A bound on the rank of weighted graphs in terms of girth

Let (Formula presented.) be a connected weighted graph and (Formula presented.) be its adjacency matrix. The rank (Formula presented.) of (Formula presented.) is defined as the rank of its adjacency matrix (Formula presented.). For a given underlying graph G, the girth (Formula presented.) (or simply g) is the length of a shortest cycle (shortest graph-theoretic distance) in G. In this paper, we establish the bound (Formula presented.) for any weighted graph (Formula presented.). Furthermore, we characterize the cases where equality holds, specifically when (Formula presented.) and (Formula presented.). Our results extend previous work on ordinary graphs by Zhou et al. (Linear Algebra Appl., 2021), signed graphs by Wu et al. (Linear Algebra Appl., 2022), and complex unit gain graphs by Khan (Linear Algebra Appl., 2024) to the setting of weighted graphs.