Classification of borderenergetic chemical graphs and borderenergetic graphs of order 12

TL;DR

This paper identifies all borderenergetic chemical graphs, revealing only three such graphs exist and providing a complete classification of borderenergetic graphs of order 12 through combined computational and theoretical methods.

Contribution

It combines computer-assisted search with theoretical proofs to classify borderenergetic chemical graphs and complete the enumeration of order 12 borderenergetic graphs.

Findings

Only three borderenergetic chemical graphs exist.

All 566 borderenergetic graphs of order 12 are identified.

The results correct and extend previous studies.

Abstract

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].