We give a construction of LDPC codes from Deza graphs with parameters $(v,k,1,0)$ . The Tanner graphs of these LDPC codes do not contain cycles of length 4. Special attention is given to the construction of LDPC codes from the Moore graphs with diameter 2. We identify and enumerate the smallest absorbing sets in the Tanner graphs of the obtained LDPC codes and analyze their structures. Furthermore, we describe a construction, based on the protograph operation, of an infinite family of LDPC codes, whose Tanner graphs have girth at least 6, obtained from each $(v,k,1,0)$ Deza graph. We give an expression for the variance of a syndrome weight of the constructed LDPC codes, and also present simulation results.
