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\title{The \koni Bridge Problem and the Friendship Theorem}
\author{Wenda Li}
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\begin{abstract}
  This development provides a formalization of undirected graphs and simple graphs, which are based on Benedikt Nordhoff and  Peter Lammich's simple formalization of labelled directed graphs \cite{website:isabelle_archive_shortest_path} in the archive. Then, with our formalization of graphs, we have shown both necessary and sufficient conditions for Eulerian trails and circuits \cite{koni_proof_lecture_notes} as well as the fact that the \koni Bridge problem does not have a solution. In addition, we have also shown the Friendship Theorem in simple graphs\cite{friend_combitorial,mertzios2008friendship}.  
\end{abstract}

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