Title: Distance-regular graphs that support a uniform structure
| Speaker: | Roghayeh Maleki |
| Affiliation: | University of Primorska |
| Location: | Please contact Sabrina Lato for Zoom link. |
Abstract: Given a connected bipartite graph $G$, the adjacency matrix $A$ of $G$ can be decomposed as $A=L+R$, where $L=L(x)$ and $R=R(x)$ are respectively the lowering and the raising matrices with respect to a certain vertex $x$. The graph $G$ has a \textit{uniform structure} with respect to $x$ if the matrices $RL^2$, $LRL$, $L^2R$, and $L$ satisfy a certain linear dependency.
Let $\Gamma=(X,E)$ be a connected non-bipartite graph. Fix a vertex $x\in X$ and let $\Gamma_f=(X,E_f)$ be the bipartite graph, where $E_f=E \setminus \{yz \mid \partial(x,y) = \partial(x,z)\}$ and $\partial$ is the distance function in $\Gamma$. The graph $\Gamma$ is said to support a uniform structure whenever $\Gamma_f$ has a uniform structure with respect to $x$.
In this talk, I will present some classification results of non-bipartite distance-regular graphs with classical parameters $(D,q,\alpha,\beta)$, that support a uniform structure.
Joint work with: Blas Fernández, Štefko Miklavič, and Giusy Monzillo.