# Smoothness of Graph Energy in Chemical Graphs
## 3 Computational Details and Results
### Structure Sensitivity and Abruptness of Graph Energy  
We now consider the structure sensitivity and the abruptness of the graph energy of alkanes and molecular graphs in the dataset \(\mathcal{M}\), which has a graph edit distance equal to 2. More precisely, for an alkane \(G\), let \(\mathcal{S}(G)\) be the set of all molecular graphs resulting from \(G\) with the addition of an edge and its endpoint. For example, for hexane \(G\), there are three molecular graphs from our dataset with GED = 2, that is, the molecular graphs of 2-methylhexane, 3-methylhexane and heptane form the set \(\mathcal{S}(G)\) (see Figure 4). The structure sensitivity and abruptness of graph energy are then calculated for each alkane \(G\). The obtained values are shown in Table A1 in Appendix A.  
Visualisation of the data from Table A1 from Appendix A is shown in Figure 5 for structure sensitivity and in Figure 6 for abruptness of graph energy. It can be seen that both structure sensitivity and abruptness of graph energy decrease with the order of a molecule.  
Figure 4: Hexane \(G\) and graphs in the set \(\mathcal{S}(G)\) with GED = 2 with respect to \(G\).  
Figure 5: Distribution of structure sensitivity of graph energy.  
Figure 3: Correlation between graph energy of methyl-alkanes and methyl-cycloalkanes.  
Finally, we consider the modified structure sensitivity \(SS^{*}\) and the modified abruptness \(\mathrm{Abr}^{*}\) as introduced by [PERSON] et al. in [14], where the authors aim to determine the structure sensitivity of molecular descriptors by grouping similar molecules using the Tanimoto index and the Morgan circular fingerprints. The same concept is also applied in [15], where the authors investigated structural sensitivity of some eigenvalue-based topological indices, including the graph energy. In this way we now consider isomeric molecules in the database instead of using the concept of graph edit distance.  
Our dataset is again the set \(\mathcal{M}\) without the smallest molecule ethane. The dataset is divided into 18 subsets of mutually isomeric molecules with respect to the number of carbon atoms, so the smallest subset contains molecules with 3 carbon atoms and the last subset contains molecules with 20 carbon atoms. The modified structure sensitivity and modified abruptness are considered separately in 18 subsets. For example, in Figure 7 we see isomers with five carbon atoms: pentane, 2-methylbutane, cyclopentane and methylcyclobutane. In each of these subsets, each molecule is set once as a referent molecule \(G\) and then the modified structure sensitivity and modified abruptness are computed for \(G\), so that the calculations must be performed for all 134 molecules from \(\mathcal{M}\). Moreover, the average values are calculated for modified structure sensitivity and modified abruptness in each of the 18 subgroups of isomers. Due to the complexity of the calculations, an algorithm was developed in the Python programming language. The results are summarised in Tables A2-A4 in Appendix A.  
In Figures 8 and 9, it can be seen that the modified sensitivity decreases with the order of the isomers considered, while the modified abruptness also decreases, but more slowly and depending on the parity of the vertices.  
Figure 6: Distribution of abruptness of graph energy.  
Figure 7: Isomers with five carbon atoms in the dataset.