On transmission-irregular graphs and long pendent paths

Abstract

The transmission of a vertex in a connected graph is the sum of distances from that vertex to all other vertices. A graph is transmission-irregular (TI) if no two of its vertices have the same transmission. Xu et al. (2023) [3] recently asked to establish methods for constructing new TI graphs from the existing ones and also about the existence of chemical TI graphs on every even order. We show that, under certain conditions, new TI graphs can be obtained from the existing TI graph G either by attaching pendent paths of equal length to every vertex of G or by attaching two pendent paths of consecutive lengths to one vertex of G. We also show the existence of chemical TI graphs for almost all even orders.