On bipartite $Q$-polynomial graphs of diameter not greater than $5$

V. V. Bitkina^a, A. A. Makhnev<sup>bc

a</sup> North Ossetian State University, 44–46 Vatutin St., Vladikavkaz 362025, Russia
^b Hainan Provincial University, 58 Renmin Ave., Haikou 570228, Hainan, China
^c N. N. Krasovskii Institute of Mathematics and Mechanics, 16 S. Kovalevskaja St., Ekaterinburg 620990, Russia

Abstract: Let $u$ be a vertex of a bipartite $Q$-polynomial distance-regular graph $\Gamma$ of diameter $D\ge 3$, $\Sigma=\Gamma_D(u)$, and $\Lambda=\Sigma_2$. Then $\Lambda$ is a distance-regular $Q$-polynomial graph. In the cases $D=4$ and $D=5$ the graph $\Lambda$ is strongly regular $Q$-polynomial. The half graph $\Gamma_2$ is strongly regular and $\Lambda$ is a neighbourhood of a vertex in the complement of $\Gamma_2$. Therefore, a necessary condition for $Q$-polynomiality of $\Gamma$ is the strong regularity of neighbourhoods and antineighbourhoods of vertices in $\Lambda$. A bipartite distance-regular graph $\Gamma$ of diameter $D\in \{4,5\}$ is called almost $Q$-polynomial if neighbourhoods and antineighbourhoods of vertices in its half-graph are strongly regular. There are two admissible intersection arrays of $Q$-polynomial graphs: $\{10,9,8,7,6;1,2,3,4,10\}$ (a folded $10$-cube) and $\{55,54,50,35,10;1,5,20,45,55\}$. These graphs have strongly regular graphs $\Lambda$ (parameters $(126,25,8,4)$ and $(210,99,48,45)$) and neighbourhoods of vertices in $\Lambda$ (parameters $(25,8,4,2)$ and $(99,48,22,24)$). There are two admissible intersection arrays corresponding to graphs on $704$ vertices: $\{26,25,24,2,1;1,2,24,25,26\}$ and $\{36,34,32,4,1;1,4,32,34,36\}$. In this manuscript we study almost $Q$-polynomial graphs of diameter $5$. We obtain that distance-regular graphs with intersection arrays $\{26,25,24,2,1;1,2,24,25,26\}$ and $\{36,35,32,4,1;1,4,32,35,36\}$ do not exist.

Key words: distance-regular graph, $Q$-polynomial graph, bipartite graph.

UDC: 519.17

MSC: 05E30, 05C50

Received: 23.02.2025

DOI: 10.46698/y5679-0662-9249-a