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.

In this paper, we develop a new method to produce explicit formulas for the number f G (n) of rooted spanning forests in the circulant graphs G = C n (s 1 , s 2 , . . . , s k ) and G = C 2n (s 1 , s 2 , . . . , s k , n). These formulas are expressed through Chebyshev polynomials. We prove that in both cases the number of rooted spanning forests can be represented in the form f G (n) = p a(n) 2 , where a(n) is an integer sequence and p is a certain natural number depending on the parity of n. Finally, we find an asymptotic formula for f G (n) through the Mahler measure of the associated Laurent polynomial P (z) = 2k + 1 − k i=1 (z si + z −si ).

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For any given graph G, consider the graph Ĝ which is a cone over G. We study two important invariants of such a cone, namely, the complexity (the number of spanning trees) and the Jacobian of the graph. We prove that complexity of graph Ĝ coincides with the number of rooted spanning forests in G and the Jacobian of Ĝ is isomorphic to the cokernel of the operator I + L(G), where L(G) is the Laplacian of G and I is the identity matrix. As a consequence, one can calculate the complexity of Ĝ as det(I + L(G)). As an application, we establish general structural theorems for the Jacobian of Ĝ in the case when G is a circulant graph or cobordism of two circulant graphs.