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

Abstract

A nonempty $k$-regular graph $\Gamma$ 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 $\Gamma$ has precisely either $b$ or $a$ common neighbours. The quantities $n$, $k$, $b$, and $a$ are called the parameters of $\Gamma$ 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 $(n,k,b,a)$, where its quantities satisfy the conditions $k=b+1$ and $\frac{k(k-1)-a(n-1)}{b-a}>1$.

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