14 problems

Lin et al.'s conjecture. The complete $k$-partite graph $K_{n_{1},n_{2},\ldots,n_{k}}$ is determined by its $D$-spectrum.

Let $G$ be a graph minimizing the spectral radius among graphs in the relevant class, and let $[?]$ denote its minimum and maximum degrees. Hong's conjecture. The minimum and maxim…

Finite-order completeness conjecture. For every finite $n$, there exists a finite $k_{\max}$ such that the collection of $k$-correlations for all $k<k_{\max}$ forms a complete tran…

Let $G_1$ and $G_2$ be graphs with $B_1\subseteq V(G_1)$ and $B_2\subseteq V(G_2)$. Coalescing conjecture. If coalescing the same connected rooted graph onto every vertex of $B_1$…

Diagonalizability conjecture. Every vertex-primitive arc-transitive digraph is diagonalizable.

Let $\alpha\in[0,1)$. For a graph $G$, write $\rho_{A_\alpha}(G)$ for its $A_\alpha$-spectral radius, and let $\Psi(\alpha)$ be the function defined in the paper. A sequence of gra…

Let $G_1$ and $G_2$ be non-isomorphic graphs on $n$ vertices, with adjacency spectra … and define their spectral distance by … Let $P_n$ and $C_n$ denote the path and cycle on $n$…

Dehmer–Pickl–Shi–Yu conjecture. For every such pair of trees,

Dehmer–Pickl–Shi–Yu conjecture. For every such pair of trees,

Let $K_r$ be the complete graph on $r$ vertices, let $sK_t$ be the disjoint union of $s$ copies of $K_t$, and let $K_r\triangledown sK_t$ denote their join. A graph is determined b…

Let $G$) be a connected irregular graph with $n$ vertices, maximum degree $Delta$ and diameter $D$. Its adjacency spectral radius is denoted by $\rho_0(G)$, and its signless-Laplac…

Downer-vertex conjecture. In any chain graph, every vertex is downer with respect to every non-zero eigenvalue.

Let $Kite_p^q$ denote the Kite graph with parameters $p$ and $q$. Kite graph spectral determination conjecture. The graph $Kite_p^q$ is determined by its adjacency spectrum for all…

Let $\Gamma$ be a finite subgroup of $GL(2,\mathbb{Z})$, let $(\mathcal{G}^{\Gamma}_{\rho_1},\mathcal{G}^{\Gamma}_{\rho_2})$ be the associated pair of graphs, let $\mathbb{T}^2$ be…