The Wiener index of a network, introduced by the chemist Harry [30], is sum distances between all pairs nodes in network. This index, originally used chemical graph representations non-hydrogen atoms molecule, considered to be fundamental and useful network descriptor. We study problem constructing geometric networks on point sets Euclidean space that minimize index: given set P n points $$\mat...

Completely independent spanning trees in a graph G are of such that for any two distinct vertices G, the paths between them pairwise edge-disjoint and internally vertex-disjoint. In this paper, we present tight lower bound on maximum number completely L(G), where L(G) denotes line G. Based new characterization with k trees, also show complete $$K_n$$ order $$n \ge 4$$ , there $$\lfloor \frac{n+...