\documentclass[12pt]{article}
\usepackage{amsmath,mathrsfs,bbm}
\usepackage{amssymb}
\textwidth=4.825in
\overfullrule=0pt
\thispagestyle{empty}

\begin{document}
\noindent
%
%
{\bf Steve Butler and Jason Grout }
%
%

\medskip
\noindent
%
%
{\bf A Construction of Cospectral Graphs for the Normalized Laplacian}
%
%

\vskip 5mm
\noindent
%
%
%
%
We give a method to construct cospectral graphs for the normalized Laplacian by a local modification in some graphs with special structure.  Namely, under some simple assumptions, we can replace a small bipartite graph with a cospectral mate without changing the spectrum of the entire graph.  We also consider a related result for swapping out biregular bipartite graphs for the matrix $A+tD$.

We produce (exponentially) large families of non-bipartite, non-regular graphs which are mutually cospectral, and also give an example of a graph which is cospectral with its complement but is not self-complementary.

\end{document}

.