I've been doing some research into the Wiener and Szeged Index for consideration into future work and I've been having some trouble with some very basic properties.
We know that the Wiener index formula is W(G) =$\displaystyle\sum_{e \, \in \, E(G)} n_1(e|G)\, n_{2}(e|G)$
And similarly the Szeged index is defined as $Sz(G) \displaystyle\sum_{e \, \in \, E(G)} n_1(e|G)\, n_{2}(e|G)$
- Where $n_1 (e|G)$ counts the vertices of G lying closer to one endpoint $(x)$ of the edge $e$ than to its other endpoint $(y)$.
- Vertices equidistant to $x$ and $y$ are not counted
- Vertices belonging to components of $G$ different than the component containing $x$ and $y$ are not counted.
My question is that the formulas of both how is a sample calculation done for a simple graph with 2 vertices for example?
Suppose that we have a simple straight line graph with only 2 vertices $u$ and $v$ on each end say. Then the Szeged index would be 1.
This is since $n_1 =1$ and $n_2 =1$
Is this reasoning correct?
How would this work for a graph with 3 vertices in the shape of the letter T ?
For reference this is a breakdown of all equations used and their respective meanings
Your reasoning is correct for the graph with $n=2$.
I've seen this index used in some papers but without this name.
What do you call a graph with 3 vertices in a shape of a T ? I can do some kind of T with 4 vertices but not with 3.
For a path of length 2 (3 vertices, 2 edges), the index is 4.
For a 'T' which I guess you mean a 3-star (n=4,m=3), I get 3*1+3*1+3*1 = 12.