Incidences between planes over finite fields

We use methods from spectral graph theory to obtain bounds on the number of incidences between $k$-planes and $h$-planes in ${\mathbb{F}}_q^d$ which generalize a recent result given by Bennett, Iosevich, and Pakianathan (2014). More precisely, we prove that the number of incidences between a set $\mathcal{P}$ of $k$-planes and a set $\mathcal{H}$ of $h$-planes with $h\ge 2k+1$,whichisdenotedby$I($\mathcal{P}$,$\mathcal{H}$)$, satisfies $$\mid I(\mathcal{P},\mathcal{H})-\frac{\mathrm{\mid }\mathcal{P}\mathrm{\mid }\mathrm{\mid }\mathcal{H}\mathrm{\mid }}{q^{(d-h)(k+1)}}\mid \lesssim q^{\frac{(d-h)h+k(2h-d-k+1)}{2}}\sqrt{\mathrm{\mid }\mathcal{P}\mathrm{\mid }\mathrm{\mid }\mathcal{H}\mathrm{\mid }}\mathrm{.}$$