Abstract
Let ℬ ( n , g ) be the class of bicyclic graphs on n vertices with girth g . Let ℬ 1 ( n , g ) be the subclass of ℬ ( n , g ) consisting of all bicyclic graphs with two edge-disjoint cycles and ℬ 2 ( n , g ) = ℬ ( n , g ) ∖ ℬ 1 ( n , g ) . This paper determines the unique graph with the maximal Laplacian spectral radius among all graphs in ℬ 1 ( n , g ) and ℬ 2 ( n , g ) , respectively. Furthermore, the upper bound of the Laplacian spectral radius and the extremal graph for ℬ ( n , g ) are also obtained.
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