Peripheral Wiener Index of a Graph

The eccentricity of a vertex v is the maximum distance between v and any other vertex. A vertex with maximum eccentricity is called a peripheral vertex. The peripheral Wiener index PW(G) of a graph G is defined as the sum of the distances between all pairs of peripheral vertices of G. In this paper, we initiate the study of the peripheral Wiener index and investigate its basic properties. In particular, we determine the peripheral Wiener index of the cartesian product of two graphs and trees.