Drawing simple edge, no arrow

There is no arrow with edge by default, but you have turned on arrows for all lines in the tikzpicture with the -> argument:

\begin{tikzpicture}[->,auto,node distance=3cm,
   thick,main node/.style={draw,font=\bfseries}]

Remove -> and there are no arrows.

Or as commented by Jake, use - as an argument to the \path (or edge) to remove the arrows locally.

\path[-]
(111) edge (112)
    edge (122)
    edge (1n2)
(121) edge (112)
    edge (122)
    edge (1n2)
(1n1) edge (112)
    edge (122)
    edge (1n2)
(121) edge[dashed] (1n1);

You could also put the \path drawing the lines in a scope environment and use the same argument. Could be convenient if you had several \path/\draw commands for which you wanted to apply this.

\begin{scope}[-]
  \path
  (111) edge (112)
      edge (122)
      edge (1n2)
    (121) edge (112)
      edge (122)
      edge (1n2)
  (1n1) edge (112)
      edge (122)
       edge (1n2)
  (121) edge[dashed] (1n1);
\end{scope}

It's also beneficial to use matrix nodes to reduce the typing labor. I used to a small Excel file to strip down the node names and to populate the matrix.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}
\matrix[matrix of math nodes, nodes={draw,font=\bfseries},row sep ={1cm,between origins},column sep={2cm,between origins}] (m) {
X_{n,n}^{(1)} &[0.5cm] X_{n,n}^{(2)} &[2cm] X_{n,n}^{(k)} &[1cm] X_{n,n}^{(n-2)} &[0.5cm] X_{n,n}^{(n-1)}\\[1cm]
X_{n,2}^{(1)} &        X_{n,2}^{(2)} &      X_{n,2}^{(k)} &      X_{n,2}^{(n-2)} &        X_{n,2}^{(n-1)}\\
X_{n,1}^{(1)} &        X_{n,1}^{(2)} &      X_{n,1}^{(k)} &      X_{n,1}^{(n-2)} &        X_{n,1}^{(n-1)}\\[1cm]
X_{2,n}^{(1)} &        X_{2,n}^{(2)} &      X_{2,n}^{(k)} &      X_{2,n}^{(n-2)} &        X_{2,n}^{(n-1)}\\[1cm]
X_{2,2}^{(1)} &        X_{2,2}^{(2)} &      X_{2,2}^{(k)} &      X_{2,2}^{(n-2)} &        X_{2,2}^{(n-1)}\\
X_{2,1}^{(1)} &        X_{2,1}^{(2)} &      X_{2,1}^{(k)} &      X_{2,1}^{(n-2)} &        X_{2,1}^{(n-1)}\\[1cm]
X_{1,n}^{(1)} &        X_{1,n}^{(2)} &      X_{1,n}^{(k)} &      X_{1,n}^{(n-2)} &        X_{1,n}^{(n-1)}\\[1cm]
X_{1,2}^{(1)} &        X_{1,2}^{(2)} &      X_{1,2}^{(k)} &      X_{1,2}^{(n-2)} &        X_{1,2}^{(n-1)}\\
X_{1,1}^{(1)} &        X_{1,1}^{(2)} &      X_{1,1}^{(k)} &      X_{1,1}^{(n-2)} &        X_{1,1}^{(n-1)}\\
};
  \path[thick]
    (m-7-1) edge (m-7-2)
        edge (m-8-2)
        edge (m-9-2)
    (m-8-1) edge (m-7-2)
        edge (m-8-2)
        edge (m-9-2)
    (m-9-1) edge (m-7-2)
        edge (m-8-2)
        edge (m-9-2)
    (m-7-1) edge[-,dashed] (m-8-1);
\end{tikzpicture}
\end{document}