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The structural stability and magnetic properties of the icosahedral Ni13, Ni+131 and Ni13-1 clusters have been obtained by utilizing all-electron density functional theory with the generalized gradient approximations for the exchangecorrelation energy. The calculated results show that the ground states of neutral and charged clusters all favour a D3d structure, a distorted icosahedron, due to the Jahn-Teller effect. The radial distortions caused by doping one electron and by doping one hole are opposite to each other. Doping one electron will result in a 1/2 decrease and doping one hole will result in a 1/2 increase of the total spin. Both increasing interatomic spacing and decreasing coordination will lead to an enhancement of the spin magnetic moments for Ni13 clusters.