A New Feasibility Condition for the AT4 Family

Let $\Gamma$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $\theta_0>\theta_1>\theta_2>\theta_3>\theta_4$. Then $\Gamma$ is tight in the sense of Juri\v{s}i\'{c}, Koolen, and Terwilliger [12] whenever $\Gamma$ is locally strongly regular with nontrivial eigenvalues $p:=\theta_2$ and $-q:=\theta_3$. Assume that $\Gamma$ is tight. Then the intersection numbers of $\Gamma$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $\Gamma$. We denote $\Gamma$ by $\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph. Using this condition, we prove that there does not exist $\mathrm{AT4}(q^3-2q,q,2)$ for $q\equiv3$ $(\mathrm{mod}~4)$. We discuss the $\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$.