The IBS Discrete Mathematics Group welcomes Dr. Eero Räty, Dr. Xiaofan Yuan, and Dr. Xin Wei, new research fellows at the IBS Extremal Combinatorics and Probability Group, starting January 1, 2026.
Dr. Eero Räty received his Ph.D. from the University of Cambridge under the supervision of Prof. Imre Leader. Until recently, he was a postdoctoral researcher at Umeå University, Sweden. He is interested in extremal and probabilistic combinatorics.
Dr. Xiaofan Yuan received her Ph.D. from the Georgia Institute of Technology under the supervision of Prof. Xingxing Yu. She is interested in graph theory and extremal combinatorics. Until recently, she was a postdoctoral researcher at Arizona State University.
Dr. Xin Wei received his Ph.D. from the University of Science and Technology of China under the supervision of Prof. Xiande Zhang. His research interests include combinatorics, coding theory, and graph theory and their interactions.
On April 9, 2024, Eero Räty from Umeå University gave a talk at the Discrete Math Seminar on lower bounds for the positive discrepancy of graphs of average degree d. The title of his talk was “Positive discrepancy, MaxCut and eigenvalues of graphs.” He is currently visiting the IBS Extremal Combinatorics and Probability Group for 2 months.
The positive discrepancy of a graph $G$ of edge density $p$ is defined as the maximum of $e(U) – p|U|(|U|-1)/2$, where the maximum is taken over subsets of vertices in G. In 1993 Alon proved that if G is a $d$-regular graph on $n$ vertices and $d = O(n^{1/9})$, then the positive discrepancy of $G$ is at least $c d^{1/2}n$ for some constant $c$.
We extend this result by proving lower bounds for the positive discrepancy with average degree d when $d < (1/2 – \epsilon)n$. We prove that the same lower bound remains true when $d < n^(2/3)$, while in the ranges $n^{2/3} < d < n^{4/5}$ and $n^{4/5} < d < (1/2 – \epsilon)n$ we prove that the positive discrepancy is at least $n^2/d$ and $d^{1/4}n/log(n)$ respectively.
Our proofs are based on semidefinite programming and linear algebraic techniques. Our results are tight when $d < n^{3/4}$, thus demonstrating a change in the behaviour around $d = n^{2/3}$ when a random graph no longer minimises the positive discrepancy. As a by-product, we also present lower bounds for the second largest eigenvalue of a $d$-regular graph when $d < (1/2 – \epsilon)n$, thus extending the celebrated Alon-Boppana theorem.
This is joint work with Benjamin Sudakov and István Tomon.
