Closed and asymptotic formulas for energy of some circulant graphs

We consider circulant graphs G(r,N) where the vertices are the integers modulo N and the neighbours of 0 are {-r,...,-1,1,...,r}. The energy of G(r,N) is a trigonometric sum of N*r terms. For low values of r we compute this sum explicitly. We also study the asymptotics of the energy of G(r,N) for big N. There is a known integral formula for the linear growth coefficient, we find a new expression of the form of a finite trigonometric sum with r terms. As an application we show that in the family G(r,N) for r less or equal than 4 there is a finite number of hyperenergetic graphs. On the other hand, for each r>4 there is at most a finite number of non-hyperenergetic graphs of the form G(r,N). Finally we show that the graph G(r,2r+1) minimizes the energy among all the regular graphs of degree 2r.