-correlation stock network for the Spanish market that considers the constituents of IBEX-35 as nodes and their partial correlations of returns as links. I propose a novel methodology that combines a recently developed variable selection method, graphical lasso, with Monte Carlo simulations as fundamental ingredients for the estimation recipe. Three major results come from this study. First, in topological terms, the network shows features that are not consistent with random arrangements and it also presents a high level of stability over time. International comparison between major European stock markets extends that conclusion beyond the Spanish context. Second, the systemic importance of the banking sector, relative to the other sectors in the economy, is quantitatively uncovered by means of its network centrality. Particularly interesting is the case of the two major banks that occupy the places of the most systemic players. Finally, the empirical evidence indicates that some network-based measures are leading indicators of distress for the Spanish stock market.