Edge transmission irregular graphs

2026-07-12

AI Summary

This work introduces, for the first time, the concept of edge transmission irregular (ETI) graphs—connected graphs in which the transmission values of any two distinct edges are distinct, where the transmission of an edge is defined as the sum of the transmissions of its endpoints. Through graph-theoretic analysis, combinatorial constructions, and explicit computation of transmission functions, the study demonstrates that almost all graphs are not ETI. Nevertheless, it establishes that for every order \( n \geq 15 \), there exists a subcubic tree that simultaneously satisfies both vertex transmission irregularity (TI) and the newly introduced ETI property. This result not only confirms the existence and constructibility of ETI graphs but also provides chemical graph theory with a novel class of molecular structure models exhibiting both TI and ETI characteristics.

Abstract

The transmission of a vertex $v$ in a connected graph $G$ is the sum of distances from $v$ to all vertices in $G$. A transmission irregular (TI) graph is a connected graph in which any two distinct vertices have different transmissions. We extend the concept of transmission to edges by defining the transmission of an edge as the sum of the transmissions of its two endpoints. A connected graph can now be called edge transmission irregular (ETI) if any two distinct edges have different transmissions. We show that almost all graphs are not ETI and then investigate several related order realizability problems involving chemical ETI graphs. In particular, we prove that for every $n \ge 15$, there exists a subcubic tree of order $n$ that is both TI and ETI.

Problem

• edge transmission irregular
• transmission irregular
• subcubic tree
• order realizability
• chemical graphs

Innovation

• edge transmission irregular
• transmission irregular graph
• subcubic tree
• graph realizability
• chemical graph theory