On the R-Spectrum and Randić Energy of Kragujevac-like Trees

The Randić matrix R(G) of a graph G is defined by (R)ij=1/didj if vertices i and j are adjacent, and 0 otherwise. The eigenvalues of R(G) constitute the R-spectrum, and the Randić energy RE(G) is the sum of their absolute values. In this paper, we investigate the R-spectrum and Randić energy of Kragujevac-like trees (KL-trees). Two explicit expressions for the R-characteristic polynomial of a KL-tree are derived. From these, we completely determine the KL-trees with the greatest Randić energy among all KL-trees of fixed order. Our results provide a partial confirmation of a conjecture of Gutman concerning the extremal Randić energy of trees.