Let 
be a connected graph with vertex
set 
and graph distance 
. The vertex transmission,
or status, of a vertex 
is defined by
 |
(1)
|
Equivalently, if 
is the graph distance matrix of 
, then the vector of vertex transmissions is
 |
(2)
|
where 
is the all-ones vector. Thus the vertex transmissions are precisely the row sums
of the graph distance matrix.
The diagonal matrix 
having the vertex transmissions on its diagonal is used,
for example, in defining transmission-adjacency matrices such as 
and 
, where 
is the adjacency matrix
(Alfaro et al. 2023).
The graph transmission of 
is defined by
 |
(3)
|
and is therefore identical to the Wiener index 
.
For a graph on 
vertices with mean distance 
, the graph transmission satisfies
 |
(4)
|
and the average vertex transmission is
 |
(5)
|
For connected graphs with 
, the usual normalized closeness centrality 
is the reciprocal of the average distance from 
to all other vertices, and hence
 |
(6)
|
The number of distinct vertex transmissions of 
is called the transmission
dimension. A graph whose vertices all have the same vertex transmission is called
transmission-regular.
