Transmission-Regular Graph

A connected graph is transmission-regular if all its vertices have the same vertex transmission. Equivalently, a graph G is transmission-regular iff its transmission dimension is dim_T(G)=1.

Every vertex-transitive graph is transmission-regular. More generally, every distance-regular graph is transmission-regular since the number of vertices at each distance from a given vertex is independent of the chosen vertex. In particular, complete graphs, cycle graphs, hypercube graphs, the Petersen graph, and balanced complete multipartite graphs are transmission-regular.

Not every regular graph is transmission-regular. For example, the Frucht graph is regular but has multiple distinct vertex transmissions.

See also

Distance-Regular Graph, Graph Transmission, Transmission Dimension, Transmission-Minimal Regular Graph, Vertex Transmission, Vertex-Transitive Graph

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References

Alfaro, C. A.; Villagrán, R. R.; and Zapata, O. "Distinguishing Graphs with Two Integer Matrices." 27 Sep 2023. https://arxiv.org/abs/2309.15365.

Cite this as:

Weisstein, Eric W. "Transmission-Regular Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Transmission-RegularGraph.html

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