Document Type
Lecture
Publication Date
4-24-2024
Abstract
Tur´an numbers are a cornerstone of extremal graph theory. Their asymptotics are completely known when forbidding graphs with chromatic number at least three, however they remain unknown for several basic bipartite graphs. Nikiforov introduced a spectral analogue to Tur´an problems referred to as spectral Tur´ an problems. Here our objective is to maximize the spectral radius of the adjacency matrices of graphs not containing some subgraphs. Such a study may give strong upper bounds for the associated Tur´ an problems. While the asymptotics of spectral Tur´ an numbers are known for graphs with chromatic number at least three, several families of bipartite graphs remain open in this scenario too. In this talk we will share recent progress on the spectral Tur´an numbers for some families of bipartite graphs. In some cases, this will strengthen the previous upper bounds for the associated Tur´ an numbers. Conversely, a recent result for forbidden graphs with chromatic number at least three, proves that whenever the extremal graph for a Tur´an problem consists of a Tur´ an graph + finitely many edges added into it, the spectral extremal graph is also edge extremal. In this talk we will share our recent explorations where we show similar phenomena also occurring with bipartite Tur´an problems, where the role of the Tur´an graph is now replaced by a small clique joined to an independent set.
Relational Format
presentation
Recommended Citation
Desai, Dheer Noal, "A general theorem in spectral extremal graph theory" (2024). Combinatorics Seminar. 8.
https://egrove.olemiss.edu/math_combinatorics/8
