Three-branching Transmission irregular graphs

Abstract

The transmission of a vertex $v$ in a graph $G$ is the sum of distances from $v$ to  other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is transmission irregular. A vertex in a graph is a branching vertex if its degree is at least $3$. A graph $G$ is three-branching if $G$ contains exactly three branching vertices.  It is shown in this paper that, for any natural number $n\geq 11$, there exists a three-branching transmission irregular graph of order $n$, in particular, which of them are just trees for each odd $n$ or $n\equiv 4(mod~6)$.