In this paper, we investigate the energy of a weighted random graph $G_p(f)$ in $G_{n,p}(f)$, 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 $G_p$ of the Erdos--Renyi random graph model $G_{n,p}$, and $f$ is a symmetric real function on two variables. Suppose $|f(d_i,d_j)|\leq C n^m$ for some constants $C, m>0$, and $f((1+o(1))np,(1+o(1))np)=(1+o(1))f(np,np)$. Then, for almost all graphs $G_p(f)$ in $G_{n,p}(f)$, the energy of $G_p(f)$ is $(1+o(1))f(np,np)\frac{8}{3\pi}\sqrt{p(1-p)}\cdot n^{3/2},$ where $p\in(0,1)$ is any fixed and independent of $n$. Consequently, with this one basket we can get the asymptotic values of various kinds of graph energies of chemical use, such as Randic energy, ABC energy, and energies of random matrices obtained from various kinds of degree-based chemical indices.

中文翻译：

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具有基于度的权重的随机图的图能量渐近值

在本文中，我们研究了 $G_{n,p}(f)$ 中加权随机图 $G_p(f)$ 的能量，其中每条边 $ij$ 的权重为 $f(d_i,d_j)$ ，其中$d_v$为随机变量，鄂尔多斯随机图$G_p$中顶点$v$的度数--仁义随机图模型$G_{n,p}$，$f$为对称实数两个变量的函数。假设 $|f(d_i,d_j)|\leq C n^m$ 对于一些常量 $C, m>0$, 和 $f((1+o(1))np,(1+o(1)) np)=(1+o(1))f(np,np)$。那么，对于$G_{n,p}(f)$中的几乎所有图$G_p(f)$，$G_p(f)$的能量为$(1+o(1))f(np,np) \frac{8}{3\pi}\sqrt{p(1-p)}\cdot n^{3/2},$ 其中 $p\in(0,1)$ 是任何固定的且独立于 $n $. 因此，有了这个篮子，我们可以得到各种化学用途的图能量的渐近值，例如兰迪克能量、ABC 能量、