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The computation of two-point resistance of a resistor network is a classical problem in electric circuit theory which was studied extensively. In this talk, we will first introduce some of known results on the resistance distances in the content of combinatorics and electric networks, and then present some of our results on the resistance distances of graphs.

This talk is on the energy of weighted random graphs $G_{n,p}(f)$, in which each edge $ij$ takes the weight $f(d_i,d_j)$, where $d_v$ is a random variable, the degree of vertex $v$ in the random graph $G_{n,p}$ of Erd\{o}s--R\'{e}nyi model, and $f$ is a Symmetric real function on two variables.   Suppose $f((1+o(1))np,(1+o(1))np)=(1+o(1))f(np,np)$. Then, for almost all graphs $G$ in $G_{n,p}(f)$, the energy of $G$ is $(1+o(1))f(np,np) \frac{8}{3 \pi} \sqrt{p(1-p)} \cdot n^{3/2}$. Consequently, with this one basket we can get the asymptotic values of  various kinds of graph energies  of chemical use, such as Randi\'c energy, ABC energy, and energies of random matrices obtained from various kinds of degree-based chemical indices.

 责任编辑： 邹家浩