Dedicated to Misha Klin on the occasion of his retirement.Abstract. We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include constructions of new examples, structure theorems, valency constraints, and a classification of certain special families of such graphs. We also present a new example of a graph with three valencies and three eigenvalues of which there are currently only finitely many known examples.

Abstract. We classify the connected graphs with precisely three distinct eigenvalues and second largest eigenvalue at most 1.

Let be a permutation of V (G) of a connected graph G. De ne the total relativ e displacement of in G by (G) = X x;y2V (G) jd G (x; y) ? d G ((x); (y))j where d G (x; y) is the length of the shortest path between x and y in G. Let (G) be the maximum value of (G) among all permutations of V (G) and the permutation which realizes (G) is called a chaotic mapping of G. In this paper, we study the chaotic mappings of complete multipartite graphs. The problem will reduce to a quadratic integer programming. 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. key word. Chaotic mapping, complete multipartite graph. AMS(MOS) subject classi cation. 05C05

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