On Strongly Regular Graphs With $m_2 = qm_3$ and $m_3 = qm_2$ for $q=\frac{7}{2}, \frac{7}{3}, \frac{7}{4}, \frac{7}{5}, \frac{7}{6}$


Mirko Lepović




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 $\lvert S_i \cap S_j \lvert = \tau$ for any two adjacent vertices $i$ and $j$, and $\lvert S_i \cap S_j \lvert = \theta$ 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 $r$, $\lambda_1$ 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 $q=\frac{7}{2}, \frac{7}{3}, \frac{7}{4}, \frac{7}{5}, \frac{7}{6}$