Laplacian coefficients of trees with given number of leaves or vertices of degree two

Abstract

Let G be a simple undirected n -vertex graph with the characteristic polynomial of its Laplacian matrix L ( G ) , det ( λ I - L ( G ) ) = ∑ k = 0 n ( - 1 ) k c k λ n - k . It is well known that for trees the Laplacian coefficient c n - 2 is equal to the Wiener index of G , while c n - 3 is equal to the modified hyper-Wiener index of graph. Using a result of Zhou and Gutman on the relation between the Laplacian coefficients and the matching numbers in subdivided bipartite graphs, we characterize the trees with k leaves (pendent vertices) which simultaneously minimize all Laplacian coefficients. In particular, this extremal balanced starlike tree S ( n , k ) minimizes the Wiener index, the modified hyper-Wiener index and recently introduced Laplacian-like energy. We prove that graph S ( n , n - 1 - p ) has minimal Laplacian coefficients among n -vertex trees with p vertices of degree two. In conclusion, we illustrate on examples of these spectrum-based invariants that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution, and pose a conjecture on extremal unicyclic graphs with k leaves.