Quadratic Integer Programming with Application to the Chaotic Mappings of Complete Multipartite Graphs

Let α be a permutation of the vertex set V (G ) of a connected graph G. Define the total relative displacement of α in G by δα (G )G ∑ x,y∈V(G) dG (x, y)AdG (α (x), α (y)) , where dG (x, y) is the length of the shortest path between x and y in G. Let π*(G ) be the maximum value of δα (G ) among all permutations of V (G ). The permutation which realizes π*(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 log n) time, where n is the total number of vertices in a complete multipartite graph.