The Spectral Radius of the Reciprocal Distance Laplacian Matrix of a Graph

In this article, we introduce a Laplacian for the reciprocal distance matrix of a connected graph, called the reciprocal distance Laplacian. Let δ1≥δ2≥⋯≥δn-1≥δn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _1\ge \delta _2\ge \cdots \ge \delta _{n-1}\ge \delta _n$$\end{document} be the reciprocal distance Laplacian spectrum. In this short note, we show that δ1≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _1\le n$$\end{document} with equality if and only if the complement graph G¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{G}$$\end{document} of G is disconnected.