The characteristic polynomials of the Laplacian and the distance Laplacian matrices of power graphs of groups of order $ pqr $, where $ p,q $ and $ r $ are { primes,} are obtained. Further, the characteristic polynomials of these matrices for proper power graphs of cyclic and dicyclic groups are given. The important inequalities for the zeros of the distance Laplacian characteristic polynomials of power graphs of finite groups are presented in comments.

翻译：本文得到了阶为$pqr$的群的幂图的拉普拉斯矩阵和距离拉普拉斯矩阵的特征多项式，其中$p,q$和$r$为素数。进一步地，给出了循环群和二环群的真幂图这些矩阵的特征多项式。评注中给出了有限群的幂图的距离拉普拉斯特征多项式零点的重要不等式。