Some edge-grafting transformations on the eccentricity resistance-distance sum and their applications

The eccentricity resistance-distance sum of a connected graph G is defined as ξR(G)=∑{u,v}⊆VG(εG(u)+εG(v))RuvG, where εG(⋅) is the eccentricity of the corresponding vertex and RuvG is the resistance distance between u and v in graph G. In this paper some edge-grafting transformations on the eccentricity resistance-distance sum of a connected graph are studied, which is mainly focused on the monotonicity on each of these edge-grafting transformations for ξR. As applications, on the one hand, we determined the graph with the minimum ξR-value among the set of all n-vertex cacti each of which contains just t cycles, and on the other hand, sharp lower bound on ξR of graphs among the n-vertex cacti is determined. The corresponding extremal graphs are identified.