Remarks on hyperenergetic circulant graphs

We first settle an open problem of Balakrishnan from Linear Algebra Appl. 387 (2004) 287-295. Further, if Ci¯(n,k1,k2,⋯,km), n ∈ N, k1 < k2 < ⋯ < km < n/2, ki ∈ N for i = 1, 2, …, m, denotes a circulant graph with the vertex set V = {0, 1, …, n − 1} such that a vertex u is adjacent to all vertices of V⧹{u} except u ± ki (mod n), i = 1, 2, …, m, we show that for any given k1 < k2 < ⋯ < km almost all circulant graphs Ci¯(n,k1,k2,…,km) are hyperenergetic.