42 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…

Cheeger constant conjecture.

Let $\Gamma$ be a $Q$-polynomial distance-regular graph with eigenvalues $\theta_i$ and dual eigenvalues $\theta_i^*$. For $2\leq i\leq D-1$, define … Assume these expressions are equ…

Let $\Gamma$ be a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$, where $b\ne1$, and suppose $\Gamma$ is not a dual polar graph. Let $T$ be its Terwilliger alg…

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…

Symmetric generation conjecture. The subspace $\omega^\perp\cap {\rm Sym}(S)$ is the subalgebra of the Norton algebra $EV$ generated by $E\hat x+E\hat y$.

Symmetric-subspace closure conjecture. We have

Let $m \geq 2$ be a fixed integer. A geometric distance-regular graph has diameter $D \geq 3$ and intersection number $c_2 \geq 2$. Koolen–Bang's classification conjecture. Any geo…

Let $m \geq 2$ be a fixed integer. A distance-regular graph is coconnected if it is not a disjoint union of complete graphs. Let $\phi_1$ and $\tau_2$ be the clique parameters asso…

Let $\Gamma$ be a geometric distance-regular graph with diameter $D \geq 3$ and distinct eigenvalues … Let $b = \frac{b_1}{\theta_1 + 1}$. For vertices $x,y$ at distance $i$, let…

Let $\Gamma$ be a geometric distance-regular graph with diameter $D$ and distinct eigenvalues … Let $b = \frac{b_1}{\theta_1 + 1}$. For a Delsarte clique, let $\phi_i$ denote the n…

Let $\Gamma=(X,\mathcal R)$ be the bilinear forms graph, let $x,y\in X$ be adjacent vertices, and let $U=U(x,y)$ be the subspace defined earlier. Let $E=E_1$ be the primitive idemp…

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…

Diameter-bound conjecture. There is a constant $\ell$ such that if $D\geq\ell$, then one of the following statements holds:

Let $H(d,3)$ be the Hamming graph on length-$d$ words over an alphabet of size $3$, and let $\Phi_d(H(d,3))$ be the associated system of polynomials. A solution is all-nonzero if e…

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…

Diameter-boundedness conjecture. The diameter of $\Gamma$ is bounded by a function in $b$.

Let $q$ and $D$ be integers with $q \ge 2$ and $D \ge 4$. A distance-regular graph with classical parameters … is a graph whose classical parameters are given by this quadruple. No…

Let $\Gamma=(X,\mathcal R)$ be a distance-regular graph that is $Q$-polynomial with respect to a primitive idempotent $E$. For distinct vertices $x,y\in X$, let $C(x,y)$ and…

Let $q>7$ be a prime power, and let $\mathcal{P}(q^2)$ denote the Paley graph on the finite field of order $q^2$. Let $\tau(G)$ denote the maximum order of an induced forest in a g…

Let $\Gamma=(X,\mathcal R)$ be a finite, undirected, connected graph without loops or multiple edges, with diameter $D\geq 1$, and let $A$ be its adjacency matrix. An ordering…

Multiplicity-one conjecture. For every irreducible $\mathbb T$-module, the common eigenspaces for each of these four families all have dimension one. This conjecture would give a u…