Talks &
abstracts.

A well known problem from an excellent book of Lovász states that any hypergraph with the property that no pair of hyperedges intersect in exactly one vertex can be properly $2$-colored. Motivated by this as well as recent works of Keszegh and of Gyárfás et al, we study the $1$-intersection graph of a hypergraph. The $1$-intersection graph encodes those pairs of hyperedges in a hypergraph that intersect in exactly one vertex. We prove for $k\in\{2,4\}$ that all hypergraphs whose $1$-intersection graph is $k$-partite can be properly $k$-colored, and on the other hand give a construction for every odd $k$ such that the hypergraph cannot be properly colored with $k$ colors although the $1$-intersection graph is $k$-colorable.

A $3\times3\times3$ tensor over a finite field $\mathbb{F}_q$ can be viewed projectively as a point of $\mathrm{PG}(26,q)$, where the rank-one tensors form the Segre variety. Contraction spaces of $3\times3\times3$ tensors correspond to points, lines and planes contained in $\mathrm{PG}(\mathbb{F}_q^3\otimes\mathbb{F}_q^3)$, and their orbits under the natural group action have strong links with finite geometry and semifield theory. About a decade ago, lines in this space were classified by Lavrauw and Sheekey in [3] using a geometric approach: there are fourteen orbits $o_4,\dots,o_{17}$ of lines in $\mathrm{PG}(8,q)$. Lavrauw and Popiel in [2] determined which orbits admit representatives consisting of symmetric matrices (symmetric representatives): exactly three of the fourteen orbits of lines, namely $o_4$, $o_7$ and $o_{11}$, admit no symmetric representative. More recently, substantial progress has been made on the classification of (partially) symmetric $3\times3\times3$ tensors, which correspond to planes in the space of the Veronese surface [1], where the geometry is more rigid and additional structure is available.

In this talk, we explain the connection between $3\times3\times3$ tensors and contraction planes and discuss the splitting and merging behaviour of orbits when passing from the symmetric setting to the full matrix space. Let $q$ be odd, and let $G$ be the stabiliser of the Segre variety $\mathcal{S}_{2,2}$ in $\mathrm{PGL}(9,q)$. We show that the symmetric planes meeting the Veronese surface give rise to fourteen $G$-orbits of planes meeting $\mathcal{S}_{2,2}$.

We then consider $G$-orbits of planes admitting no symmetric representative. In particular, we classify the planes meeting $\mathcal{S}_{2,2}$ and containing a line from one of the orbits $o_4$, $o_7$, or $o_{11}$, obtaining twenty-five further $G$-orbits. We provide matrix representatives and rank distributions, describe the relevant line stabilisers, and resolve all orbit identifications. The main part of the argument concerns planes through an $o_7$-line, where the geometry of the associated hyperbolic quadrics determines a geometric configuration that we use to describe the action of the line stabiliser and classify the planes. Finally, we discuss the remaining case of planes admitting no symmetric representative although all their lines do, and the problem of completing the classification of planes meeting the Segre variety.

Distance-regular graphs arise naturally in finite geometry and algebraic combinatorics. Among them, $Q$-polynomial distance-regular graphs possess an additional algebraic structure encoded by a so-called dual adjacency matrix. To what extent can this structure be recovered without distance-regularity?

In this talk, we explore a generalization of the $Q$-polynomial property to graphs with a distinguished vertex. One approach to finding such graphs begins with a dual adjacency matrix candidate: a diagonal matrix, constant on distance layers, that satisfies a tridiagonal relation with the adjacency matrix. This relation provides a starting point for establishing the desired action on the eigenspaces of the adjacency matrix (as required by the definition of the generalized $Q$-polynomial property).

Our main focus is a characterization, under suitable parameter conditions, of dual adjacency matrix candidates for bipartite graphs in terms of weakly uniform structures. These structures express linear dependencies among counts of walks of length three relative to the distance partition, providing a combinatorial interpretation of the matrix relation.

We illustrate this connection with bipartite graphs obtained from Hamming and dual polar graphs by deleting edges within distance layers. For these families, the candidates yield dual adjacency matrices and establish the generalized $Q$-polynomial property.

The talk is based on joint work with B. Fernández, R. Maleki, and Š. Miklavič.