New transmission irregular chemical graphs

TL;DR

This paper investigates transmission irregular chemical graphs, proving existence for all odd orders above 7, and introduces methods to construct and characterize such graphs, including conditions for preserving irregularity when adding edges.

Contribution

It establishes the existence of transmission irregular chemical trees for all odd orders ≥7 and provides new constructions and characterizations of such graphs.

Findings

Existence of transmission irregular chemical trees for all odd n ≥ 7

New construction methods for transmission irregular chemical trees

Conditions for maintaining transmission irregularity when adding edges

Abstract

The transmission of a vertex $v$ of a (chemical) 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 a transmission irregular graph. It is shown that for any odd number $n\geq 7$ there exists a transmission irregular chemical tree of order $n$. A construction is provided which generates new transmission irregular (chemical) trees. Two additional families of chemical graphs are characterized by property of transmission irregularity and two sufficient condition provided which guarantee that the transmission irregularity is preserved upon adding a new edge.