Let G be a simple undirected graph with the characteristic polynomial of its Laplacian matrix L(G), P(G,x)=∑k=0n(-1)kckxn-k. Aleksandar Ilić [A. Ilić, Trees with minimal Laplacian coefficients, Comput. Math. Appl. 59 (2010) 2776–2783] identified n-vertex trees with given matching number q which simultaneously minimize all Laplacian coefficients. In this paper, we give another proof of this result. Generalizing the approach in the above paper, we determine n-vertex trees with given matching number q which have the second minimal Laplacian co...
