The first two cacti with larger multiplicative eccentricity resistance-distance

Abstract

For a connected graph G, the multiplicative eccentricity resistance-distance ξ * R (G) is defined as ξ * R (G) = {x,y}⊆V(G) ε(x) • ε(y)R G (x, y), where ε(•) is the eccentricity of the corresponding vertex and R G (x, y) is the effective resistance between vertices x and y. A cactus is a connected graph in which any two simple cycles have at most one vertex in common. Let Cat(n; t) be the set of cacti possessing n vertices and t cycles, where 0 ≤ t ≤ n−1 2. In this paper, we first introduce some edge-grafting transformations which will increase ξ * R (G). As their applications, the extremal graphs with maximum and second-maximum ξ * R (G)-value in Cat(n; t) are characterized, respectively.