Geodesic
The first Dirichlet eigenvalue of bicyclic graphs
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 441-451
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In this paper, we have investigated some properties of the first Dirichlet eigenvalue of a bicyclic graph with boundary condition. These results can be used to characterize the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicyclic graphs with a given graphic bicyclic degree sequence with minor conditions. Moreover, the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicycle graphs with fixed $k$ interior vertices of degree at least 3 are obtained.
In this paper, we have investigated some properties of the first Dirichlet eigenvalue of a bicyclic graph with boundary condition. These results can be used to characterize the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicyclic graphs with a given graphic bicyclic degree sequence with minor conditions. Moreover, the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicycle graphs with fixed $k$ interior vertices of degree at least 3 are obtained.
DOI : 10.1007/s10587-012-0038-1
Classification : 05C35, 05C50
Keywords: first Dirichlet eigenvalue; bicyclic graph; degree sequence 
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Zhang, Guang-Jun; Zhang, Xiao-Dong. The first Dirichlet eigenvalue of bicyclic graphs. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 441-451. doi: 10.1007/s10587-012-0038-1

[1] koğlu, T. Bıyı, Leydold, J.: Faber-Krahn type inequalities for trees. J. Comb. Theory, Ser. B 97 (2007), 159-174. | DOI | MR

[2] Friedman, J.: Some geometric aspects of graphs and their eigenfunctions. Duke Math. J. 69 (1993), 487-525. | DOI | MR | Zbl

[3] Leydold, J.: The geometry of regular trees with the Faber-Krahn property. Discrete Math. 245 (2002), 155-172. | DOI | MR | Zbl

[4] Pruss, A. R.: Discrete convolution-rearrangement inequalities and the Faber-Krahn inequality on regular trees. Duke Math. J. 91 (1998), 463-514. | DOI | MR | Zbl

[5] Zhang, G. J., Zhang, J., Zhang, X. D.: Faber-Krahn Type Inequality for Unicyclic Graphs. Linear and Multilinear Algebra, DOI: 10.1080/03081087.2011.651722. | DOI

[6] Zhang, X. D.: The Laplacian spectral radii of trees with degree sequences. Discrete Math. 308 (2008), 3143-3150. | DOI | MR | Zbl

[7] Zhang, X. D.: The signless Laplacian spectral radius of graphs with given degree sequences. Discrete Appl. Math. 157 (2009), 2928-2937. | DOI | MR | Zbl

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