Spectral Extrema for Graphs: The Zarankiewicz Problem
Keyword(s):
Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).
Keyword(s):
2021 ◽
Vol 2090
(1)
◽
pp. 012127
Keyword(s):
2019 ◽
Vol 11
(06)
◽
pp. 1950070
Keyword(s):
2019 ◽
Vol 19
(04)
◽
pp. 2050068
Keyword(s):
Keyword(s):
Keyword(s):