| The Kirchhof index of a graph G is defned as Kf (G)=1∑n∑n2i=1j=1rij, where rijis the resistance distance between vian vj. Let G be a simple bipartite graph, G thecomplement graph of G, S(G) the subdivision graph of G. First, we get a relation betweenthe Kirchhof index of the complement of G and the numbers of closed walks of S(G), thatis, Kf (G)=n m1M2+∑2k≥02k(S(G))nk1. Next, by comparing the numbers of closed walks ofthe corresponding subdivision graph, we determine the trees with the frst three maximumand the frst two minimum values of Kf (T), and bipartite unicyclic graphs with the frst andsecond maximum and frst minimum values of Kf (G). Finally, we compare the Kirchhofindex of the complement of a bipartite graph with the Estrada index and Laplacian Estradaindex of that graph itself, and fnd highly similarity among their expressions. |
