Proving a conjecture on chromatic polynomials by counting the number of acyclic orientations

The chromatic polynomial \(P(G,x)\) of a graph \(G\) of order \(n\) can be expressed as \(\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i\), where \(a_i\) is interpreted as the number of broken-cycle free spanning subgraphs of \(G\) with exactly \(i\) components. The parameter \(\epsilon(G)=\sum\limits_{i=1}^n (n-i)a_i/\sum\limits_{i=1}^n a_i\) is the mean size of a broken-cycle-free spanning subgraph of \(G\). In this article, we confirm and strengthen a conjecture proposed by Lundow and Markstr\"{o}m in 2006 that \(\epsilon(T_n)< \epsilon(G)