On the Total Distance and Diameter of Graphs

Abstract

The total distance (or Wiener index) of a connected graph $G$ is the sum of all distances between unordered pairs of vertices of $G$. DeLaViña and Waller [‘Spanning trees with many leaves and average distance’, Electron. J. Combin.15(1) (2008), R33, 14 pp.] conjectured in 2008 that if $G$ has diameter $D>2$ and order $2D+1$, then the total distance of $G$ is at most the total distance of the cycle of the same order. In this note, we prove that this conjecture is true for 2-connected graphs.

Extremal results on stepwise transmission irregular graphs

“…This concept arose in different contexts, hence it is not surprising that it is also known as the total distance, the farness, and the vertex Wiener value, cf. [1,22,[24][25][26]. The transmission also led to the Wiener complexity (the number of different transmissions) [3], is closely related to other topological indices [26], and characterizes the distance-balanced property and the opportunity index [9,13].…”