Constructing new families of transmission irregular graphs

Abstract

The transmission of a vertex v of a graph G is the sum of distances from v to all the other vertices in G. A graph is transmission irregular if all of its vertices have pairwise different transmissions. A starlike tree T(k1,…,kt) is a tree obtained by attaching to an isolated vertex t pendant paths of lengths k1,…,kt, respectively. It is proved that if a starlike tree T(a,a+1,…,a+k), k≥2, is of odd order, then it is transmission irregular. T(1,2,…,ℓ), ℓ≥3, is transmission irregular if and only if ℓ∉{r2+1:r≥2}. Additional infinite families among the starlike trees and bi-starlike trees are determined. Transmission irregular unicyclic infinite families are also presented, in particular, the line graph of T(a,a+1,a+2), a≥2, is transmission irregular if and only if a is even.