Dan Hu, Hajo Broersma, Xueliang Li, Shenggui Zhang
We consider the random graph model $G(\mathbf{w})$ for a given expected degree sequence $\mathbf{w} = (w_1, w_2,\ldots, w_n)$, where the probability $p_{ij}$ of an edge between vertices $v_i$ and $v_j$ is defined as $w_iw_j\rho$ with $\rho=\frac{1}{\sum_{i=1}^{n}w_i}$. In this paper, we apply a matrix concentration inequality to derive an upper bound on the largest eigenvalue of the adjacency matrix of the random graph with given expected degrees. We further analyze the expectation of the largest eigenvalue of the adjacency matrix and establish its concentration properties. Additionally, by utilizing an extension of the matrix Chernoff inequality that incorporates an intrinsic dimension parameter, we investigate the expectation and tail behavior of the largest eigenvalue of the Laplacian matrix for the random graph model $G(\mathbf{w})$.
