The ordering of unicyclic graphs with the smallest algebraic connectivity |
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Authors: | Ying Liu Yue Liu |
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Institution: | a College of Mathematics and Information, Shanghai Lixin University of Commerce, Shanghai, 201620, China b Department of Mathematics, Tongji University, Shanghai, 200092, China |
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Abstract: | Fielder M. Fielder, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298-305] has turned out that G is connected if and only if its algebraic connectivity a(G)>0. In 1998, Fallat and Kirkland S.M. Fallat, S. Kirkland, Extremizing algebraic connectivity subject to graph theoretic constraints, Electron. J. Linear Algebra 3 (1998) 48-74] posed a conjecture: if G is a connected graph on n vertices with girth g≥3, then a(G)≥a(Cn,g) and that equality holds if and only if G is isomorphic to Cn,g. In 2007, Guo J.M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711] gave an affirmatively answer for the conjecture. In this paper, we determine the second and the third smallest algebraic connectivity among all unicyclic graphs with vertices. |
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Keywords: | Laplacian matrix Algebraic connectivity Fielder vector Unicyclic graph Characteristic polynomial |
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