48 problems

Under Assumption, let $D$ be the diameter and let ${\tilde V}^{\downarrow \downarrow}_{ij}$, ${\tilde V}^{\uparrow \uparrow}_{rs}$, ${\tilde V}^{\downarrow \uparrow}_{ij}$, and…

Terwilliger systems' rank-generating polynomial factorization conjecture. There exist positive integers $d_1,d_2,\ldots,d_n$ whose sum is $d$ such that

Terwilliger systems' subdegree bound conjecture. The inequality above holds for every TD system.

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 $\Gamma$ 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…

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

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

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…

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

Consider the $Q$-polynomial distance-regular graph $\Gamma=(X,\mathcal R)$, with standard module $V$, diameter $D$, and Krein parameters $q^h_{i,j}$. Let $\Lambda$ be the fundament…

Let $\Gamma=(X,\mathcal R)$ be a $Q$-polynomial distance-regular graph with diameter $D\geq 3$, and let $E$ be a $Q$-polynomial primitive idempotent of $\Gamma$. Assume that the se…

Let ${\mathcal P}$ be a rank $d$ polar space, and let $Y$ be a family of lines with characteristic vector $\chi_Y \in \langle \bm{j}\rangle + V_{10} + V_{11}$. Consider the example…

A standard planar scaffold is a weakly connected planar scaffold whose edge weights are among the adjacency matrices $A_0,\dots,A_c$ of an association scheme. For sequences of such…

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

Eigenvalue pattern conjecture. The eigenvalues of the class $[2k]$ corresponding to these modules are, respectively,

Least-eigenvalue conjecture. The row sum and least eigenvalue of $\hat{B}_{3}(2k)$ are, respectively,

Bannai–Ito conjecture. For sufficiently large diameter, a primitive association scheme is metric if and only if it is cometric.

Hamming distinct-eigenvalue conjecture. If $H(d,q,j)$ is connected, it has more than $d/2$ distinct eigenvalues.