An $r$-graph is an $r$-regular graph in which every odd set of vertices is connected to its complement by at least $r$ edges. A central question regarding $r$-graphs is determining the maximum number of pairwise disjoint perfect matchings they can contain. This talk explores how edge connectivity influences this parameter. For ${0 \leq \lambda \leq r}$, let $m(\lambda,r)$ denote the maximum number $s$ such that every $\lambda$-edge-connected $r$-graph contains $s$ pairwise disjoint perfect matchings. The values of $m(\lambda,r)$ are known only in limited cases; for example, $m(3,3)=m(4,r)=1$, and $m(r,r) \leq r-2$ for all $r \not = 5$, with $m(r,r) \leq r-3$ when $r$ is a multiple of $4$. In this talk, we present new upper bounds for $m(\lambda,r)$ and examine connections between $m(5,5)$ and several well-known conjectures for cubic graphs. This is joint work with Davide Mattiolo, Eckhard Steffen, and Isaak H. Wolf.

TBA

For a positive real number $p$, the $p$-norm $\|G\|_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices, focusing on the case where $F$ is a bipartite graph. It is natural to conjecture that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $\mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions. We determine the exact value of $p_{F}$, under a mild assumption on the growth rate of $\mathrm{ex}(n,F)$. Our results extend to $r$-uniform hypergraphs as well. We also prove a general upper bound that is tight up to a $\log n$ factor for $\mathrm{ex}_{p}(n,F)$ when $p = p_{F}$. We conjecture that this $\log n$ factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles. This is a joint work with Xizhi Liu, Jie Ma and Oleg Pikhurko.

An edge-weighted graph $G$, possibly with loops, is said to be antiferromagnetic if it has nonnegative weights and at most one positive eigenvalue, counting multiplicities. The number of graph homomorphisms from a graph $H$ to an antiferromagnetic graph $G$ generalises various important parameters in graph theory, including the number of independent sets and proper vertex colourings. We obtain a number of new homomorphism inequalities for antiferromagnetic target graphs $G$. In particular, we prove that, for any antiferromagnetic $G$, $|\mathrm{Hom}(K_d, G)|^{1/d} ≤ |\mathrm{Hom}(K_{d,d} \setminus M, G)|^{1/(2d)}$ holds, where $K_{d,d} \setminus M$ denotes the complete bipartite graph $K_{d,d}$ minus a perfect matching $M$. This confirms a conjecture of Sah, Sawhney, Stoner and Zhao for complete graphs $K_d$. Our method uses the emerging theory of Lorentzian polynomials due to Brändén and Huh, which may be of independent interest. Joint work with Jaeseong Oh and Jaehyeon Seo.