Ordering connected graphs by their Kirchhoff indices.

The Kirchhoff indexof a graphGis the sum of resistance distances between all unordered pairs of vertices, which was introduced by Klein and Randić. In this paper, we characterize all extremal graphs with respect to Kirchhoff index among all graphs obtained by deletingpedges from a complete graphwithand obtain a sharp upper bound on the Kirchhoff index of these graphs. In addition, all the graphs with the first to ninth maximal Kirchhoff indices are completely determined among all connected graphs of order. [ABSTRACT FROM PUBLISHER]