On a poset of trees revisited

Abstract

This contribution gives an extensive study on the Wiener indices, the number of closed walks, the coefficients of some graph polynomials (the adjacency polynomial, the Laplacian polynomial, the edge cover polynomial and the independence polynomial) of trees. Csikvári (2010) [4] introduced the generalized tree shift, which keeps the number of vertices of trees. Applying the generalized tree shifts and recurrence relation, we extend the works of Csikvári (2010) [4] and Csikvári (2013) [5]. Using a unified approach, we obtain the following main results: Firstly, for all n and ℓ, we characterize the unique tree having the maximum (resp. minimum) Wiener index and the unique tree having the maximum (resp. minimum) number of closed walks of length ℓ among the trees of order n which are neither a path nor a star. Secondly, we characterize the unique tree whose adjacency polynomial (resp. Laplacian polynomial, independence polynomial) has the maximum (resp. minimum) coefficients in absolute value among the trees of order n which are neither a path nor a star, respectively. At last, we identify all the n-vertex trees whose edge cover polynomial has the minimum coefficients in absolute value and we also determine the unique tree having the maximum coefficients in absolute value among the trees of order n which are neither a path nor a star.