The number of spanning trees in directed circulant graphs with non-fixed jumps

Let Cpn(a1,a2,…,ak,q1n,q2n,…,qmn)Cpn(a1,a2,…,ak,q1n,q2n,…,qmn) be a directed circulant graphs with pn   vertices and some non-fixed jumps, where 1⩽a1⩽a2⩽⋯⩽ak(⩽n-1)1⩽a1⩽a2⩽⋯⩽ak(⩽n-1),1⩽q1⩽q2⩽⋯⩽qm⩽p-1,1⩽q1⩽q2⩽⋯⩽qm⩽p-1, are fixed integers, and an integer n varies. In this paper, a formula, asymptotic behaviors and linear recurrence relations for the number of its spanning trees are obtained.