Distance in Graphs: Trees

The distance of a vertex \(u\) in a connected graph \(G\) is defined by \(\sigma_G(u) := \sum_{v \in V(G)} d(u, v)\), and the distance of \(G\) is given by \(\sigma(G) = \frac{1}{2} \sum_{u \in V(G)} \sigma(u) (= \sum_{\{u,v\} \subseteq V(G)} d(u, v)\). Thus, the average distance between vertices in a connected graph \(G\) of order \(n\) is \(\frac{\sigma(G)}{\binom{n}{2}}\). These graph invariants have been studied for the past fifty years. Here, we discuss some known properties and present a few new results, together with several open problems. We focus on trees.