Group photo will be added here. We would love to see you on it!
Venue: College of Science Room SC4009-1, National Sun Yat-sen University Venue: [ trasportation campus map ]
Announcements:
The conference's primary objective is to create a conducive environment where researchers can exchange findings, insights, and challenges within the fields of graph theory, combinatorics, and their practical applications. It encompasses a comprehensive range of topics in discrete mathematics and their real-world implementations, with a particular focus on the intersection with artificial intelligence. This conference represents a continuation of previous gatherings at NSYSU, paying tribute to the individuals who have made substantial contributions to our shared endeavors.
Each talk is 40 minutes for presentation and 10 minuntes for questions.
Assume $G$ is a bridgeless graph. A cycle cover of $G$ is a collection of cycles of $G$ such that each edge of $G$ is contained in at least one of the cycles. The length of a cycle cover of $G$ is the sum of the lengths of the cycles in the cover. The minimum length of a cycle cover of $G$ is denoted by $cc(G)$. It was proved independently by Alon and Tarsi and by Bermond, Jackson, and Jaeger that $cc(G)\le \frac{5}{3}m$ for every bridgeless graph $G$ with $m$ edges. This remained the best-known upper bound for $cc(G)$ for 40 years. In this paper, we prove that if $G$ is a bridgeless graph with $m$ edges and $n_2$ vertices of degree $2$, then $cc(G) < \frac{29}{18}m+ \frac 1{18}n_2$. As a consequence, we show that $cc(G) \le \frac 53 m - \frac 1{42} \log m$. The upper bound $ cc(G) < \frac{29}{18}m \approx 1.6111 m$ for bridgeless graphs $G$ of minimum degree at least 3 improves the previous known upper bound $1.6258m$. A key lemma used in the proof confirms Fan's conjecture that if $C$ is a circuit of $G$ and $G/C$ admits a nowhere zero 4-flow, then $G$ admits a 4-flow $f$ such that $E(G)-E(C)\subseteq \text{supp} (f)$ and $|\textrm{supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|$. This is a joint work with Deping Song.
A conflict-avoiding code (CAC) $\mathcal{C}$ of length $L$ with weight $w$ is a collection of $w$-subsets of $\mathbb{Z}_L$ such that $d^*(S_1)\cap d^*(S_2)=\emptyset$ for any two distinct $w$-subsets $S_1, S_2\in\mathcal{C}$, where $d^*(S)=\{a-b\ (\bmod\, L):\,a,b\in S, a\neq b\}$. A CAC is a deterministic transmission scheme for asynchronous multiple-access without feedback. When the number of simultaneously active users is less than or equal to $w$, a CAC of length $L$ with weight $w$ can provide a hard guarantee that each active user has at least one successful transmission within every consecutive $L$ time slots. The design goal of CACs is to determine the maximum code size, denoted by $K(L,w)$, for given $L$ and $w$. A CAC is called optimal if its code size achieves the value $K(L,w)$. In this talk, we will provide a series of optimal CACs by the help of Kneser’s Theorem and some other techniques in Additive Combinatorics. Mixed-weight CACs and multichannel CACs, two class of natural generalization of CACs, will be discussed as well.
The spectral radius of a square matrix is the maximum of the magnitude of its eigenvalues. The spectral radius of a graph, which is the spectral radius of its adjacency matrix, is related to many properties of the graph and is an important and extensively studied topic in spectral graph theory. This talk will introduce the extremal values of spectral radius under various constraints, and the tools used to address these problems.
To Be Announced
Neural networks, a cornerstone of machine learning, have profoundly impacted the modern world. Their foundation lies in a blend of mathematical disciplines, including matrix theory, gradient descent optimization, matrix derivatives, randomized algorithms, and more. This harmonious integration of fields has contributed to the remarkable success of neural networks in various applications. In this talk, we will explore the details of a neural network prototype and demonstrate how to construct one from scratch.
There is no default arrangement for dinner. You are welcome to stay, and we may have an informal gathering at our own cost.