论文标题
$ n \ leq3 $
Two-geodesic transitive graphs of order $p^n$ with $n\leq3$
论文作者
论文摘要
如果$ v $均与$ u $和$ w $相邻，并且$ w $与$ w $相邻，则图表的顶点三重$（u，v，w）$被称为$ 2 $ -GEODESIC。据说，如果其自动形态组在$ 2 $ -GEODESICS的集合中及时，则图表为$ 2美元。在本文中，为每个Prime $ P $和$ n \ leq 3 $提供了$ 2 $ - 斜角的及时分类。事实证明，所有这些图形都由三个小图组成：完整的双方图$ k_ {4,4,4} $ $ 8 $的订单，订单$ 27 $及其补充的Schläfli图和第四次无限族：循环$ C_P，C_P，C_P，C_P，C_ {P^2} $ and $ C_ {P^3} $和$ c_ { $K_{p^3}$, the complete multipartite graphs $K_{p[p]}$, $K_{p[p^2]}$ and $K_{p^2[p]}$, the Hamming graph $H(2,p)$ and its complement, the Hamming graph $H(3,p)$, and two infinite families of normal Cayley graphs on extraspecial group of order $ p^3 $和指数$ p $。
A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to be $2$-geodesic transitive if its automorphism group is transitive on the set of $2$-geodesics. In this paper, a complete classification of $2$-geodesic transitive graphs of order $p^n$ is given for each prime $p$ and $n\leq 3$. It turns out that all such graphs consist of three small graphs: the complete bipartite graph $K_{4,4}$ of order $8$, the Schläfli graph of order $27$ and its complement, and fourteen infinite families: the cycles $C_p, C_{p^2}$ and $C_{p^3}$, the complete graphs $K_p, K_{p^2}$ and $K_{p^3}$, the complete multipartite graphs $K_{p[p]}$, $K_{p[p^2]}$ and $K_{p^2[p]}$, the Hamming graph $H(2,p)$ and its complement, the Hamming graph $H(3,p)$, and two infinite families of normal Cayley graphs on extraspecial group of order $p^3$ and exponent $p$.
