Quadratic integer programming with application to the chaotic mappings of complete multipartite graphs

Let a be a permutation of the vertex set V(G) of a connected graph G. Define the total relative displacement of alpha in G by [GRAPHIC] where d(G) (x, y) is the length of the shortest path between x and y in G. Let pi*(G) be the maximum value of delta (alpha)(G) among all permutations of V(G). The permutation which realizes pi*(G) is called a chaotic mapping of G. In this paper, we study the chaotic mappings of complete multipartite graphs. The problem is reduced to a quadratic integer programming problem. We characterize its optimal solution and present an algorithm running in O(n(5) log n) time, where n is the total number of vertices in a complete multipartite graph.