On the ordering of trees by the Laplacian coefficients

Abstract We generalize the results from [X.-D. Zhang, X.-P. Lv, Y.-H. Chen, Ordering trees by the Laplacian coefficients, Linear Algebra Appl. (2009), doi:10.1016/j.laa.2009.04.018 ] on the partial ordering of trees with given diameter. For two n -vertex trees T 1 and T 2 , if c k ( T 1 ) ⩽ c k ( T 2 ) holds for all Laplacian coefficients c k , k = 0 , 1 , … , n , we say that T 1 is dominated by T 2 and write T 1 ⪯ c T 2 . We proved that among n vertex trees with fixed diameter d , the caterpillar C n , d has minimal Laplacian coefficients c k , k = 0 , 1 , … , n . The number of incomparable pairs of trees on ⩽ 18 vertices is presented, as well as infinite families of examples for two other partial orderings of trees, recently proposed by Mohar. For every integer n , we construct a chain { T i } i = 0 m of n -vertex trees of length n 2 4 , such that T 0 ≅ S n , T m ≅ P n and T i ⪯ c T i + 1 for all i = 0 , 1 , … , m - 1 . In addition, the characterization of the partial ordering of starlike trees is established by the majorization inequalities of the pendent path lengths. We determine the relations among the extremal trees with fixed maximum degree, and with perfect matching and further support the Laplacian coefficients as a measure of branching.