On Deza graphs with parameters of lattice graphs

A Deza graph with parameters $(v,k,b,a)$, where $b\ge a$, is a $k$-regular graph on $v$ vertices in which any two vertices have either $a$ or $b$ common neighbors. A strongly regular graph with parameters $(v,k,\lambda,\mu)$ is a $k$-regular graph on $v$ vertices in which any two adjacent vertices have exactly $\lambda$ common neighbors and any two nonadjacent vertices have $\mu$ common neighbors. An strictly Deza graph is a Deza graph of diameter 2 that is not strongly regular. If a strongly regular graph has an involutive automorphism that transposes nonadjacent vertices only, then it is known that this automorphism can be used to obtain a Deza graph with the parameters of the initial strongly regular graph. We find all the automorphisms of strongly regular lattice $n\times n$ graphs with $n\ge3$ that satisfy the above condition. It turns out that there is exactly one such automorphism for odd $n$ and exactly two automorphisms for even $n$. Neighborhoods of exact Deza graphs obtained by means of this automorphism are found and a characterization of such strictly Deza graph with respect to its parameters and the structure of neighborhoods is obtained.