On the index of broken wheels

Abstract

The index of a graph G is the largest eigenvalue of a (0,1)-adjacency matrix of G. Let denote the set of all graphs which can be obtained from an n-cycle by joining and additional ('central') vertex to k of the vertices of the cycle (Such graphs are called broken wheels.) By using a result of Schwenk's to compare the characteristic polynomials of graphs in . we identify the graphs with greatest and least index. We show in fact that the index is greatest when the k 'spokes' are bunched together as closely as possible, and is least when they are spread out as evenly as possible.