On strictly Deza graphs with parameters (n,k,k-1,a)

A nonempty $k$-regular graph $\mathrm{\Gamma }$ on $n$ vertices is called a Deza graph if there exist constants $b$ and $a$ $(b\ge a)$ such that any pair of distinct vertices of $\mathrm{\Gamma }$ has precisely either $b$ or $a$ common neighbours. The quantities $n$, $k$, $b$, and $a$ are called the parameters of $\mathrm{\Gamma }$ and are written as the quadruple $(n,k,b,a)$. If a Deza graph has diameter 2 and is not strongly regular, then it is called a strictly Deza graph. In the paper we investigate strictly Deza graphs with parameters $ (n, k, b, a) $,whereitsquantitiessatisfytheconditions$k = b + 1$and$\frac{k(k - 1) - a(n - 1)}{b - a} > 1$.