The Xu index is a graph index defined for a connected graph 
on 
vertices 
, with vertex degrees 
and vertex
transmissions 
,
by
![X(G)=sqrt(n)ln(sum_(i)d_i[T(v_i)]^2)/(sum_(i)d_iT(v_i)),](/images/equations/XuIndex/NumberedEquation1.svg) |
where the vertex transmission is the corresponding row sum of the graph distance matrix (Ren 1999, Mangaldan).
Thus, the Xu index is not itself a graph transmission, but a degree-weighted natural-logarithmic descriptor formed from the vertex transmissions. The usual graph distance/graph transmission-based definition is not applicable to disconnected graphs.
Ren (1999) introduced the index for alkane QSPR (quantitative structure-property relationship) as a topological index based on the adjacency matrix and graph distance matrix. Ren et al. (1999) applied it to organic-compound QSPR/QSAR (quantitative structure-property/activity relationship).
The Xu index of a graph or molecule is implemented in the Wolfram Function Repository as ResourceFunction["XuIndex"][g]. For molecules, hydrogen atoms are ignored by default (Mangaldan).
Mangaldan, J. "XuIndex." Wolfram Function
Repository.