This analysis demonstrates the existence of transmission irregular graphs with three branching vertices, highlighting their unique characteristics in graph theory.
The transmission of a vertex v in a 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 transmission irregular. A vertex in a graph is a branching vertex if its degree is at least 3. A graph G is three-branching if G contains exactly three branching vertices. It is shown in this paper that, for any natural number n ? 11 with n < {12, 14}, there exists a three-branching transmission irregular graph of order n. In particular, there exists a three-branching transmission irregular tree for each odd n or n ? 4 (mod 6).
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Anran Xu (2025) studied this question.
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