On the Extended Adjacency Eigenvalues of a Graph

On the Extended Adjacency Eigenvalues of a Graph

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> be a graph of order <i>n</i> with <i>m&...

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Bibliographic Details
Main Authors:	Alaa Altassan, Hilal A. Ganie, Yilun Shang
Format:	Article
Language:	English
Published:	MDPI AG 2024-09-01
Series:	Information
Subjects:	graphs eigenvalues spectral radius extended adjacency eigenvalues extended adjacency energy
Online Access:	https://www.mdpi.com/2078-2489/15/10/586
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Summary:	Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> be a graph of order <i>n</i> with <i>m</i> edges. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub><mo>=</mo><mi>d</mi><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> be the degree of the vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula>. The extended adjacency matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> matrix defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>d</mi><mi>i</mi></msub><msub><mi>d</mi><mi>j</mi></msub></mfrac></mstyle><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>d</mi><mi>j</mi></msub><msub><mi>d</mi><mi>i</mi></msub></mfrac></mstyle></mfenced></mrow></semantics></math></inline-formula>, whenever <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>j</mi></msub></semantics></math></inline-formula> are adjacent and equal to zero otherwise. The largest eigenvalue of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is called the extended adjacency spectral radius of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> and the sum of the absolute values of its eigenvalues is called the extended adjacency energy of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>. In this paper, we obtain some sharp upper and lower bounds for the extended adjacency spectral radius in terms of different graph parameters and characterize the extremal graphs attaining these bounds. We also obtain some new bounds for the extended adjacency energy of a graph and characterize the extremal graphs attaining these bounds. In both cases, we show our bounds are better than some already known bounds in the literature.
ISSN:	2078-2489