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

Abstract

Let $\Gamma$ denote a bipartite graph with vertex set X and color partitions Y, $Y'$. For a nonnegative integer i and a vertex $x\in X$, let $\Gamma _{i}(x)$ denote the set of vertices in X that are at distance i from x. For $x\in Y$, $y \in \Gamma _2(x)$ and $z \in \Gamma _{i}(x)\cap \Gamma _i(y)$, let $\gamma _i(x,y,z)$ denote the number of common neighbors of x and y which are at distance $i-1$ from z (i.e., let $\gamma _i(x,y,z):=|\Gamma _1(x)\cap \Gamma _{1}(y)\cap \Gamma _{i-1}(z)|$). For the moment assume that every vertex in Y has eccentricity $D\ge 3$. Graph $\Gamma$ is almost 2-Y-homogeneous whenever for all $i \; (1\le i \le D-2)$ and for all $x\in Y$, $y \in \Gamma _2(x)$ and $z \in \Gamma _{i}(x)\cap \Gamma _i(y)$, the number $\gamma _i(x,y,z)$ 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. In this paper, we study the combinatorial structure of distance-biregular graphs. We give sufficient and necessary conditions under which a distance-biregular graph is (almost) 2-Y-homogeneous. Moreover, for a 2-Y-homogeneous distance-biregular graph we write the intersection numbers of the color class Y in terms of three parameters.