Classification of borderenergetic chemical graphs and borderenergetic graphs of order 12

The energy $E(G)$ of a simple graph $G$ is the sum of absolute values of the eigenvalues of its adjacency matrix. A borderenergetic graph of order $n\in \mathbb{N}$ is any noncomplete graph~$G$ such that $E(G)=E(K_n)=2n-2$. Here we combine two-phase computer-assisted search with theoretical arguments to show that there are only three borderenergetic chemical graphs, thus completing the earlier findings of Li, Wei and Zhu [MATCH Commun. Math. Comput. Chem. 77 (2017), 25-36]. We perform two-phase computer-assisted search to also find all $566$ borderenergetic graphs of order~$12$, thereby correcting and extending the results from a previous search performed by Furtula and Gutman [Iranian J. Math. Chem. 8(4) (2017), 339-344].