On Strongly Regular Graphs With $m_2 = qm_3$ AND $m_3 = qm_2$ for $q = 5,6,7,8$

Abstract

We say that a regular graph $G$ of order $n$ and degree $r\ge 1$ (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 describe the parameters $n$, $r$, $\tau$ and $\theta$ for strongly regular graphs with $m_2 = qm_3$ and $m_3 = qm_2$ for $q = 5,6,7,8$.