71 problems

Second-largest eigenvalue conjecture. If $\mu$ has at least two parts of size $1$, or if $\mu=[n-1,1]$ with $n\neq 4$, then the second largest eigenvalue of $A_\mu$ occurs on the i…

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…

Godsil's conjecture. If $G$ is a connected graph which is a colour class in an association scheme, then

Brouwer's conjecture. The vertex-connectivity of $G$ equals its degree:

Koolen's conjecture. For almost all finite connected simple graphs, the Terwilliger algebras coincide with the full matrix algebras.

Grassmann eigenvalue conjecture. (i) If $(n,q) eq (2d,2)$, then

A 3-class association scheme is an association scheme with three nontrivial relations; here, consider a certain such scheme on $64$ points. Cohn's universal optimality conjecture.…

Let $Y=(X,\{R_i\}_{0\leq i\leq D})$ be a symmetric, $D$-class association scheme, and let $E_0,\ldots,E_D$ be an ordering of its primitive idempotents with respect to which $Y$ is…

Association-scheme conjecture for 15 points. The graphs $G_i$, for $i=2,\dots,13$, form an association scheme on $\mathcal{U}^{15}_3$.

Association-scheme conjecture for 12 points. The graphs … form an association scheme on $\mathcal{U}^{12}_3$.

Let $C$ be the described $40$-point code on the unit sphere in $\mathbb{R}^{10}$, whose maximal inner product is $1/6$. Uniqueness conjecture for the 40-point code. This code is th…

Let $\Gamma$ be a taut distance-regular graph with odd diameter $D\geq 5$. By the source's stated theorem, $\Gamma$ is an antipodal $2$-cover; hence its antipodal quotient is defin…

Let $\Gamma$ be a bipartite distance-regular graph, and let $T$ denote its subconstituent algebra. Assume that, up to isomorphism, there exist exactly two irreducible $T$-modules w…

Let ${\cal L}(q^2)$ be the set of lines under consideration, and let $H(q^2)$ denote the 4-class fusion obtained by restricting the relations $R_k(\epsilon,\phi)$ to the hyperbolic…

Hollmann's conjecture. If $m$ is an odd prime, then $({\cal E},\{\Delta_a\})$ is pseudocyclic.

Hollmann's conjecture. If $m$ is an odd prime, then the orbital association scheme of ${\rm P\Gamma L}(2,2^m)$ on ${\cal E}$ is pseudocyclic.

Let ${\cal F}$ be a field, let $V$ be a finite-dimensional nonzero vector space over ${\cal F}$, and let $\Phi$ be a TD system on $V$ with diameter $d$. Let $R$ and $L$ be the maps…

Consider the Clark–Liang association scheme $CL(15,2)$ on $X=\mathbb{Z}_{15}$, with relation classes determined by $X_1=C_1\cup C_7$, $X_2=C_3$, and $X_3=C_5$, and adjacency matric…

Largest-diameter conjecture. The diameter of $X_{[2,1^{n-2}]}$ is the largest diameter among all graphs in the perfect matching association scheme.

Smallest spectral-gap conjecture. Among the adjacency matrices

Let $n,k,t$ be non-negative integers with $k\leq n$ and $t\leq\min(k,n-k)$. Let $m_1(n,k,t)$ denote the first matching number associated with the paired construction for Boolean fu…

LP optimum conjecture for $D_n$. If $n$ and $d$ are odd integers with $1\leq d\leq n$, then

Let $r,s$ be positive integers, let $n:=r+s$, and consider the Johnson scheme $J(2n,n)$. A tight symmetric inner distribution code is a code for which equality holds in the relevan…

Let $n\geq 6$, and let $\mathfrak{X}(\operatorname{Sym}(n))$ be the conjugacy class association scheme of the symmetric group. Let $T$ be its Terwilliger algebra with respect to th…

Consider the $Q$-polynomial distance-regular graph $\Gamma=(X,\mathcal R)$ of diameter $D$, its standard module $V$, and the fundamental $\mathbb T$-submodule…