Abstract
The Wiener index is defined as the sum of all distances between all pairs of unordered vertices in a connected graph. Replacing the ordinary distance by resistance distance in Wiener index, one gets the Kirchhoff index which is defined as the sum of all
resistance distances between all pairs of unordered vertices in a connected graph. This two distance-based invariants are viewed as important measures associated with a (molecular) network which correlate nicely to chemical and physical
properties, and have been studied extensively in the past decades. In this paper, we determine respectively the graphs which have the maximum Wiener index and Kirchhoff index among all connected graphs of order \(n\) with girth \(g\) and
maximum degree \(\Delta\). The corresponding extremal graphs are characterized completely.