Kirchhoff Index and Kirchhoff Energy

The Kirchhoff energy and Kirchhoff Laplacian energy for Kirchhoff matrix are examined in this paper‎. ‎The Kirchhoff index with Kirchhoff Laplacian eigenvalues is defined and different inequalities including the distances‎, ‎the vertices and the edges are obtained‎. ‎Indeed‎, ‎some bounds for the degree Kirchhoff index associated with its eigenvalues are found.‎