Hilbert polynomials of configuration spaces over graphs of circumference at most 1

The $ k $-configuration space $ B_k\Gamma $ of a topological space $ \Gamma $ is the space of sets of $ k $ distinct points in $ \Gamma $. In this paper, we consider the case where $ \Gamma $ is a graph of circumference at most $1$. We show that for all $ k\ge0 $, the $ i $-th Betti number of $ B_k\Gamma $ is given by a polynomial $P_\Gamma^i(k)$ in $ k $, called the Hilbert polynomial of $ \Gamma $. We find an expression for the Hilbert polynomial $P_\Gamma^i(k)$ in terms of those coming from the canonical $1$-bridge decomposition of $ \Gamma $. We also give a combinatorial description of the coefficients of $P_\Gamma^i(k)$.