On (almost) $2$-$Y$-homogeneous distance-biregular graphs

Abstract

Let $\Gamma$ denote a bipartite graph with vertex set $X$, color partitions $Y$, $Y'$, and assume that every vertex in $Y$ has eccentricity $D\ge 3$. For $z\in X$ and a non-negative integer $i$, let $\Gamma_{i}(z)$ denote the set of vertices in $X$ that are at distance $i$ from $z$. Graph $\Gamma$ is almost $2$-$Y$-homogeneous whenever for all $i \; (1\leq i \leq D-2)$ and for all $x\in Y$, $y \in \Gamma_2(x)$ and $z \in \Gamma_{i}(x)\cap\Gamma_i(y)$, the number of common neighbours of $x$ and $y$ which are at distance $i-1$ from $z$ is independent of the choice of $x$, $y$ and $z$. In addition, if the above condition holds also for $i=D-1$, then we say that $\Gamma$ is $2$-$Y$-homogeneous. Now, let $\Gamma$ denote a distance-biregular graph. In this paper we study the intersection arrays of $\Gamma$ and we give sufficient and necessary conditions under which $\Gamma$ is (almost) $2$-$Y$-homogeneous. In the case when $\Gamma$ is $2$-$Y$-homogeneous we write the intersection numbers of the color class $Y$ in terms of three parameters.