On the Laplacian coefficients of unicyclic graphs with prescribed matching number

Let ϕ ( G , λ ) = ∑ k = 0 n ( − 1 ) k c k ( G ) λ n − k be the characteristic polynomial of the Laplacian matrix of a graph G of order n . We give some transformations of connected graphs that decrease all Laplacian coefficients c k ( G ) , we then derive the unicyclic graphs with the minimum Laplacian coefficients in the set of all connected unicyclic graphs with prescribed order and matching number. Furthermore, we determine the unique connected unicyclic graph with the minimal Laplacian coefficients among all connected unicyclic graphs of order n except S n ′ , where S n ′ is the unicyclic graph obtained from the n -vertex star S n by joining two of its pendent vertices with an edge.