\documentclass[12pt]{article} \usepackage{amsmath,mathrsfs,bbm} \usepackage{amssymb} \textwidth=4.825in \overfullrule=0pt \thispagestyle{empty} \begin{document} \noindent % % {\bf David Perkinson, Nick Salter and Tianyuan Xu} % % \medskip \noindent % % {\bf A Note on the Critical Group of a Line Graph} % % \vskip 5mm \noindent % % % % This note answers a question posed by Levine. The main result shows that under certain circumstances a critical group of a directed graph is the quotient of a critical group of its directed line graph. \end{document} .