kirchhoff index and degree kirchhoff index of complete multipartite   graph
  The Kirchhoff index of a graph is defined as half of the sum of all effective resistance distances between any two vertices. Assuming a complete multipartite graph G, by methods from linear algebra we explicitly formulate effective resistance distances between any two vertices of G, and its Kirchhoff index. In rest of paper we explore extremal value of Kirchhoff index for multipartite graphs 
Ravindra B. Bapat, Masoud Karimi, Jia-Bao Liu
math.CO