Some new results on energy of graphs with self loops

摘要

Abstract The graph Gσdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G_sigma $$end{document} is obtained from graph G by attaching self loops on σdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$sigma $$end{document} vertices. The energy E(Gσ)documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ E(G_sigma )$$end{document} of the graph Gσdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G_sigma $$end{document} with order n and eigenvalues λ1,λ2,⋯,λndocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$lambda _1,lambda _2,dots ,lambda _n$$end{document} is defined as E(Gσ)=∑i=1nλi-σndocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ E(G_sigma )= displaystyle sum _{i=1}^nbigg lambda _i-dfrac{sigma }{n}bigg $$end{document}. It has been proved that if σ=0orndocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$sigma =0; or; n$$end{document} then E(G)=E(Gσ)documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ E(G)=E(G_sigma ) $$end{document}. The obvious question arise: Are there any graph such that E(G)=E(Gσ)documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$E(G)=E(G_sigma )$$end{document} and 0<σ