We consider a family of graphs Hn(s1, . . . , s k ; t1, . . . , t ℓ ), which is a generalization of the family of I-graphs, which in turn, includes the generalized Petersen graphs and the prism graphs. We present an explicit formula for the number fH (n) of rooted spanning forests in these graphs in terms of Chebyshev polynomials and find its asymptotics. Also, we show that the number of rooted spanning forests can be represented in the form fH (n) = p a(n) 2 , where a(n) is an integer sequence and p is a prescribed integer depending on the number of odd elements in the sequence s1, . . . , s k , t1, . . . , t ℓ and the parity of n.