At first, a genetic algorithm in combination with either the parametrized density-functional tight-binding method or a Gupta-potential is used to determine the putative global minimum energy structures of mixed Ag $$_{n-m}$$ n - m Rh $$_{m}$$ m and Ag $$_{m}$$ m Rh $$_{n-m}$$ n - m clusters with $$n\le 20$$ n ≤ 20 and $$m=0,1$$ m = 0 , 1 . Subsequently, the resulting structures are re-optimized with a first-principles method. The results demonstrate that the exchange of a single silver atom by rhodium leads to compact core-shell-like structures with structural motifs well known from the Lennard-Jones system. For the systems of the present study, AgRh $$_{n-1}$$ n - 1 clusters retain their cube-based structural motif and the silver atoms typically avoid the corner positions within a cube if possible. Population analysis of both cluster systems shows that the total magnetic moment is mainly due to unpaired electrons on the rhodium atoms with a small ferro-magnetic contribution of the silver host in Ag $$_{n-1}$$ n - 1 Rh and virtually no contribution to the total magnetic moment from the single silver atom in AgRh $$_{n-1}$$ n - 1 clusters.