Given a graph $G = (V, E)$, we can calculate its chromatic polynomial $P(G, k)$, and it has $n$ (complex) roots, also known as chromatic roots. It is a well-known fact that the sum of chromatic roots of the graph $G$ is equal to the number of edges $e(G)$.

I am working on a problem and it somehow is related to the following pattern, which I also experiment with SageMath:

**Want to prove**: Given a bipartite graph $G$ of minimum degree $2$ and $G$ is not a forest, with its chromatic polynomial $P(G, k)$ and chromatic roots $r_{1}, r_{2}, \ldots, r_{n}$, then we have

$$\sum_{i = 1}^{n} r_{i}^{2} = e(G)$$

However, I could not seem to prove it. I tried some Cauchy-Schwarz inequality calculations. I also couldn't find relevant literatures containing this result, or relevant research on (complex) chromatic roots of bipartite graphs.

Thank you!