具有唯一定长路的有向图的一个注记

Note on directed graphs with unique paths of fixed length

Ｌａｍ和ｖａｎ Ｌｉｎｔ构造了一类具有唯一定长路的有向图Ｄ（ｃ，ｋ），其阶为ｎ＝ｃ＾ｋ＋１，并证明Ｄ（ｃ，ｋ）的自同群包含一个２（ｃ＋１）阶二面体群，其中ｃ为大于１的整数，ｋ为大于１的奇数。本文利用（０，１）矩阵方程的性质证明，对任意的整数ｃ＞１和奇数ｋ＞１，存在ψ（ｋ）（ψ为Ｅｕｌｅｒ函数）个ｎ＝Ｃ＾ｋ＋１阶具有唯一定长路的有路的有向图；

Lam and van Lint construct a directed graph D(c,k) of order n = c^k+l with unique paths of fixed length and show that the automorphism group for D(c,k) contains a dihedral group of order 2(c + 1),where c≥ is an integer and k≥l is odd. Using the properties of (0,1)-matrix equations,the authors have proved that,for every integer c≥1 and odd integer k≥1,there are ψ(k) (ψ Euler function) directed graphs of order n =c^k+1 with unique paths of fixed length, which are not j isomorphic to each other, and each of them has a dihedral group of order 2 (c+1) as its full automorphism group.