Vertex Transmission

Let G be a connected graph with vertex set V(G) and graph distance d(u,v). The vertex transmission, or status, of a vertex u in V(G) is

Equivalently, if D=(d_(ij)) is the graph distance matrix of G, then the vector of vertex transmissions is

where 1 is the all-ones vector, so vertex transmissions are the row sums of the graph distance matrix.

The graph transmission of G is the half-sum of all vertex transmissions,

and is therefore identical to the Wiener index W(G).

For a graph on n vertices with all-distance-matrix-entry mean distance d^_, the average vertex transmission is nd^_. For connected graphs with n>1, the usual normalized closeness centrality C(u) is

See also

Connected Graph, Graph Distance, Graph Distance Matrix, Graph Transmission, Mean Distance, Transmission-Minimal Regular Graph, Wiener Index

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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.Sharon, J. O. and Rajalaxmi, T. M. "Transmission in Certain Trees." Procedia Comput. Sci. 172, 193-198, 2020. https://doi.org/10.1016/j.procs.2020年05月03日0.

Cite this as:

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

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