  In a 2018 paper, Cameron and Semeraro posed the problem of finding all
group-graph reciprocal pairs. In this paper, we make a significant contribution
to finding all such pairs. A group and graph form a reciprocal pair if they
satisfy the relation $$P_{\Gamma,G}(x)=(-1)^nF_G(-x)$$ where $P_{\Gamma,G}(x)$
is the orbital chromatic polynomial of a graph $\Gamma$ and $F_G(x)$ is the
cycle polynomial of a finite permutation group. We define a set of graphs to be
\textit{$k$-stars} and prove that they satisfy a reciprocality relation with
some group depending on $k$. These graphs are comprised of a complete graph
with $k$ vertices and a further $\alpha$ `points' which are only connected to
each vertex in the centre. This group is a subgroup of $S_k\times S_\alpha$,
which is the automorphism group of a \textit{$k$-star} and $\alpha$ is the
number of points on the star. We conjecture a list of group-graph reciprocal
pairs.
