On Schatten p-norm of the distance matrices of graphs

摘要

Abstract For a connected simple graph G, the generalized distance matrix is defined by Dα:=αTr(G)+(1-α)D(G),0≤α≤1documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ D_{alpha }:= alpha Tr(G)+(1-alpha ) D(G), ~0le alpha le 1 $$end{document}, where Tr(G) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix. For particular values of αdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ alpha $$end{document}, we obtain the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix and other uncountable distance based matrices. Let ∂1≥∂2≥⋯≥∂ndocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ partial _{1}ge partial _{2}ge dots ge partial _{n} $$end{document} be the Dαdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ D_{alpha } $$end{document} eigenvalues of G and p≥2documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ pge 2 $$end{document} a real number, the Schatten p-norm is the p-th root of the sum of p-th powers of eigenvalues of Dα,α∈12,1documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ D_{alpha }, ~alpha in frac{1}{2},1 $$end{document}, that is, ‖Dα‖pp=∂1p+∂2p+⋯+∂npdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ Vert D_{alpha }Vert _{p}^{p} =partial _{1}^{p}+partial _{2}^{p}+dots +partial _{n}^{p}$$end{document}. In this paper, we obtain various bounds for ‖Dα‖ppdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ Vert D_{alpha }Vert _{p}^{p} $$end{document} in terms of different graph parameters and characterize the corresponding extremal graphs.