Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

Building upon the notion of the Gutman index SGut(G), Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index SGutk (G) of G is defined by SGutk (G) = ∑S⊆V(G),|S|=k (∏v∈S degG (v)) dG (S), in which dG (S) is the Steiner distance of S and degG (v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk . We obtain sharp upper and lower bounds of SGutk (G) + SGutk (G) and SGutk (G) · SGutk (G) for a connected graph G of order n, m edges, maximum degree ∆ and minimum degree δ

Inverse Problems Related to the Wiener and Steiner-Wiener Indices

In a graph, the generalized distance between multiple vertices is the minimum number of edges in a connected subgraph that contains these vertices. When we consider such distances between all subsets of $k$ vertices and take the sum, it is called the Steiner $k$-Wiener index and has important applications in Chemical Graph Theory. In this thesis we consider the inverse problems related to the Steiner Wiener index, i.e. for what positive integers is there a graph with Steiner Wiener index of that value