<p>The sum of the absolute values of the eigenvalues of the extended adjacency matrix <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ A_{ex}(\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mrow> <mi mathvariant="italic">ex</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (called the extended adjacency eigenvalues of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation>) of a graph <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation> is called the extended adjacency energy <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {E}_{ex}(\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">E</mi> <mrow> <mi mathvariant="italic">ex</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation>. In this paper, we obtain some sharp upper and lower bounds for the extended adjacency energy in terms of different graph parameters and characterize the extremal graphs attaining these bounds. We show that our bounds are better than some already known bounds in the literature.</p>

Extremal results for the extended adjacency energy

期刊论文
发表日期：

2025年12月5日

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Hilal A. Ganie,
Alaa Altassan

摘要

The sum of the absolute values of the eigenvalues of the extended adjacency matrix $ A_{ex}(\mathcal {G})$ $A_{ex} (G)$ (called the extended adjacency eigenvalues of $\mathcal {G}$ $G$ ) of a graph $\mathcal {G}$ $G$ is called the extended adjacency energy $\mathcal {E}_{ex}(\mathcal {G})$ $E_{ex} (G)$ of $\mathcal {G}$ $G$ . In this paper, we obtain some sharp upper and lower bounds for the extended adjacency energy in terms of different graph parameters and characterize the extremal graphs attaining these bounds. We show that our bounds are better than some already known bounds in the literature.