Spectral and Structural Control of Convergence in Greedy Max-Cut

We develop a structural framework for analyzing the convergence behavior of the greedy vertex-flip algorithm for the Max-Cut problem. We show that spectral properties of the graph constrain the distribution of local gains and derive quantitative relationships linking maximum gain, total positive gain, and expected improvement under a randomized greedy rule. These results imply that the algorithm cannot sustain stagnation unless all local gains are uniformly small. Using these structural bounds, we establish high-probability guarantees on cumulative improvement and characterize convergence as a two-phase process: a structural phase of forced improvement followed by a terminal low-gain regime. Our results provide a unified perspective connecting spectral graph theory, gain distribution, and stochastic dynamics, yielding new insight into how graph structure governs convergence behavior in greedy Max-Cut.