Geodesic
Spectral extrema for graphs: the Zarankiewicz problem
The electronic journal of combinatorics, Tome 16 (2009) no. 1
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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).
DOI : 10.37236/212
Classification : 05C50, 05C35
Mots-clés : spectral radius, Kövari Turan Sos upper bound, Zarankiewicz problem 
@article{10_37236_212,
     author = {L\'aszl\'o Babai and Barry Guiduli},
     title = {Spectral extrema for graphs: the {Zarankiewicz} problem},
     journal = {The electronic journal of combinatorics},
     year = {2009},
     volume = {16},
     number = {1},
     doi = {10.37236/212},
     zbl = {1186.05079},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/212/}
}
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László Babai; Barry Guiduli. Spectral extrema for graphs: the Zarankiewicz problem. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/212

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