Some new notions of centrality measures in trees

Several notions of centrality in graphs have been defined over the years that are important from both theoretical and practical points of view. We introduce some new concepts based on ratios and show that these concepts coincide for a tree. With each centrality concept one may naturally associate a function defined on the vertices of the tree. These functions based on ratios are shown to possess certain convexity and quasiconvexity properties.