Remarks on hyperenergetic circulant graphs
The {energy} of a graph \(G\) is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of \(K_n\) is \(2(n-1)\). Motivated by a refuted conjecture that \(K_n\) is the graph of order \(n\) with maximum energy, graphs with energy greater than \(2(n-1)\) are called hyperenergetic. The circulant graph \(C(n,k_1,\dots,k_m)\) is the graph with vertex set \(\{0,\dots,n-1\}\) such that two of its vertices are joined by an edge if their difference modulo \(n\) is not equal to any of the numbers \(k_1,\dots,k_m\). The authors show that for every \(k_1,\dots,k_m\), there exists an integer \(n_0\) such that every circulant graph \(C(n,k_1,\dots,k_m)\) with \(n>n_0\) is hyperenergetic. This settles a problem of \textit{R. Balakrishnan} [Linear Algebra Appl. 387, 287--295 (2004; Zbl 1041.05046)] whether the complements of cycles are hyperenergetic.
scientific article; zbMATH DE number 2173036