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Bounds for the Energy of Graphs

Author

Listed:
  • Slobodan Filipovski

    (FAMNIT, University of Primorska, 6000 Koper, Slovenia)

  • Robert Jajcay

    (Department of Algebra and Geometry, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia)

Abstract

Let G be a graph on n vertices and m edges, with maximum degree Δ ( G ) and minimum degree δ ( G ) . Let A be the adjacency matrix of G , and let λ 1 ≥ λ 2 ≥ … ≥ λ n be the eigenvalues of G . The energy of G , denoted by E ( G ) , is defined as the sum of the absolute values of the eigenvalues of G , that is E ( G ) = | λ 1 | + … + | λ n | . The energy of G is known to be at least twice the minimum degree of G , E ( G ) ≥ 2 δ ( G ) . Akbari and Hosseinzadeh conjectured that the energy of a graph G whose adjacency matrix is nonsingular is in fact greater than or equal to the sum of the maximum and the minimum degrees of G , i.e., E ( G ) ≥ Δ ( G ) + δ ( G ) . In this paper, we present a proof of this conjecture for hyperenergetic graphs, and we prove an inequality that appears to support the conjectured inequality. Additionally, we derive various lower and upper bounds for E ( G ) . The results rely on elementary inequalities and their application.

Suggested Citation

  • Slobodan Filipovski & Robert Jajcay, 2021. "Bounds for the Energy of Graphs," Mathematics, MDPI, vol. 9(14), pages 1-10, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:14:p:1687-:d:596442
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    References listed on IDEAS

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    1. Jahanbani, Akbar, 2017. "Some new lower bounds for energy of graphs," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 233-238.
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