50 problems

Let $q=p^{\ell}$ be a prime power, let $m$ be a positive integer, and let $N$ divide $(q^m-1)/(q-1)$. Let $C_0^{(N,q^m)}$ be the subgroup of $\mathbb{F}_{q^m}^*$ of index $N$, and…

Let $\mathcal{R}$ be an association scheme in which every relation is strongly regular. Ivanov's conjecture. The association scheme $\mathcal{R}$ must be amorphic. This would exten…

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…

Strong regularity conjecture. For even dimension $m$, the graph of integral distances $\mathfrak{G}_{m,q}$ is a strongly regular graph. The conjecture predicts that the common-neig…

Let $q$ be a prime power, and let the collinearity graph of the polar space $\mathcal{Q}^{-}(5,q)$ be the graph whose vertices are the points of $\mathcal{Q}^{-}(5,q)$, with two ve…

Let a strongly-regular graph have Latin square or negative Latin square parameters, and suppose that its subconstituents are strongly regular. Cameron's conjecture. These are the o…

Let $G=(V,E)$ be a strongly regular conference graph with parameters $(4\gamma+1,2\gamma,\gamma-1,\gamma)$, where $\gamma\geq 2$, and let $\kappa(x,y)$ denote the Lin--Lu--Yau curv…

Emms–Hancock–Severini–Wilson conjecture.

Maximum subgraph conjecture. If $U$ spans a complete subgraph or an empty subgraph of order $q$ in $V_q$, then $U$ is a line in $F_q\times F_q$.

Let $D_{\ell,j,k}$ be the partial difference sets constructed in the paper, and let the associated strongly regular Cayley graphs be the Cayley graphs arising from $D_{\ell,j,k}$.…

Let $\Gamma$ be a graph as in Hypothesis; in particular, $\Gamma$ is a strongly-regular graph in the setting of the paper. Complete classification conjecture. The graph $\Gamma$ is…

Let $G$ be a connected $k$-regular graph with $n$ vertices and co-edge-regular with parameter $\mu$, having four distinct eigenvalues. Let $t\geq 2$ be an integer. Tan et al.'s con…

Let $r$ be an odd prime power, and let $P(r^2)$ denote the Paley graph of order $r^2$. A clique is a set of pairwise adjacent vertices, and two cliques are in the same orbit when t…

A graph $Gamma$ satisfies the $t$-vertex condition if, for every positive integer $j\leq t$, the number of $j$-vertex subgraphs of each isomorphism type containing a given pair…

Cohn–de Laat–Leijenhorst conjecture. Three-point semidefinite programming bounds prove that $C$ is a maximal spherical code.

Uniquely $C_{4}^{+}$-saturation conjecture. A graph $G$ is nontrivial uniquely $C_{4}^{+}$-saturated if and only if $G$ is a strongly regular graph with parameters $(n,k,0,2)$ or…

Thas's conjecture. J. A. Thas conjectured that the Segre construction for $q=3$ was the only example of a hemisystem on the Hermitian surface $H(3,q^{2})$.

Let $G$ be a strongly regular graph with parameters $\lambda=1$ and $\mu=2$, order $n$, and valency $k$. Let $p_6$ denote the number of hexagons in $G$. Exact hexagon-count conject…

Let an association scheme be one whose relations are all strongly regular graphs. Ivanov's conjecture. Every association scheme in which all relations are strongly regular is amorp…

Strongly regular graph gap problem. Find all strongly regular graphs for which the gap between the lower and upper bounds in Theorem Q12 is $O(n)$.

Let $G$ be an edge-regular graph. The Greaves–Soicher conjecture. The clique adjacency bound for $G$ is at least as good as the Hoffman ratio bound of the complement graph…

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

Ivanov's conjecture. The graphs $\Sigma^{(m)}$ satisfy the 5-vertex condition, and the graphs ${\rm T}^{(m)}$ and $\Upsilon^{(m)}$ satisfy the 4-vertex condition.

Cameron–Goethals–Seidel conjecture. Every non-grid example with Latin square or negative Latin square parameters has parameters as in the lemma, or has a complement with these para…

Faudree–Rousseau–Sheehan's conjecture. There exists a constant $c\ge2$ such that for every strongly regular graph $srg(\nu,k,\lambda,\mu)$,