Hu, Qi and Shao's monotonicity conjecture for Laplacian eigenvalues of power hypergraphs

Let $G$ be an ordinary graph, let $k=2r$ be even, and let $G^{k}$ be the $k$-th power hypergraph of $G$. For a tensor $\mathcal{T}$, an H-eigenvalue is an eigenvalue having a real eigenvector, and $\lambda(\mathcal{T})$ denotes its largest H-eigenvalue. The Laplacian and signless Laplacian tensors of $G^k$ are denoted by $\mathcal{L}(G^k)$ and $\mathcal{Q}(G^k)$, respectively.

Hu, Qi and Shao's conjecture. The sequence

$$\{\lambda(\mathcal{L}(G^{k}))=\lambda(\mathcal{Q}(G^{k}))\}$$

is strictly decreasing.

Hu, Qi and Shao established this behavior for cycles and stars and conjectured it for every ordinary graph $G$. The paper under consideration is devoted to proving this conjecture, so the claim is mathematically resolved, although the supplied parser status is unknown.