Maximum spread of graphs and bipartite graphs
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- by Jane Breen, Alex W. N. Riasanovsky, Michael Tait and John Urschel
- Comm. Amer. Math. Soc. 2 (2022), 417-480
- DOI: https://doi.org/10.1090/cams/14
- Published electronically: December 22, 2022
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Abstract:
Given any graph $G$, the spread of $G$ is the maximum difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers $n$, the $n$-vertex graph $G$ that maximizes spread is the join of a clique and an independent set, with $\lfloor 2n/3 \rfloor$ and $\lceil n/3 \rceil$ vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all $n$ sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over $\mathscr {L}^2[0,1]$. The second conjecture claims that for any fixed $m \leq n^2/4$, if $G$ maximizes spread over all $n$-vertex graphs with $m$ edges, then $G$ is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we construct an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower-order error terms.References
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Bibliographic Information
- Jane Breen
- Affiliation: Faculty of Science, Ontario Tech University, 2000 Simcoe St N, Oshawa, Ontario L1G 0C5 Canada
- MR Author ID: 1121809
- ORCID: 0000-0002-6048-9368
- Email: jane.breen@ontariotechu.ca
- Alex W. N. Riasanovsky
- Affiliation: Department of Mathematics, Iowa State University, 411 Morrill Road, Ames, Iowa 50011
- MR Author ID: 1009037
- Email: a.riasanovsky@gmail.com
- Michael Tait
- Affiliation: Department of Mathematics & Statistics, SAC Room 305, Villanova University, 800 Lancaster Avenue, Villanova, Pennsylvania 19085
- MR Author ID: 929797
- ORCID: 0000-0003-3695-6883
- Email: michael.tait@villanova.edu
- John Urschel
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- MR Author ID: 1017439
- Email: urschel@mit.edu
- Received by editor(s): October 25, 2021
- Received by editor(s) in revised form: May 20, 2022
- Published electronically: December 22, 2022
- Additional Notes: The work of the first author was supported in part by NSERC Discovery Grant RGPIN-2021-03775. The work of the second author was supported in part by NSF award DMS-1839918 (RTG). The work of the third author was supported in part by NSF award DMS-2011553. The work of the fourth author was supported in part by ONR Research Contract N00014-17-1-2177. This work was supported by the Institute for Advanced Study and the National Science Foundation under Grant No. DMS-1926686.
- © Copyright 2022 by the authors under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)
- Journal: Comm. Amer. Math. Soc. 2 (2022), 417-480
- MSC (2020): Primary 05C50, 15A18, 15A60
- DOI: https://doi.org/10.1090/cams/14
- MathSciNet review: 4525711