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The past decade has seen growing importance being attached to the Completely Independent Spanning Trees (CISTs). The CISTs can facilitate many network functionalities, and the existence and construction schemes of CISTs in various networks can be an indicator of the network's robustness. In this paper, we establish the number of CISTs that can be constructed in the line graph of the complete graph $K_n$ K n (denoted $L(K_n)$ L (K n) , for $n\geq 4$ n ≥ 4 ), and present an algorithm to construct the optimal (i.e., maximal) number of CISTs in $L(K_n)$ L (K n) . The $L(K_n)$ L (K n) is a special class of SWCube [13], an architectural model proposed for data center networks. Our construction algorithm is also implemented to verify its validity. [ABSTRACT FROM AUTHOR]