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The Laplacian matrix of a graph G is denoted by  where  and  is the adjacency matrix of G. A one-edge connection of two graphs  and  is a graph  with  and  where  and  In this article, we study the existence and multiplicity of Laplacian eigenvalue 2 in the bicyclic graph  based on existence and multiplicity of this special Laplacian eigenvalue in  and  Examples to illustrate and delimit the results are provided.

Laplacian eigenvalue, characteristic polynomial, multiplicity, unicyclic graph, bicyclic graph.