Research on spectral Turán problems under coexistence constraints of matching and color-critical graphs

This paper investigates the spectral Turán problem for graphs simultaneously excluding a matching M_s+1 of size s+1 and a color-critical graph F with chromatic number r+1 . By introducing combinatorial parameters such as the covering number and independent covering number of the graph, this paper develops a unified framework for analyzing spectral extremal problems of degenerate forbidden graph families. Based on this framework, it is proved that, for sufficiently large s , the maximum spectral radius of an n -vertex \M_s+1,F\\text-free graph is attained by the complete r -partite graph G(n,r,s) , and this extremal graph is uniquely determined. This result not only extends the edge-extremal results of Alon and Frankl to the spectral setting, but also completely determines the spectral extremal structure under the coexistence constraints of matchings and color-critical graphs in this setting.