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.