Improved lower bound on the energy of line graphs. (English) Zbl 1518.05107
Summary: The energy of a graph is defined as the sum of absolute values of all eigenvalues of its adjacency matrix. For a nonempty graph \(G\), S. Akbari et al. [Linear Algebra Appl. 636, 143–153 (2022; Zbl 1480.05083)] proposed a conjecture: The energy of the line graph of \(G\) is at least \(|E(G)|+\Delta(G)-3\), where \(E(G)\) is the edge set of \(G\) and \(\Delta(G)\) is the maximum degree of \(G\). In this paper, we give a proof confirming the conjecture, and present a lower bound and an upper bound for the energy of line graphs of regular graphs.
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C35 | Extremal problems in graph theory |
| 05C07 | Vertex degrees |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
Citations:
Zbl 1480.05083References:
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