Spectral Properties of Dense Barab\'asi-Albert Graphs

Preferential attachment graphs model networks whose growth produces highly uneven degree distributions, describing many real-world systems. Their adjacency spectra are important because they allow graph-theoretic questions to be studied through the eigenvalues of matrices. We analyze the adjacency matrix of a dense Barab\'asi-Albert (B-A) multigraph, where the number of edges added at each step is proportional to the final number of vertices. First, we compute the large-$n$ limit of the expected adjacency matrix and show that it is described by a rank-one limiting kernel, viewed as a continuous analogue of the adjacency matrix. After centering and scaling, the fluctuations form a random matrix with a computable variance profile. Using the quadratic vector equation approach, we derive the limiting bulk spectral distribution. We also determine the asymptotic location of the leading eigenvalue generated by the rank-one mean component.