Characterization of transmission irregular starlike and double starlike trees

The transmission of a vertex in a connected graph is the sum of its distances to all the other vertices. A graph is transmission irregular, or TI for short, when all of its vertices have mutually distinct transmissions. In an earlier paper, Al-Yakoob and Stevanović (Appl. Math. Comput. 380:125257, 2020) gave the full characterization of TI starlike trees with three branches. Here, we improve these results by using a different approach to provide the complete characterization of all TI starlike trees. Moreover, we find the precise conditions under which a double starlike tree is TI. Finally, we implement the aforementioned conditions in order to find several infinite families of TI starlike trees and TI double starlike trees.