For a graph G, the resistance distance $r_G(x,y)$ is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index $R^{\ast }(G)={\sum }_{\{x,y\}\subset V(G)}{d}_G(x)d_G(y)r_G(x,y)$ , where $d_G(x)$ is the degree of vertex x, and $V(G)$ denotes the vertex set of G. L. Feng et al. obtained the element in $Cact(n;t)$ with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on $R^{\ast }(G)$ , and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively.

In the actual network the information collected through routing protocols is uncertain with the expansion of network size and complexity. In this paper, the QOS routing problem is investigated in the networks environment with uncertain parameters. The fuzzy set theory is used to define the relationship between two fuzzy numbers, and construct a QOS routing model of multi-objective optimization. A multi-objective evolutionary algorithm (MOEA) is utilized to solve the problem. Simulation shows the MOEA can provide an available approach to QoS routing in networks environment with uncertain parameters.