Kourovka Notebook Problem 20.54

Definitions and mathematical statement

Let $\Gamma=(V,E)$ be a finite, connected, simple, undirected graph, with graph distance $d$ and diameter $3$. It is distance-regular if, whenever $d(x,y)=i$, the numbers \[ c_i=\left|\{z\in V:z\sim y,\ d(x,z)=i-1\}\right|,\qquad b_i=\left|\{z\in V:z\sim y,\ d(x,z)=i+1\}\right| \] depend only on $i$, wherever the indicated distance layers exist. Put $c_0=b_3=0$, let $k=b_0$ be the valency, and set $a_i=k-b_i-c_i$. The intersection array is \[ \{b_0,b_1,b_2;c_1,c_2,c_3\}. \] Write the distinct eigenvalues of the adjacency matrix in decreasing order as $k=\theta_0>\theta_1>\theta_2>\theta_3$. A distance-regular graph of diameter $3$ is a Shilla graph if $\theta_1=a_3$. The supplied source states that, for a Shilla graph, $a=a_3$ divides $k$; it defines $b(\Gamma)=k/a$.

The source attributes to Koolen and Park the result that there are twelve feasible intersection arrays for Shilla graphs with $b(\Gamma)=3$. It further reports that any such graph must have one of the arrays \[ \{12,10,3;1,3,8\},\qquad \{12,10,5;1,1,8\},\qquad \{15,12,6;1,2,10\}. \] It identifies the first two with the Doro graph and the nonisotropic unitary graph for $q=4$, respectively. These are supplied background assertions, not independently verified here; feasibility alone is not an existence assertion.

Existence question (catalogue entry). Does there exist a distance-regular graph $\Gamma$ of diameter $3$ with intersection array \[ \{15,12,6;1,2,10\}? \] This is the remaining existence question in the source's discussion of Shilla graphs with $b(\Gamma)=3$. The source also says that A. Makhnev and N. Zyulyarkina found the automorphisms of the last graph. Because existence is explicitly posed as a question, that wording is retained as a qualified source report and is not treated as establishing existence.

Short English statement

Determine whether a distance-regular graph of diameter $3$ exists with intersection array $\{15,12,6;1,2,10\}$. The supplied record places this question in the classification of Shilla graphs with $b(\Gamma)=3$ and labels it open. That status and the background literature claims have not been independently verified.

Sources

• Local UnsolvedMath registry: permanent ID $701712$, source problem code catalogue entry, title Kourovka Notebook Problem 20.54. The supplied record labels the problem open and gives its update timestamp as 2026-10-16T14:00:00Z; this is registry metadata, not independent literature verification.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 20.54, 20th issue (2022), also cited in the supplied record through the 21st issue edition (2026), arXiv identifier 1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. Proposed by A. A. Makhnev and D. O. Revin in 2022.
• Koolen and Park: credited in the supplied statement with proving that twelve feasible intersection arrays occur as candidates for Shilla graphs with $b=3$. No paper title, publication details, or link is supplied for this attribution.
• A. Makhnev and N. Zyulyarkina, Doklady Maths., volume 84, no. 1 (2011), pp. 510--514: cited by the supplied statement concerning automorphisms of the graph with intersection array $\{15,12,6;1,2,10\}$. No article title or link is supplied, and the precise scope of the cited result has not been verified.