\documentclass[tikz]{standalone}

\usetikzlibrary{decorations.pathreplacing,decorations.markings}
\usetikzlibrary{patterns}

\standaloneconfig{margin=12pt}

\begin{document}

\begin{tikzpicture}

%diagonals + diagonal labels
\draw[gray,thick,dashed] (0,1) to node[midway,right,yshift=2pt,xshift=-2pt]{ $b$} (1,0);
\draw[gray,thick,dashed] (0,2) to node[midway,right,yshift=2pt,xshift=-2pt]{ $1$} (1,1);
\draw[gray,thick,dashed] (1,2) to node[midway,right,yshift=2pt,xshift=-2pt]{ $1$} (2,1);
\draw[gray,thick,dashed] (2,2) to node[midway,right,yshift=2pt,xshift=-2pt]{ $b$} (3,1);


%snake graph outline
\draw[thick] (0,0) to (1,0) to node[midway,scale=0.75,right,xshift=-1pt]{$1$} (1,1) to node[midway,above,scale=0.75,yshift=-1pt]{$a$}  (0,1) to (0,0);
\draw[thick] (0,1) to node[midway,left,scale=0.75,xshift=1pt]{$b$} (0,2) to (1,2) to node[midway,right,scale=0.75,xshift=-2pt]{$\mu$}  (1,1);
\draw[thick] (1,2) to node[midway,above,scale=0.75,yshift=-1pt]{$b$}  (2,2) to node[midway,right,scale=0.75,xshift=-1pt]{$a$}  (2,1) to (1,1);
\draw[thick] (2,2) to (3,2) to (3,1) to node[midway,below,scale=0.75,yshift=1pt]{$b$}  (2,1);


\end{tikzpicture}

\end{document}