AI 中文总结
研究边传输不规则图，通过扩展顶点传输概念到边，证明几乎所有图非ETI，探讨相关阶可实现性问题，尤其证明对\(n\geq15\)存在阶为\(n\)的亚立方树既是TI又是ETI。
AI 中文摘要
在连通图\(G\)中，顶点\(v\)的传输是\(v\)到\(G\)中所有顶点距离之和。传输不规则（TI）图是任意两个不同顶点传输不同的连通图。通过将边的传输定义为其两个端点传输之和，把传输概念扩展到边。若任意两条不同边传输不同，则连通图为边传输不规则（ETI）图。证明几乎所有图不是ETI，并研究了涉及化学ETI图的几个相关阶可实现性问题。特别证明对于每个\(n\geq15\)，存在阶为\(n\)的亚立方树既是TI又是ETI。
英文摘要
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.
