Complexity of the circulant foliation over a graph

Abstract

In the present paper, we investigate the complexity of infinite family of graphs \(H_n=H_n(G_1,\,G_2,\ldots ,G_m)\) obtained as a circulant foliation over a graph H on m vertices with fibers \(G_1,\,G_2,\ldots ,G_m.\) Each fiber \(G_i=C_n(s_{i,1},\,s_{i,2},\ldots ,s_{i,k_i})\) of this foliation is the circulant graph on n vertices with jumps \(s_{i,1},\,s_{i,2},\ldots ,s_{i,k_i}.\) This family includes the family of generalized Petersen graphs, I-graphs, sandwiches of circulant graphs, discrete torus graphs and others. We obtain a closed formula for the number \(\tau (n)\) of spanning trees in \(H_n\) in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as \(n\rightarrow \infty .\)