  Analyzing the spectral behavior of random matrices with dependency among
entries is a challenging problem. The adjacency matrix of the random
$d$-regular graph is a prominent example that has attracted immense interest. A
crucial spectral observable is the extremal eigenvalue, which reveals useful
geometric properties of the graph. According to the Alon's conjecture, which
was verified by Friedman, the (nontrivial) extremal eigenvalue of the random
$d$-regular graph is approximately $2\sqrt{d-1}$.
  In the present paper, we analyze the extremal spectrum of the random
$d$-regular graph (with $d\ge 3$ fixed) equipped with random edge-weights, and
precisely describe its phase transition behavior with respect to the tail of
edge-weights. In addition, we establish that the extremal eigenvector is always
localized, showing a sharp contrast to the unweighted case where all
eigenvectors are delocalized. Our method is robust and inspired by a
sparsification technique developed in the context of Erd\H{o}s-R\'{e}nyi graphs
(Ganguly and Nam, '22), which can also be applied to analyze the spectrum of
general random matrices whose entries are dependent.
