12 problems

Let $H$ be a connected $k$-uniform hypergraph, and let $\rho$ be the spectral radius of its adjacency tensor $\mathcal{A}(H)$. Let $\mathcal{L}(H)$ denote its Laplacian tensor. The…

Let $\mathcal{M}$ be an order $h$ hypermatrix of dimension $n$, and let $\mu$ be an eigenvalue of $\mathcal{M}$. Write $am_{\mathcal{M}}(\mu)$ for its algebraic multiplicity and…

Zheng's conjecture. For every $k$-uniform hypertree $T$ with $k\geq 3$, the multiplicity of the zero Laplacian eigenvalue is $k^{|E(T)|(k-2)}$. This conjecture concerns the Laplaci…

Let $(T,\mathbf{w})$ be a weighted $k$-tree, and let $T'$ be a subtree of $T$. Write $\mu(T',\mathbf{w},x)$ for the weighted matching polynomial of $T'$ and let…

Let $\mathcal{H}$ be a planar $3$-uniform hypergraph on $n$ vertices, and let its shadow be the graph obtained by replacing each hyperedge by the edges joining its pairs of vertice…

The $NC(m,d)$ extremal conjecture. For $G\in NC(m,d)$,

Let $k$ be a positive integer, and let the $k$-uniform loose path of length $3$ be the hypergraph considered in Theorem. The conjecture. Theorem is true for the $k$-uniform loose p…

Let $G$ be a connected hypergraph, let $G^{m,s}$ denote its generalized power hypergraph with parameters $m$ and $s$, and let $c(G)$ denote the cyclic index of $G$. Spectral index…

Strict inequality conjecture.

Hu, Qi and Shao's conjecture. The sequence

Let $G$ be a graph, let $k$ be even, and let $G^k$ denote the $k$-uniform hypergraph associated with $G$. Write $\mathcal{Q}_{G^k}$ for its signless Laplacian tensor and…

Let $H$ be a connected $k$-uniform hypergraph. A vertex labeling is a map $f:V(H)\rightarrow [k]$, where $[k]=\{1,\ldots,k\}$. The hypergraph $H$ is odd-bipartite when $k$ is even…