The maximum spectral radius of k ⋅ P₃-free graphs

Let k⋅P₃ denote the graph consisting of k vertex-disjoint paths on three vertices. The Brualdi–Solheid–Turán type problem asks for the maximum spectral radius of an n -vertex graph that does not contain a given subgraph. In this paper, we study the Brualdi–Solheid–Turán type problem for k⋅P₃-free graphs. More precisely, for all positive integers n and k≥2, we determine the maximum spectral radius of an n -vertex k⋅P₃-free graph and completely characterize the corresponding extremal graphs. Our results show that, although the spectral extremal graphs belong to graph families similar to those arising in the edge-extremal setting, the underlying extremal mechanism is fundamentally different. In particular, we identify a sharp structural transition at n=7k−1, obtain a spectral analogue of the classical Turán type result for k⋅P₃-free graphs, and strengthen earlier results by removing the restriction on n .