On the Laplacian coefficients of unicyclic graphs with prescribed matching number

Let ϕ(G,λ)=∑k=0n(−1)kck(G)λn−k be the characteristic polynomial of the Laplacian matrix of a graph GG of order nn. We give some transformations of connected graphs that decrease all Laplacian coefficients ck(G)ck(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 nn except Sn′, where Sn′ is the unicyclic graph obtained from the nn-vertex star SnSn by joining two of its pendent vertices with an edge.