Non-existence of two infinite families of strongly regular graphs

Abstract

For a positive integer $t$, a putative strongly regular graph $G$ with parameters $(n,k,\lambda ,\mu )=(1+k+\frac{k(k-1-\lambda )}{\mu },2t(4t+1)\mu ,(2t+1)(32t^3+4t-1),(2t+1)(8t^2+1))$ satisfies both the Krein condition and the absolute bound. Also the multiplicities of the eigenvalues of the graph $G$ are integers. This may mean that such a strongly regular graph exists. However, Koolen and Gebremichel proved that such a strongly regular graph does not exist for $t=1$. In this paper, we generalize their method for all $t\ge 1$ and rule out the infinite family of such strongly regular graphs. In order to do so, we find a restriction on the orders of two large maximal cliques intersecting in many vertices. And we also look at the case where the equality of the claw-bound holds to find an upper bound on the order of a coclique in a local graph (when $G$ is not Terwilliger). In a similar fashion, we note that one can also rule out another infinite family of putative strongly regular graphs with parameters $(n,k,\lambda ,\mu )=(1+k+\frac{k(k-1-\lambda )}{\mu },(2t+1)(4t+3)\mu ,(2t+2)(32t^3+64t^2+44t+9),(2t+2)(8t^2+12t+5))$. With the generalized method we are able to rule out two infinite families of putative strongly regular graphs. We are sure that this generalized method can be applied to rule out more putative strongly regular graphs.