On 2-distance primitive graphs of prime valency (Q6542029)

This paper is a contribution to the classification of \(2\)-distance primitive graphs of prime valency. Let \(\Gamma\) be a graph with vertex set \(V(\Gamma)\) and full automorphism group \(A:=\Aut(\Gamma)\). The graph \(\Gamma \) is said to be vertex transitive if \(A\) acts transitively on \(V(\Gamma)\). Let \(u\) and \(v\) be two distinct vertices of \(\Gamma\). Then the smallest length \(d_{\Gamma}(u, v)\) of paths between \(u\) and \(v\) is called the distance of \(u\) and \(v\), and the maximum \(d:=\mathrm{diam}(\Gamma)\) of \(d_{\Gamma}(u, v)\) for distinct \(u,v\in V(\Gamma)\) is the diameter of \(\Gamma\). Let \(u\in V(\Gamma)\), and for a positive integer \(i\leq d\), let \(\Gamma_{i}(u)\) be the set of vertices at distance \(i\), and set \(\Gamma(u):=\Gamma_{1}(u)\). The graph \(\Gamma\) is called \(s\)-distance transitive if for each \(i \leq s \leq d\), the vertex stabilizer \( A_{u}\) is transitive on \(\Gamma_{i}(u)\), and \(\Gamma\) is called distance transitive if it is \(d\)-distance transitive. All distance transitive graphs with diameter at least \(2\) are \(2\)-distance transitive, but the converse is not true, see [\textit{C. Li} et al., Sci. China, Math. 54, No. 4, 845--854 (2011; Zbl 1217.05106)]. Let \(\Gamma\) be an \(s\)-distance transitive graph with \( 1 \leq s \leq d\), and let \(u\in V(\Gamma)\). For \(1\leq i\leq s\), the number of edges from \(v\) to \(\Gamma_{i-1}(u)\), \(\Gamma_{i}(u)\) and \(\Gamma_{i+1}(u)\) are denoted by \(c_{i}\), \(a_{i}\) and \(b_{i}\), respectively, and these numbers do not depend on the choice of \(v\). A non-complete vertex transitive graph \(\Gamma\) is called \(2\)-distance primitive, if for each \(u \in V (\Gamma)\), the stabilizer \(A_{u}\) is primitive on both \(\Gamma(u)\) and \(\Gamma_{2}(u)\) (The notion of \(2\)-distance primitive graphs is introduced in [\textit{W. Jin} et al., Appl. Math. Comput. 357, 310--316 (2019; Zbl 1428.05328); \textit{C. E. Praeger}, J. Lond. Math. Soc., II. Ser. 17, 67--73 (1978; Zbl 0378.20003)].\N\N\textit{W. Jin} et al. [Electron. J. Comb. 27, No. 4, Research Paper P4.53, 15 p. (2020; Zbl 1456.05045)] proved that for a \(2\)-distance primitive graph \(\Gamma\), if \(\Gamma\) is neither a cycle, nor a complete bipartite graph, then \(\Gamma\) has girth at most \(4\) and the vertex stabilizer \(A_{u}\) acts faithfully on both \(\Gamma(u)\) and \(\Gamma_{2}(u)\). \textit{W. Jin} et al. [Appl. Math. Comput. 357, 310--316 (2019; Zbl 1428.05328)] studied prime valent \(2\)-distance primitive graphs, and proved that if \(\Gamma\) is such a graph of valency \(p\), then either it belongs to (i) one of the seven known families, or (ii) \(c_{2}\in\{2,3,4,6\}\), \(A_{u}\) is \(2\)-transitive with soc\((A_{u})=PSL_{d}(q)\) [loc. cit.] for more details.\N\NIn this paper, the authors studied the graphs in case (ii) and proved that such a graph is either the folded \(5\)-cube, or it has girth \(4\) and contains no \(5\)-cycles and \(A_u=PSL(2,q)\). They also revisited the main results in [loc. cit.], and gave a shorter proof of the main result in [loc. cit.] with some new graphs that are not listed in [loc. cit.]. They proposed a problem (Problem 1.4) for further study: Classify \(2\)-distance primitive graphs of prime valency \(p\) and girth \(4\) such that \((a_{2},b_{2},c_{2}) = (0, p - 2,2)\) and \((p,\mathrm{soc}(A_{u}))\in \{(11,PSL(2,11)), (11,M_{11}), (23,M_{23}), (q+1,PSL(2,q))\}\).