On the largest eigenvalue of non-regular graphs. (English) Zbl 1125.05062
Summary: We study the spectral radius of connected non-regular graphs. Let \(\lambda _{1}(n,\varDelta )\) be the maximum spectral radius among all connected non-regular graphs with \(n\) vertices and maximum degree \(\varDelta \). We prove that \(\varDelta - \lambda _{1}(n,\varDelta )=\varTheta (\varDelta /n^{2})\). This improves two recent results by Stevanović and Zhang, respectively; see D. Stevanović [J. Comb. Theory, Ser. B 91, 143–146 (2004; Zbl 1048.05061)] and X.-D. Zhang [Linear Algebra Appl. 409, 79–86 (2005; Zbl 1072.05040)].
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C35 | Extremal problems in graph theory |
References:
| [1] | Cvetković, D.; Petrić, M., A table of connected graphs on six vertices, Discrete Math., 50, 1, 37-49 (1984) · Zbl 0533.05052 |
| [2] | Cvetković, D.; Doob, M.; Gutman, I.; Torgašev, A., Recent Results in the Theory of Graph Spectra, Ann. Discrete Math., vol. 36 (1988), North-Holland: North-Holland Amsterdam · Zbl 0634.05054 |
| [3] | Cvetković, D.; Doob, M.; Sachs, H., Spectra of Graphs—Theory and Applications (1995), Johann Ambrosius Barth: Johann Ambrosius Barth Heidelberg · Zbl 0824.05046 |
| [4] | Cvetković, D.; Rowlinson, P.; Simić, S., Eigenspaces of Graphs (1997), Cambridge Univ. Press · Zbl 0878.05057 |
| [5] | Latham, G. A., A remark on Minc’s maximal eigenvector bound for positive matrices, SIAM J. Matrix Anal. Appl., 16, 1, 307-311 (1995) · Zbl 0816.15019 |
| [6] | Minc, H., On the maximal eigenvector of a positive matrix, SIAM J. Numer. Anal., 7, 424-427 (1970) · Zbl 0219.15008 |
| [7] | Papendieck, B.; Recht, P., On maximal entries in the principal eigenvector of graphs, Linear Algebra Appl., 310, 1-3, 129-138 (2000) · Zbl 0966.05047 |
| [8] | Stevanović, D., The largest eigenvalue of nonregular graphs, J. Combin. Theory Ser. B, 91, 143-146 (2004) · Zbl 1048.05061 |
| [9] | Zhang, X. D., Eigenvectors and eigenvalues of non-regular graphs, Linear Algebra Appl., 409, 79-86 (2005) · Zbl 1072.05040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
