  We investigated the magnetic and electric properties of nanometer-sized
vanadium dioxide (VO$_2$) particles. VO$_2$ nanoparticles were formed by
milling VO$_2$ powder. We measured the magnetic field dependence of the
magnetization of the VO$_2$ powder and nanoparticles. The VO$_2$ powder did not
exhibit ferromagnetism, whereas the VO$_2$ nanoparticles exhibited
ferromagnetism. In addition, we fabricated samples by bridging between
electrodes with the VO$_2$ nanoparticles, and the temperature dependence of
their resistance was measured. Metal-insulator transitions (MITs) were
observed, and the temperature range where the MIT occurred was wider than that
in a typical bulk VO$_2$. The VO$_2$ nanoparticles exhibited these properties
of ferromagnetism and MIT possibly because of the surface and size effects of
the VO$_2$ nanoparticles. These results indicate the first observation of the
competitive coexistence of ferromagnetism and MIT of VO$_2$ nanoparticles.
