A theoretical study has been carried out to determine the stability of various isomers of the lanthanum(-) carbon clusters LaCx+. Included in this study are linear isomers and several classes of cyclic and graphitic structures. Geometries and energies of the proposed structures are calculated using the B3LYP density functional method. Cyclic isomers are clearly established as more stable than linear isomers over a wide range of cluster sizes. The results reveal that two classes of cyclic isomers, which we denote "open rings" and "closed rings", have a pattern of relative thermodynamic stability that correlates to the pattern of cluster intensities in the ion mobility observations of the LaCx+ clustering process. The competitive factors that provide stability to lanthanum-carbon clusters are elucidated, such as ring strain and variations in the lanthanum bonding environment. Graphitic LaCx+ with 18 or 20 carbon atoms are shown to be more stable than their cyclic counterparts, and B3LYP is shown to underestimate the stability of graphitic LaCx+.