Abstract
We discuss how graph expansion is related to the behavior of Lp-functions on the covering tree. We show that the non-trivial eigenvalues of the adjacency operator on a (q + 1)-regular graph are bounded by q1/p + q(p-1)/p-the Lp-norm of the operator on the covering tree—if and only if properly averaged lifts of functions from the graph to the tree lie in Lp+ε for every ε > 0. We generalize the result to operators on edges and to bipartite graphs.
The work is based on a combinatorial interpretation of representationtheoretic ideas.
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Acknowledgments
This work was done as part of an M.Sc. thesis submitted to the Hebrew University of Jerusalem. The author would like to thank his adviser Prof. Alex Lubotzky for his guidance, support and patience regarding the author’s English, and the referee for multiple corrections and improvements to the text.
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Kamber, A. Lp-Expander Graphs. Isr. J. Math. 234, 863–905 (2019). https://doi.org/10.1007/s11856-019-1938-7
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DOI: https://doi.org/10.1007/s11856-019-1938-7