  We present our theoretical results on the electronic state in vanadium spinel
oxide AlV$_2$O$_4$. The material is a mixed-valent system with the average
valence V$^{2.5+}$, and V cations constitute a frustrated pyrochlore structure.
It shows a structural transition at $\sim 700$ K, leading to the formation of
seven V-sites clusters -- heptamers. We study the electronic state of the
heptamer by explicitly taking account of orbital degree of freedom as well as
electron correlations. We show that the ground state of the heptamer for
realistic parameters becomes spin-singlet because of strong $\sigma$-type
bonding states of $t_{2g}$ orbitals. The temperature dependence of the magnetic
susceptibility in experiments is naturally understood by this singlet formation
in heptamers. Our results indicate that in AlV$_2$O$_4$ orbital physics is
relevant to stabilize a cluster-type singlet state instead of a previously
proposed charge-ordered state with valence skipping.
