On Strongly Regular Graphs With $M_2=Qm_3$ and $M_3=Qm_2$ Where $q\in\mathbb Q$

Mirko Lepović

We say that a regular graph $G$ of order $n$ and degree $r\geq 1$ (which is not the complete graph) is strongly regular if there exist non-negative integers $\tau$ and $\theta$ such that $|S_i\cap S_j|=\tau$ for any two adjacent vertices $i$ and $j$, and $|S_i\cap S_j|=\theta$ for any two distinct non-adjacent vertices $i$ and $j$, where $S_k$ denotes the neighborhood of the vertex $k$. Let $\lambda_1=r$, $\lambda_2$ and $\lambda_3$ be the distinct eigenvalues of a connected strongly regular graph. Let $m_1=1$, $m_2$ and $m_3$ denote the multiplicity of $r$, $\lambda_2$ and $\lambda_3$, respectively. We here describe the parameters $n$, $r$, $\tau$ and $\theta$ for strongly regular graphs with $m_2=qm_3$ and $m_3=qm_2$ for $q=\frac32,\,\frac43,\frac52,\,\frac53,\,\frac54,\,\frac65$.