9 problems

Zhou's distance Laplacian conjecture. For every integer $r$,

Let $n\geq 2$ be an integer, and let $\mathcal{P}(\mathbb{Z}_{n})$ denote the power graph of the cyclic group $\mathbb{Z}_{n}$. Write $\partial_{1}^{L}$ for the relevant distance L…

Let $G$ be a graph on $n$ vertices, and let $\mathcal{D}^\mathcal{L}(G)$ be its normalized distance Laplacian matrix. Write $KPK_{n_1,n_2,n_3}$ for the graph obtained by connecting…

Let $G$ be a graph on $n$ vertices, and let $\partial_n^\mathcal{L}$ denote the largest eigenvalue of its normalized distance Laplacian matrix $\mathcal{D}^\mathcal{L}(G)$. Reinhar…

Let $G$ be a connected graph on $n\geq 2$ vertices, let $K_n$ be the complete graph, and let $\rho(\mathcal D^{\mathcal L}(G))$ denote the spectral radius of the normalized distanc…

For integers $n_1,n_3\geq 1$ and $n_2\geq 2$, let $KPK_{n_1,n_2,n_3}$ be formed by taking the vertex sum of a vertex in $K_{n_1}$ with one end of $P_{n_2}$ and the vertex sum of a…

Let $G$ be a graph on $n\geq 2$ vertices. Write $\partial^{\mathcal L}_n$ for the largest eigenvalue of its normalized distance Laplacian and $\rho_{\mathcal D^{\mathcal L}}$ for t…

Let $G$ be a graph on $n$ vertices, let $\rho_{\mathcal D^{\mathcal L}}(G)$ denote the spectral radius of its normalized distance Laplacian, and let $KPK_{n_1,n_2,n_3}$ denote the…

Let $T$) be a tree with vertex set $V(T)=\{1,\ldots,n\}$. Let $D^L$ be its distance Laplacian, with off-diagonal entries $-d_{ij}$ and diagonal entries $\sum_{j=1}^n d_{ij}$. A Fie…