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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 ∑i=1n(−1)n−iaixi , where ai is interpreted as the number of broken‐cycle‐free spanning subgraphs of G with exactly i components. The parameter ϵ(G)=∑i=1n(n−i)ai∕∑i=1nai 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 Markström in 2006 that ϵ(Tn)