Author

Zhen-Bo Chen

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Preprint Sep 2026

Polynomial positivity cones for Coxeter roots and walks in trees

For a finite simple graph $G$ and an integer $k\ge0$, let $w_k(G)$ denote the number of walks of length $k$. We prove the conjecture of T\"aubig, Weihmann, Kosub, Hemmecke, and Mayr for every finite tree and determine all equality cases. If $T$ has $n\ge1$ vertices, then $n w_{k+1}(T)-2(n-1)w_k(T)\ge0$ for every $k\ge1...

Dong-Xiu Cai, Zhen-Bo Chen, Jia-Sheng Zeng et al. · 0 citations

Preprint Oct 2026

A finite threshold for the double-comet conjecture

For a tree $T$, let $g(T)=\lambda_1(T)-\lambda_2(T)$ be the difference between its two largest adjacency eigenvalues. A balanced double comet is obtained by attaching equally many leaves to the two endpoints of a path. Jovovi\'c, Koledin and Stani\'c conjectured that such a tree attains the minimum adjacency spectral g...

Dong-Xiu Cai, Zhen-Bo Chen, Xiao-Dong Zhang · 0 citations