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A Deza graph with parameters    (   n  ,  k  ,  b  ,  a   )    is a   k  ‐regular graph with   n   vertices, in which any two vertices have   a   or   b   (  a  ≤  b  ) common neighbours. A Deza graph is strictly Deza if it has diameter   2  , and is not strongly regular. In an earlier paper, the two last authors et al characterised the strictly Deza graphs with   b  =  k  −  1   and   β  >  1  , where   β   is the number of vertices with   b   common neighbours with a given vertex. Here, we start with a characterisation of Deza graphs (not necessarily strictly Deza graphs) with parameters    (   n  ,  k  ,  k  −  1  ,  0   )   . Then, we deal with the case   β  =  1   and   a  >  0  , and thus complete the characterisation of Deza graphs with   b  =  k  −  1  . It follows that all Deza graphs with   b  =  k  −  1  ,   β  =  1   and   a  >  0   can be made from special strongly regular graphs, and in fact are strictly Deza except for    K  2   . We present several examples of such strongly regular graphs. A divisible design graph (DDG) is a special Deza graph, and a Deza graph with   β  =  1   is a DDG. The present characterisation reveals an error in a paper on DDGs by the second author et al. We discuss the cause and the consequences of this mistake and give the required errata.

Deza graphs with parameters ( n , k , k − 1 , a ) and β = 1