The Graham–Lovász conjecture on unimodality of normalized tree coefficients

Let $T$ be a tree of order $n\geq 3$. For a graph $G$ of order $n$, write $\delta_k(G)$ for the coefficients of its distance characteristic polynomial and define the normalized coefficients

$$d_k(G)=|\delta_k(G)|/2^{n-k-2},$$

for $0\leq k\leq n-2$. A sequence is unimodal if it is nondecreasing up to some index and nonincreasing thereafter. Graham–Lovász's conjecture. The sequence $d_0(T),\dots,d_{n-2}(T)$ is unimodal and its peak occurs at $\left\lfloor n/2\right\rfloor$.

The conjecture concerns the shape of the coefficients of the distance characteristic polynomial of a tree. Its prescribed peak location was disproved by Collins in 1985, although unimodality is known for stars and paths; hence the conjecture as stated is refuted.