Let 
be a connected graph with vertex
set 
and graph distance 
. The vertex transmission, or status, of a vertex 
is
 |
(1)
|
Equivalently, if 
is the graph distance matrix of 
, then the vector of vertex transmissions is
 |
(2)
|
where 
is the all-ones vector, so vertex transmissions are the row sums of the graph distance
matrix.
The term "transmission" is descriptive: 
measures the total graph-distance cost of transmitting
from 
to all vertices of 
.
It is not an information theory quantity in
the Shannon sense; in this graph-theoretic usage, only graph distances are involved.
The graph transmission of 
is the half-sum of all vertex transmissions,
 |
(3)
|
and is therefore identical to the Wiener index 
.
For a graph on 
vertices with all-distance-matrix-entry mean distance 
, the average vertex transmission is
.
For connected graphs with 
, the usual normalized closeness closeness
centrality 
is
 |
(4)
|
For a connected vertex-transitive graph on 
vertices, vertex transmission also determines betweenness
centrality: 
when ordered pairs are counted, and half this value when unordered pairs are counted.
