On the Laplacian coefficients and Laplacian-like energy of unicyclic graphs with n vertices and m pendent vertices

Let φ(g,λ) = det(λin - l(g)) = ∑k=0 n(-1)k ck(g)λ n-k be the characteristic polynomial of the laplacian matrix of a graph g of order n. in this paper,we give four transforms on graphs that decrease all laplacian coefficients ck(g) and investigate a conjecture a. ilić and m. ilić (2009) about the laplacian coefficients of unicyclic graphs with n vertices and m pendent vertices. finally,we determine the graph with the smallest laplacian-like energy among all the unicyclic graphs with n vertices and m pendent vertices. © 2012 xinying pai and sanyang liu.