Ordering trees by their distance spectral radii ?
Let D(G) D ( G ) be the distance matrix of a connected graph G G . The distance spectral radius of G G , denoted by ?1(G) ? 1 ( G ) , is the largest eigenvalue of D(G) D ( G ) . In this paper we present a new transformation of a certain graph G G that decreases ?1(G) ? 1 ( G ) . With the transformation, we partially confirm a conjecture proposed by Stevanovi? and Ili? [17] by showing that, if ? ? ? n 2 ? , the double star S?,n?? S ? , n ? ? uniquely minimizes the distance spectral radius among all trees on n n vertices with maximum degree ? ? . Moreover, the trees on n?10 n ? 10 vertices with the fourth and fifth least distance spectral radii are characterized.
ordering trees distance spectral radii transformations