Bounds for the Energy of Graphs

Slobodan Filipovski and Robert Jajcay
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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

Mathematics, 2021, vol. 9, issue 14, 1-10

Abstract: Let G be a graph on n vertices and m edges, with maximum degree ? ( G ) and minimum degree ? ( G ) . Let A be the adjacency matrix of G , and let ? 1 ? ? 2 ? … ? ? 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 ) = | ? 1 | + … + | ? n | . The energy of G is known to be at least twice the minimum degree of G , E ( G ) ? 2 ? ( 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 ) ? ? ( G ) + ? ( 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.

Keywords: energy of graphs; conjecture; new bounds (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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