19 problems

A graph $X$ is core-complete if $X$ is isomorphic to its core $X^{\bullet}$ or its core is a complete graph. Strongly regular graphs are graphs with constant parameters governing t…

Mullin's conjecture. The Peisert graph of order $q^2$ has the strict-EKR property.

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…

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…

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 an amply regular graph with parameters $(n,d,\alpha,\beta)$, and let $\operatorname{diam}(G)$ denote its diameter. Diameter bound conjecture. When $\alpha=\beta=2$, the…

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 $X$ be a vertex-transitive graph. An automorphism of $X$ is semiregular if it fixes no vertex and all of its orbits on the vertices have the same length. The semiregular automo…

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…

The 2k minus 1 orbit conjecture. One has

Let $d$ be a positive integer, let $\Gamma$ be a connected graph of valence $d$, and let $G\leq \operatorname{Aut}(\Gamma)$ act transitively on the arcs of $\Gamma$. Suppose that t…

Let $G$ be a nontrivial finite transitive $2$-closed permutation group. A derangement is a permutation that fixes no point, and a derangement has prime order when its order is a pr…

Godsil's integrality conjecture. If an abelian Cayley graph has uniform mixing, then the graph is integral; equivalently, all eigenvalues of its adjacency matrix are integers.