# Smoothness of Graph Energy in Chemical Graphs
## 5 Conclusions  
This research gives us an insight into the relationship between graph energies of cyclic and acyclic structures and establishes the linear formulae for a very good prediction of this topological index from the order of the molecular graph. The measures of smoothness of graph energy are provided and their behaviour is studied. The next line of work is to compare the data provided here with other molecular descriptors to derive how smooth the graph energy behaves compared to other topological indices. Figure 8: Distribution of modified structure sensitivity of graph energy. Figure 9: Distribution of modified abruptness of graph energy. **Author Contributions:** All authors contributed equally to this work. Conceptualization, P.Z.P. and K.Z. ; methodology, P.Z.P. and K.Z. ; software, P.Z.P. ; formal analysis, P.Z.P. and K.Z. ; investigation, P.Z.P. and K.Z. ; resources, K.Z. ; data curation, P.Z.P. ; writing--original draft preparation, P.Z.P. and K.Z. ; writing--review and editing, P.Z.P. and K.Z. ; visualization, P.Z.P. and K.Z. ; supervision, P.Z.P. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the Slovenian Research Agency, grant number P1-0297 (P.Z.P). **Conflicts of Interest:** The authors declare no conflict of interest. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \# C & \(G\) & \(SS_{G}^{\bullet}(E)\) & \(\mathbf{Abr}_{G}^{\bullet}(E)\) & \(\overline{SS^{\bullet}}(E)\) & \(\overline{\mathbf{Abr}^{\bullet}}(E)\) \\ \hline \multirow{6}{*}{12} & dodecane & 0.43322 & 0.68215 & \\ \cline{2-5}  & 2-methylundecane & 0.65890 & 1.15977 & \\ \cline{2-5}  & 3-methylundecane & 0.40449 & 0.59723 & \\ \cline{2-5}  & 4-methylundecane & 0.58503 & 1.06267 & \\ \cline{2-5}  & 5-methylundecane & 0.40266 & 0.62438 & \\ \cline{2-5}  & 6-methylundecane & 0.56934 & 1.04079 & \\ \cline{2-5}  & cyclododecane & 0.63885 & 1.01789 & \\ \cline{2-5}  & methylcycloundecane & 0.75435 & 1.15977 & \\ \hline \multirow{6}{*}{13} & tridecane & 0.40136 & 0.84197 & \\ \cline{2-5}  & 2-methyldodecane & 0.58248 & 1.26986 & \\ \cline{2-5}  & 3-methyldodecane & 0.42327 & 0.98223 & \\ \cline{2-5}  & 4-methyldodecane & 0.50147 & 1.14839 & \\ \cline{2-5}  & 5-methyldodecane & 0.44474 & 1.03935 & \\ \cline{2-5}  & 6-methyldodecane & 0.46724 & 1.08698 & \\ \cline{2-5}  & cyclotridecane & 0.93790 & 1.26986 & \\ \cline{2-5}  & methylcyclododecane & 0.59177 & 0.85706 & \\ \hline \multirow{6}{*}{14} & tetradecane & 0.47516 & 0.84230 & \\ \cline{2-5}  & 2-methylridecane & 0.73279 & 1.50436 & \\ \cline{2-5}  & 3-methylrlreduceane & 0.46608 & 0.96408 & \\ \cline{2-5}  & 4-methylrlreduceane & 0.65694 & 1.40199 & \\ \cline{2-5}  & 5-methylrlreduceane & 0.46881 & 0.99504 & \\ \cline{2-5}  & 6-methylrlreduceane & 0.63480 & 1.37006 & \\ \cline{2-5}  & 7-methylrlreduceane & 0.46980 & 1.00269 & \\ \cline{2-5}  & cyclotetradecane & 1.04737 & 1.50436 & \\ \cline{2-5}  & methylcyclotridecane & 0.73900 & 1.14013 & \\ \hline \multirow{6}{*}{15} & pentadecane & 0.40023 & 0.82720 & \\ \cline{2-5}  & 2-methyltetradacane & 0.57807 & 1.27073 & \\ \cline{2-5}  & 3-methyltetradacane & 0.41464 & 0.96417 & \\ \cline{2-5}  & 4-methyltetradacane & 0.49861 & 1.1513 & \\ \cline{2-5}  & 5-methyltetradacane & 0.43140 & 1.01587 & \\ \cline{2-5}  & 6-methyltetradacane & 0.46638 & 1.09363 & \\ \cline{2-5}  & 7-methyltetradacane & 0.44703 & 1.05359 & \\ \hline cyclopentadecane & 0.94172 & 1.27073 & \\ \hline methylcyclotetradecane & 0.68286 & 0.97177 & \\ \hline \end{tabular}
\end{table} 
Table 3: Modified structure sensitivity and modified abruptness with average values, Part II.