Some transformations on multiplicative eccentricity resistance-distance and their applications

For a connected graph G, the multiplicative eccentricity resistance-distance is defined as ξR*(G)=∑{x,y}⊂V(G)ɛG(x)·ɛG(y)RG(x,y), where εG( · ) is the eccentricity of the corresponding vertex and RG(x, y) is the effective resistance between vertices x and y in G. A connected graph G is called a cactus if any two of its cycles have at most one common vertex. Let Cat(n; t) be the set of cacti possessing n vertices and t cycles, where 0≤t≤n−12. In this paper, we introduce some edge-grafting transformations which decrease ξR*(G). As their applications, the extremal graphs with minimum and second minimum ξR*(G)-value in Cat(n; t) are characterized.