Abstract
Let G be a graph, and let \(\lambda (G)\) denote the smallest eigenvalue of G. First, we provide an upper bound for \(\lambda (G)\) based on induced bipartite subgraphs of G. Consequently, we extract two other upper bounds, one relying on the average degrees of induced bipartite subgraphs and a more explicit one in terms of the chromatic number and the independence number of G. In particular, motivated by our bounds, we introduce two graph invariants that are of interest on their own. Finally, special attention goes to the investigation of the sharpness of our bounds in various classes of graphs as well as the comparison with an existing well-known upper bound.
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References
Alon, N., Sudakov, B.: Bipartite subgraphs and the smallest eigenvalue. Comb. Probab. Comput. 9, 1–12 (2000)
Bell, F.K., Cvetković, D., Rowlinson, P., Simić, S.K.: Graphs for which the least eigenvalue is minimal I. Linear Algebra Appl. 429, 234–241 (2008)
Bell, F.K., Cvetković, D., Rowlinson, P., Simić, S.K.: Graphs for which the least eigenvalue is minimal II. Linear Algebra Appl. 429, 2168–2179 (2008)
Bonamy, M., Lévêque, B., Pinlou, A.: Graphs with maximum degree \(\Delta \ge 17\) and maximum average degree less than 3 are list 2-distance \((\Delta +2)\)-colorable. Discrete Math. 306(317), 19–32 (2014)
Brouwer, A.E., Cioaba, S.M., Ihringer, F., McGinnis, M.: The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters. J. Comb. Theory Ser. B 133, 88–121 (2018)
Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)
Cioaba, S.M.: The spectral radius and maximum degree of irregular graphs. Electron. J. Comb. 14(1), Research Paper 38, 10 pp. (2007)
Cioaba, S.M., Elzinga, R.J., Gregory, D.A.: Some observations on the smallest adjacency eigenvalue of a graph. Discuss. Math. Graph Theory 40, 467–493 (2020)
Cvetković, D., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. London Mathematical Society Student Texts, 75. Cambridge University Press, Cambridge (2010)
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42, 1115–1145 (1995)
Kolaitis, P.G., Promel, H.J., Rothschild, B.L.: \(K_{l+1}\)-free graphs: asymptotic structure and a 0–1 law. Trans. Amer. Math. Soc. 303, 637–671 (1987)
Kwan, M., Letzter, S., Sudakov, B., Tran, T.: Dense induced bipartite subgraphs in triangle-free graphs. Combinatorica 40(2), 283–305 (2020)
Nadara, W., Smulewicz, M.: Decreasing the maximum average degree by deleting an independent set or a \(d\)-degenerate subgraph. Electron. J. Comb. 29(1), Paper No. 1.49, 15 pp. (2022)
Nikiforov, V.: The smallest eigenvalue of \(K_r\)-free graphs. Discrete Math. 306(2), 612–616 (2006)
Nikiforov, V.: The spectral radius of subgraphs of regular graphs. Electron. J. Comb. 14(1), Note 20, 4 pp. (2007)
Nikiforov, V.: Max \(k\)-cut and the smallest eigenvalue. Linear Algebra Appl. 504, 462–467 (2016)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. System Sci. 43, 425–440 (1991)
Trevisan, L.: Max cut and the smallest eigenvalue. SIAM J. Comput. 41(6), 1769–1786 (2012)
Turán, P.: Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48, 436–452 (1941)
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The second author was in part supported by a grant from IPM (No. 1403130020).
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Esmailpour, A., Saeedi Madani, S. & Kiani, D. Combinatorial upper bounds for the smallest eigenvalue of a graph. Arch. Math. (2024). https://doi.org/10.1007/s00013-024-01998-8
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DOI: https://doi.org/10.1007/s00013-024-01998-8