On hyper-Kirchhoff index

The hyper-Kirchhoff index is introduced when the hyper-Wiener operator is applied to the resistance-distance matrix of a connected graph. We give lower and upper bounds for the hyper-Kirchhoff index, and determine the nn-vertex unicyclic graphs with the smallest, the second and the third smallest as well as the largest, the second and the third largest hyper-Kirchhoff indices for n≥5n≥5. We also determine the nn-vertex unicyclic graphs of cycle length ss, 3≤s≤n3≤s≤n, with the smallest and the largest hyper-Kirchhoff indices.