A New Feasibility Condition for the AT4 Family

Abstract

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 Jurišić, 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 $AT4(p,q,r)$ 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 $AT4(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $AT4(p,q,2)$ to be an antipodal tight graph. Using this condition, we prove that there does not exist $AT4(q^3-2q,q,2)$ for $q\equiv 3$ $(\mathrm{mod}4)$. We discuss the $AT4(p,q,r)$ graphs with $r=(p+q^3)(p+q{)}^{-1}$.

15 pages, 1 table

References in corpus (1)

• Remarks on pseudo-vertex-transitive graphs with small diameter