Abstract
The (ordinary) Wiener index of a connected graph is defined to be the sum of distances between all vertex pairs in this graph. For a connected graph, its peripheral Wiener index is defined as the sum of distances between all pairs of peripheral vertices (vertices whose eccentricities are equal to diameter). More recently, Chen et al. (2018) investigated extremal problems on the peripheral Wiener index for general trees and trees with given restricted conditions, respectively. In this paper, we obtain further results on the peripheral Wiener index. First, we give sharp lower and upper bounds on the peripheral Wiener index for graphs without cut vertices. Second, we give two sharp upper bounds on the peripheral Wiener index in terms of other distance-based graph invariants. Finally, we establish sharp lower and upper bounds on the difference between the Wiener index and peripheral Wiener index for general connected graphs.
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