Extremal trees of given segment sequence with respect to some eccentricity-based invariants

Abstract A path P is a segment of a tree if the endpoints of P are of degree 1 or at least 3, and each of the rest vertices are of degree 2 in the tree. The lengths of all the segments of this tree form its segment sequence. Denote by T l the set of all trees on n vertices with the segment sequence l = ( l 1 , l 2 , … , l m ) , where l 1 ⩾ l 2 ⩾ ⋯ ⩾ l m . In this paper, the extremal structures of trees among T l , which minimize or maximize the first (resp. second) Zagreb eccentricity index, the eccentric connectivity index, and the eccentric distance sum, are characterized. Furthermore, we consider similar extremal problems for trees with fixed number of segments.