Matrices Associated with D-Distance Magic Graphs and Their Properties

Matrices associated with D-distance magic graphs are considered in the paper. Results regarding the spectral properties of these matrices have been obtained. It has been proved that if two graphs G and H of the same order have similar distance matrices AD1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {A}_{D_1} $$\end{document} and AD2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {A}_{D_2} $$\end{document} , respectively, then graph G is D1-distance magic if and only if H is a D2-distance magic graph. Graphs G and H are called magic distance-similar and their distance magic constants have been proved to coincide.