Finding the weighted parameters of the vertex adjacency matrix for a derivative of azulene

In my last question ( [Numbering in a derivative of azulene](https://chemistry.stackexchange.com/questions/105036/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
[![VAM formalism](https://i.stack.imgur.com/FSFno.png)](https://i.stack.imgur.com/FSFno.png)
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](https://pubs.acs.org/doi/suppl/10.1021/acs.chemrev.7b00733/suppl_file/cr7b00733_si_001.pdf) the VAM represented by A is:
[![VAM for azulene only](https://i.stack.imgur.com/SyasW.png)](https://i.stack.imgur.com/SyasW.png)


Now by having the correct numbering to my molecule ([Numbering in a derivative of azulene](https://chemistry.stackexchange.com/questions/105036/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:
[![Finding the VAM for this molecule](https://i.stack.imgur.com/u1KOd.png)](https://i.stack.imgur.com/u1KOd.png)


To find VAM the only way was to study the VAM of weighted Graphs as shown below:
[![VAM for weighted graphs](https://i.stack.imgur.com/hQxIx.png)](https://i.stack.imgur.com/hQxIx.png)


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](https://www.researchgate.net/publication/253516227_Topological_Indices_Based_on_Topological_Distances_in_Molecular_Graph) descriptions are still [reviewed](http://euro-math-soc.eu/review/graph-theoretical-matrices-chemistry) ([2nd edition](https://www.crcpress.com/Graph-Theoretical-Matrices-in-Chemistry/Janezic-Milicevic-Nikolic-Trinajstic/p/book/9781138894280)), and [apparently](https://www.sciencedirect.com/science/article/pii/S0972860017301408)  the [Open Molecular Generator](https://link.springer.com/article/10.1186/1758-2946-4-21) were one tool to generate them.


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