Indices for special classes of trees. (English) Zbl 1282.05148
Summary: In this article we study the index (the largest eigenvalue of the adjacency matrix) in two special classes of trees, namely: starlike trees and double brooms. For each class, we determine conditions for the index to be integer. We examine conditions in order to compare the index of a starlike tree with \(\sqrt{\Delta + 1}\), where \({\Delta}\) is the maximum degree of the graph. We also prove that indices of a broom-like tree and a double broom, except for some cases, are between \(\sqrt{\Delta}\) and \(\sqrt{\Delta + 1}\), characterizing when occurs one equality. Furthermore, we build in these classes, infinite families of non-integral trees with integer index.
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