The spectrum and toughness of regular graphs

In 1995, Brouwer proved that the toughness of a connected $k$-regular graph $G$ is at least $k/\lambda-2$, where $\lambda$ is the maximum absolute value of the non-trivial eigenvalues of $G$. Brouwer conjectured that one can improve this lower bound to $k/\lambda-1$ and that many graphs (especially graphs attaining equality in the Hoffman ratio bound for the independence number) have toughness equal to $k/\lambda$. In this paper, we improve Brouwer's spectral bound when the toughness is small and we determine the exact value of the toughness for many strongly regular graphs attaining equality in the Hoffman ratio bound such as Lattice graphs, Triangular graphs, complements of Triangular graphs and complements of point-graphs of generalized quadrangles. For all these graphs with the exception of the Petersen graph, we confirm Brouwer's intuition by showing that the toughness equals $k/(-\lambda_{min})$, where $\lambda_{min}$ is the smallest eigenvalue of the adjacency matrix of the graph.Comment: 15 pages, 1 figure, accepted to Discrete Applied Mathematics, special issue dedicated to the "Applications of Graph Spectra in Computer Science" Conference, Centre de Recerca Matematica (CRM), Bellaterra, Barcelona, June 16-20, 201

Edge-disjoint spanning trees and eigenvalues of regular graphs

Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of $k$ edge-disjoint spanning trees in a regular graph, when $k\in \{2,3\}$. More precisely, we show that if the second largest eigenvalue of a $d$-regular graph $G$ is less than $d-\frac{2k-1}{d+1}$, then $G$ contains at least $k$ edge-disjoint spanning trees, when $k\in \{2,3\}$. We construct examples of graphs that show our bounds are essentially best possible. We conjecture that the above statement is true for any $k<d/2$.Comment: 4 figure

Spanning trees, toughness, and eigenvalues of regular graphs

Spectral graph theory is a branch of graph theory which finds relationships between structural properties of graphs and eigenvalues of matrices corresponding to graphs. In this thesis, I obtain sufficient eigenvalue conditions for the existence of edge-disjoint spanning trees in regular graphs, and I show this is best possible. The vertex toughness of a graph is defined as the minimum value of [special characters omitted], where S runs through all subsets of vertices that disconnect the graph, and c(G\S ) denotes the number of components after deleting S. I obtain sufficient eigenvalue conditions for a regular graph to have toughness at least 1, and I show this is best possible. Furthermore, I determine the toughness value for many families of graphs, and I classify the subsets S of each family for when this value is obtained

Science goals and mission architecture of the Europa Lander mission concept

© The Author(s), 2022. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Hand, K., Phillips, C., Murray, A., Garvin, J., Maize, E., Gibbs, R., Reeves, G., San Martin, A., Tan-Wang, G., Krajewski, J., Hurst, K., Crum, R., Kennedy, B., McElrath, T., Gallon, J., Sabahi, D., Thurman, S., Goldstein, B., Estabrook, P., Lee, S. W., Dooley, J. A., Brinckerhoff, W. B., Edgett, K. S., German, C. R., Hoehler, T. M., Hörst, S. M., Lunine, J. I., Paranicas, C., Nealson, K., Smith, D. E., Templeton, A. S., Russell, M. J., Schmidt, B., Christner, B., Ehlmann, B., Hayes, A., Rhoden, A., Willis, P., Yingst, R. A., Craft, K., Cameron, M. E., Nordheim, T., Pitesky, J., Scully, J., Hofgartner, J., Sell, S. W., Barltrop, K. J., Izraelevitz, J., Brandon, E. J., Seong, J., Jones, J.-P., Pasalic, J., Billings, K. J., Ruiz, J. P., Bugga, R. V., Graham, D., Arenas, L. A., Takeyama, D., Drummond, M., Aghazarian, H., Andersen, A. J., Andersen, K. B., Anderson, E. W., Babuscia, A., Backes, P. G., Bailey, E. S., Balentine, D., Ballard, C. G., Berisford, D. F., Bhandari, P., Blackwood, K., Bolotin, G. S., Bovre, E. A., Bowkett, J., Boykins, K. T., Bramble, M. S., Brice, T. M., Briggs, P., Brinkman, A. P., Brooks, S. M., Buffington, B. B., Burns, B., Cable, M. L., Campagnola, S., Cangahuala, L. A., Carr, G. A., Casani, J. R., Chahat, N. E., Chamberlain-Simon, B. K., Cheng, Y., Chien, S. A., Cook, B. T., Cooper, M., DiNicola, M., Clement, B., Dean, Z., Cullimore, E. A., Curtis, A. G., Croix, J-P. de la, Pasquale, P. Di, Dodd, E. M., Dubord, L. A., Edlund, J. A., Ellyin, R., Emanuel, B., Foster, J. T., Ganino, A. J., Garner, G. J., Gibson, M. T., Gildner, M., Glazebrook, K. J., Greco, M. E., Green, W. M., Hatch, S. J., Hetzel, M. M., Hoey, W. A., Hofmann, A. E., Ionasescu, R., Jain, A., Jasper, J. D., Johannesen, J. R., Johnson, G. K., Jun, I., Katake, A. B., Kim-Castet, S. Y., Kim, D. I., Kim, W., Klonicki, E. F., Kobeissi, B., Kobie, B. D., Kochocki, J., Kokorowski, M., Kosberg, J. A., Kriechbaum, K., Kulkarni, T. P., Lam, R. L., Landau, D. F., Lattimore, M. A., Laubach, S. L., Lawler, C. R., Lim, G., Lin, J. Y., Litwin, T. E., Lo, M. W., Logan, C. A., Maghasoudi, E., Mandrake, L., Marchetti, Y., Marteau, E., Maxwell, K. A., Namee, J. B. Mc, Mcintyre, O., Meacham, M., Melko, J. P., Mueller, J., Muliere, D. A., Mysore, A., Nash, J., Ono, H., Parker, J. M., Perkins, R. C., Petropoulos, A. E., Gaut, A., Gomez, M. Y. Piette, Casillas, R. P., Preudhomme, M., Pyrzak, G., Rapinchuk, J., Ratliff, J. M., Ray, T. L., Roberts, E. T., Roffo, K., Roth, D. C., Russino, J. A., Schmidt, T. M., Schoppers, M. J., Senent, J. S., Serricchio, F., Sheldon, D. J., Shiraishi, L. R., Shirvanian, J., Siegel, K. J., Singh, G., Sirota, A. R., Skulsky, E. D., Stehly, J. S., Strange, N. J., Stevens, S. U., Sunada, E. T., Tepsuporn, S. P., Tosi, L. P. C., Trawny, N., Uchenik, I., Verma, V., Volpe, R. A., Wagner, C. T., Wang, D., Willson, R. G., Wolff, J. L., Wong, A. T., Zimmer, A. K., Sukhatme, K. G., Bago, K. A., Chen, Y., Deardorff, A. M., Kuch, R. S., Lim, C., Syvertson, M. L., Arakaki, G. A., Avila, A., DeBruin, K. J., Frick, A., Harris, J. R., Heverly, M. C., Kawata, J. M., Kim, S.-K., Kipp, D. M., Murphy, J., Smith, M. W., Spaulding, M. D., Thakker, R., Warner, N. Z., Yahnker, C. R., Young, M. E., Magner, T., Adams, D., Bedini, P., Mehr, L., Sheldon, C., Vernon, S., Bailey, V., Briere, M., Butler, M., Davis, A., Ensor, S., Gannon, M., Haapala-Chalk, A., Hartka, T., Holdridge, M., Hong, A., Hunt, J., Iskow, J., Kahler, F., Murray, K., Napolillo, D., Norkus, M., Pfisterer, R., Porter, J., Roth, D., Schwartz, P., Wolfarth, L., Cardiff, E. H., Davis, A., Grob, E. W., Adam, J. R., Betts, E., Norwood, J., Heller, M. M., Voskuilen, T., Sakievich, P., Gray, L., Hansen, D. J., Irick, K. W., Hewson, J. C., Lamb, J., Stacy, S. C., Brotherton, C. M., Tappan, A. S., Benally, D., Thigpen, H., Ortiz, E., Sandoval, D., Ison, A. M., Warren, M., Stromberg, P. G., Thelen, P. M., Blasy, B., Nandy, P., Haddad, A. W., Trujillo, L. B., Wiseley, T. H., Bell, S. A., Teske, N. P., Post, C., Torres-Castro, L., Grosso, C. Wasiolek, M. Science goals and mission architecture of the Europa Lander mission concept. The Planetary Science Journal, 3(1), (2022): 22, https://doi.org/10.3847/psj/ac4493.Europa is a premier target for advancing both planetary science and astrobiology, as well as for opening a new window into the burgeoning field of comparative oceanography. The potentially habitable subsurface ocean of Europa may harbor life, and the globally young and comparatively thin ice shell of Europa may contain biosignatures that are readily accessible to a surface lander. Europa's icy shell also offers the opportunity to study tectonics and geologic cycles across a range of mechanisms and compositions. Here we detail the goals and mission architecture of the Europa Lander mission concept, as developed from 2015 through 2020. The science was developed by the 2016 Europa Lander Science Definition Team (SDT), and the mission architecture was developed by the preproject engineering team, in close collaboration with the SDT. In 2017 and 2018, the mission concept passed its mission concept review and delta-mission concept review, respectively. Since that time, the preproject has been advancing the technologies, and developing the hardware and software, needed to retire risks associated with technology, science, cost, and schedule.K.P.H., C.B.P., E.M., and all authors affiliated with the Jet Propulsion Laboratory carried out this research at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (grant No. 80NM0018D0004). J.I.L. was the David Baltimore Distinguished Visiting Scientist during the preparation of the SDT report. JPL/Caltech2021