On Spectral Radius and Energy of a Graph with Self-Loops

The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron–Frobenius theory, it will be one among its eigenvalues. In this paper, Perron–Frobenius theory for adjacency matrix of graph with self-loops <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>A</a:mi><a:mfenced open="(" close=")" separators="|"><a:mrow><a:msub><a:mi>G</a:mi><a:mi>S</a:mi></a:msub></a:mrow></a:mfenced></a:math> will be explored. Specifically, it discusses the nontrivial existence of Perron–Frobenius eigenvalue and eigenvector pair in the matrix <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"><f:mi>A</f:mi><f:mfenced open="(" close=")" separators="|"><f:mrow><f:msub><f:mi>G</f:mi><f:mi>S</f:mi></f:msub></f:mrow></f:mfenced><f:mo>−</f:mo><f:mfrac><f:mi>σ</f:mi><f:mi>n</f:mi></f:mfrac><f:mi>I</f:mi></f:math> , where <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M3"><k:mi>σ</k:mi></k:math> denotes the number of self-loops. Also, Koolen–Moulton type bound for the energy of graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M4"><m:msub><m:mi>G</m:mi><m:mi>S</m:mi></m:msub></m:math> is explored. In addition, the existence of a graph with self-loops for every odd energy is proved.