On the transmission irregular trees with the maximum Wiener index

Paper

Abstract

The transmission of a vertex

in a (chemical) graph

is the sum of distances from

to other vertices in

. If any two vertices of

have different transmissions, then

is transmission irregular. The Wiener index

of a graph

is the sum of all distances between all unordered pairs of vertices in

, which has another formula as the half of the sum of transmissions of all vertices of

. In this paper, we consider the Wiener index maximization problem on the set of transmission irregular trees of a given order

. We solve the problem for all odd values of

and for almost all even values of

. Each resolved extremal problem has a unique solution that is a chemical tree.