  Let $P$ be a finite point set in $\mathbb{R}^2$ with the set of distance
$n$-chains defined as $$
\Delta_n(P)=\{(|p_1-p_2|,|p_2-p_3|,\ldots,|p_n-p_{n+1}|):p_i \in P\}.$$ We show
that for $2\leq n=O_{|P|}(1)$ we have $$|\Delta_n(P)|\gtrsim
\frac{|P|^{n}}{\log^{\frac{13}{2}(n-1)}|P|}.$$ Our argument uses the energy
construction of Elekes and a general version of Rudnev's rich-line bound
implicit in Rudnev's recent hinge paper which allows one to iterate efficiently
on highly intersecting nested subsets of Guth-Katz lines. Let $G$ is a simple
connected graph on $m=O(1)$ vertices with $m\geq 2$. Define the graph-distance
set $\Delta_G(P)$ as $$ \Delta_G(P) = \{ (|p_{i}-p_{j}|)_{\{i,j\}\in E(G)} :
p_i,p_j \in P\}.$$
  Combining with results of Guth and Katz and Rudnev with the above, if $G$ has
a Hamiltonian path we have $$ |\Delta_G(P)| \gtrsim
\frac{|P|^{m-1}}{\text{polylog}|P|}. $$ \end{abstract}
