A Soltes' digraph is a directed graph whose total distance (Wiener index) is unchanged when any single vertex is removed, and a z-Soltes' digraph shifts that total by a fixed integer z. This work constructs, from circulant digraphs D(n,S) with carefully chosen difference sets, infinitely many z-Soltes' digraphs for every integer z, and exhibits one whose automorphism group is trivial. It shows how modular difference-set constructions and asymptotic distance estimates yield extremal graph families with prescribed vertex-deletion distance behaviour.
The total distance (Wiener index) of a directed graph is the sum of shortest-path distances over all ordered pairs of vertices. A z-Soltes' digraph is one for which deleting any single vertex changes…