Super Edge-connectivity and Zeroth-order General Randić Index for −1 ≤ α < 0

Let G be a connected graph with order n, minimum degree δ = δ(G) and edge-connectivity λ = λ(G). A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randic index $$R_\alpha ^0\left( G \right) = \sum\limits_{x \in V\left( G \right)} {d_G^\alpha \left( x \right)} $$ , where dG(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randic index for −1 ≤ α < 0, respectively.