Improved bounds for embedding certain configurations in subsets of vector spaces over finite fields

The fourth listed author and Hans Parshall (\cite{IosevichParshall}) proved that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, and $G$ is a connected graph on $k+1$ vertices such that the largest degree of any vertex is $m$, then if $|E| \ge C q^{m+\frac{d-1}{2}}$, for any $t>0$, there exist $k+1$ points $x^1, \dots, x^{k+1}$ in $E$ such that $||x^i-x^j||=t$ if the $i$'th vertex is connected to the $j$'th vertex by an edge in $G$. In this paper, we give several indications that the maximum degree is not always the right notion of complexity and prove several concrete results to obtain better exponents than the Iosevich-Parshall result affords. This can be viewed as a step towards understanding the right notion of complexity for graph embeddings in subsets of vector spaces over finite fields.