Constructing new families of transmission irregular graphs

TL;DR

This paper introduces new families of transmission irregular graphs, especially starlike and bi-starlike trees, providing conditions for their transmission irregularity and expanding the known classes of such graphs.

Contribution

The paper constructs and characterizes new infinite families of transmission irregular graphs, including specific starlike, bi-starlike, and unicyclic graphs, with precise conditions for irregularity.

Findings

Odd order starlike trees with certain parameters are transmission irregular.

Characterization of when linear starlike trees are transmission irregular.

Identification of infinite families of transmission irregular unicyclic 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 $T(k_1,\ldots,k_t)$ is a tree obtained by attaching to an isolated vertex $t$ pendant paths of lengths $k_1,\ldots,k_t$, respectively. It is proved that if a starlike tree $T(a,a+1,\ldots,a+k)$, $k\ge 2$, is of odd order, then it is transmission irregular. $T(1,2,\ldots,\ell)$, $\ell \ge 3$, is transmission irregular if and only if $\ell \notin \{r^2 + 1:\ r\ge 2\}$. 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 $T(a,a+1,a+2)$, $a\ge 2$, is transmission irregular if and only if $a$ is even.