Graph 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 defined by
Equivalently, if D=(d_(ij)) is the graph distance matrix of G, then the vector of vertex transmissions is
| T=D1, |
(2)
|
where 1 is the all-ones vector. Thus the vertex transmissions are precisely the row sums of the graph distance matrix.
The diagonal matrix trs(G) having the vertex transmissions on its diagonal is used, for example, in defining transmission-adjacency matrices such as trs(G)-A and trs(G)+A, where A is the adjacency matrix (Alfaro et al. 2023).
The graph transmission of G is defined by
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 graph transmission satisfies
| [画像: T(G)=W(G)=(n^2)/2d^_, ] |
(4)
|
and the average vertex transmission is
For connected graphs with n>1, the usual normalized closeness centrality C(u) is the reciprocal of the average distance from u to all other vertices, and hence
| [画像: C(u)=(n-1)/(T(u)). ] |
(6)
|
The number of distinct vertex transmissions of G is called the transmission dimension. A graph whose vertices all have the same vertex transmission is called transmission-regular.
A transmission-minimal regular graph minimizes graph transmission among regular graphs of fixed order and degree.
See also
Graph Distance, Graph Distance Matrix, Mean Distance, Transmission Dimension, Transmission-Minimal Regular Graph, Transmission-Regular Graph, Vertex Transmission, Wiener IndexExplore with Wolfram|Alpha
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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. "Graph Transmission." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GraphTransmission.html