Finding the weighted parameters of the vertex adjacency matrix for a derivative of azulene In my last question ( Numbering in a derivative of azulene ), I asked about the numbered of a molecule in order to find the Vertex Adjacency Matrix (VAM) of the same molecule. To explain my question properly, I will do my best to show my effort from A to Z as follows:
In general in order to find the VAM we use the formalism described by

To apply it onto the molecule I am studying, I first applied it to azulene and with agreement with the paper Quantum Interference, Graphs, Walks, and Polynomials the VAM represented by A is:

Now by having the correct numbering to my molecule (Numbering in a derivative of azulene), it also should be a 10 x 10 matrix. The difference is instead of hydrogen we now have Oxygen and hydrogen Sulfide involved in positions 1,5 and 2,6 as shown:

To find VAM the only way was to study the VAM of weighted Graphs as shown below:

Here are my questions:
1- Should I include  the parameter h only for the weighted carbons since the numbering of the molecule I am studying includes only carbon?
2- If 1 is no, how should I include the k parameters described before hand?
Any other suggestions to find VAM is appreciated.
As a comment: The matrix-representation you search emerges from graph theory, and mathematics.  The numbers drawn in the structures earlier (1 for the carbonyl carbon, 2 to the carbon adjacent to the -SH group) a synthetic chemist refers to (e.g., "let us introduce an alkyl chain in the 4-position") do not (necessarily) relate to the numbers to denote the vertices in these nets.  These are seasoned descriptions are still reviewed (2nd edition), and apparently  the Open Molecular Generator were one tool to generate them.