Theoretical and computational methods to minimize Kirchhoff index of graphs with a given edge k-partiteness

Let G be a connected graph. The edge k-partiteness of G is the minimum number of edges whose deletion from G yields a k-partite graph. The Kirchhoff index Kf(G) of G is the sum of the resistance distance between all unordered pairs of vertices. In this paper, we study the problem to determine the minimum Kirchhoff index of graphs with a given edge k-partiteness. First, we theoretically characterize the graphs with minimum Kirchhoff index in this graph family when the edge k-partiteness is not big compared to the number of vertices of G. Then we propose an exhaustive search algorithm to find the optimal graphs. At last, three strategies are used to reduce the computation complexity of our algorithm and several performance comparisons of these strategies are given.