first Dirichlet eigenvalue; bicyclic graph; degree sequence
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.
 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 10.1080/03081087.2011.651722