# Neighbour-transitive incidence structures in odd graphs.

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Eugenia O'Reilly-Regueiro

IMUNAM

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**Minisymposium:**
POLYTOPES AND GRAPHS

**Content:**
Given $k>0$ and a set $X$ of size $2k+1$, the odd graph $O_{k+1}$ is a $(k+1)$-regular graph whose vertices are the $k$-sets of $X$ and any two are adjacent if and only if they are disjoint. For a subset $\Gamma$ of $V(O_{k+1})$, there is an incidence structure $D$ between the elements of $X$ and those of $\Gamma$. If we define $\Gamma_1$ as the set of neighbours of $\Gamma$ in $D$, (that is, $\Gamma_1$ is the set of vertices of $O_{k+1}$ that are not in $\Gamma$ and are adjacent in the graph to at least one element of $\Gamma$), then a group of automorphisms of $D$ is neighbour-transitive if it is transitive on $\Gamma_1$. In this talk we will present some results and examples of these incidence structures, which are part of ongoing joint work with Brian Corr, Wei Jin, Cheryl Praeger, and Csaba Schneider.