The transmission of a vertex in a connected graph is the sum of distances from that vertex to all other vertices. A graph is transmission-irregular (TI) if no two of its vertices have the same transmission. Xu et al. (2023) recently asked to establish methods for constructing new TI graphs from the existing ones and also about the existence of chemical TI graphs on every even order. We show that, under certain conditions, new TI graphs can be obtained from the existing TI graph either by attaching pendent paths of equal length to every vertex of or by attaching two pendent paths of consecutive lengths to one vertex of . We also show the existence of chemical TI graphs for almost all even orders.

中文翻译：

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关于传输-不规则图和长悬垂路径


连通图中顶点的传输是从该顶点到所有其他顶点的距离之和。如果图的任何两个顶点都没有相同的透射率，则该图是不规则透射率 (TI)。徐等人。 (2023) 最近要求建立从现有的 TI 图构建新的 TI 图的方法，以及每个偶数阶上化学 TI 图的存在性。我们证明，在某些条件下，可以通过将相等长度的悬垂路径附加到 的每个顶点，或者将两个连续长度的悬垂路径附加到 的一个顶点，从现有 TI 图获得新的 TI 图。我们还展示了几乎所有偶数阶的化学 TI 图的存在。