Kourovka Notebook Problem 21.90

Definitions and mathematical statement

Let $\Gamma$ be a finite connected simple graph of diameter three, on $v$ vertices. Put $A_i(x,y)=1$ if the distance from $x$ to $y$ is $i$, and zero otherwise, for $0\leq i\leq3$. Thus $A_0=I$ and $A_1$ is the adjacency matrix. The graph is distance-regular if the number of vertices $z$ with $d(x,z)=i$ and $d(y,z)=j$ depends only on $i,j,d(x,y)$. Its distance matrices span a commutative algebra with mutually orthogonal primitive idempotents $E_0,E_1,E_2,E_3$, where $E_0=J/v$, $J$ is the all-ones matrix, $E_iE_j=0$ for $i\ne j$, and $\sum E_i=I$.

It is $Q$-polynomial if these idempotents can be ordered so that, for the entrywise matrix product $\circ$, \[ E_1\circ E_i=\frac1v\sum_{j=0}^3 q_{1i}^{j}E_j, \] the coefficient matrix $(q_{1i}^{j})_{i,j=0}^3$ is irreducible tridiagonal: entries with $|i-j|>1$ vanish and every immediately adjacent off-diagonal entry is positive.

Let $\Gamma_i$ have adjacency matrix $A_i$. Does such a $Q$-polynomial distance-regular $\Gamma$ exist for which both $\Gamma_2$ and $\Gamma_3$ are strongly regular? A strongly regular graph is regular of valency $k$ with $0<k<v-1$, and has constants $\lambda,\mu$ such that adjacent vertex pairs have $\lambda$ common neighbors and distinct nonadjacent pairs have $\mu$. The two distance graphs may have different parameters.

Short English statement

Can a $Q$-polynomial distance-regular graph of diameter three have both its distance-two and distance-three graphs strongly regular?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2599 (KOU-21.90), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.90. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.
• S3 B. G. Kodalen, Linked systems of symmetric designs, Algebraic Combinatorics; the Krein-parameter criterion for a Q-polynomial ordering. https://alco.centre-mersenne.org/item/10.5802/alco.22.pdf.