14 problems

For positive integers $\ell_1\leq \ell_2\leq \ell_3$, let $T(\ell_1,\ell_2,\ell_3)$ be the tree with a unique degree-three vertex $v$ such that…

For every nontrivially unstable graph $G$, there exists an odd positive integer $k$ such that $G$ contains cycles isomorphic to both $C_k$ and $C_{2k}$.

Let $G$ be a finite transitive permutation group. Call $G$ elusive if it contains no derangements of prime order. The $2$-closure of $G$ on its permutation domain is the largest su…

Type I conjecture. Every such graph is of Type I.

Type II classification conjecture. If $\Gamma$ is nontrivially unstable and of Type II, then

Let $X$ be an antipodal, non-bipartite distance-regular graph of diameter $3$. An endomorphism of $X$ is a graph homomorphism from $X$ to itself, and a subgraph has diameter $2$ wh…

Let $G$ be a connected amply regular graph with parameters $(n,d,\alpha,\beta)$, and let $0<\varepsilon<1$ be a real number. Qiao, Park and Koolen's conjecture. There exists a cons…

Let $G$ be an amply regular graph with parameters $(n,d,\alpha,\beta)$, where $n$ is possibly infinite. Terwilliger's conjecture. If $\beta\geq 2$, then $G$ is finite. Terwilliger'…

Jin–Tan's conjecture. Every connected $2$-distance transitive dihedrant either is a known $2$-arc transitive dihedrant, is isomorphic to $\mathsf{K}_{m[b]}$ for some $m\geq 3$ and…

Let $D(k,q)$ be the algebraic bipartite graph proposed by Lazebnik and Ustimenko, where $k\geq 2$ and $q=p^m$ is a prime power. Fredi's conjecture. $D(k,q)$ has girth $k+5$ for all…

For a graph $G$, let $\Omega(G)$ denote the auxiliary graph used in the paper's generalized spectral characterization, and call $G$ controllable when it satisfies the controllabili…

Koolen–Gebremichel conjecture. Either $\mu\in\{6,9\}$ or $n\leqslant 276$.

Let $\Gamma$ be a cubic vertex-transitive graph of order $n$ that is not isomorphic to $K_{3,3}$ or a split Praeger–Xu graph, and let $g\in\operatorname{Aut}(\Gamma)$. A regular or…

All graphs considered are finite, undirected and simple. A graph is co-edge-regular with parameters $(n,k,c)$ if it is $k$-regular on $n$ vertices and every two distinct non-adjace…