arXiv2025
The well-known Turán theorem states that if $G$ is an $n$-vertex $K_{r+1}$-free graph, then $e(G)\le e(T_{n,r})$, with equality if and only if $G$ is the $r$-partite Turán graph $T_{n,r}$. A graph $F$ is called color-critical if it contains an edge whose deletion reduces its chromatic number. Extending the Turán theorem, Simonovits (1968) proved that for any color-critical graph $F$ with $χ(F)=r+1$ and sufficiently large $n$, the Turán graph $T_{n,r}$ is the unique graph with maximum number of edges among all $n$-vertex $F$-free graphs. Subsequently, Nikiforov [Electron. J. Combin., 16 (1) (2009)] proved a spectral version of the Simonovits theorem in terms of the adjacency spectral radius. In this paper, we show an extension of the Simonovits theorem for the signless Laplacian spectral radius. We prove that for any color-critical graph $F$ with $χ(F)=r+1\ge 4$ and sufficiently large $n$, if $G$ is an $F$-free graph on $n$ vertices, then $q(G)\le q(T_{n,r})$, with equality if and only if $G=T_{n,r}$. Our approach is to establish a signless Laplacian spectral version of the criterion of Keevash, Lenz and Mubayi [SIAM J. Discrete Math., 28 (4) (2014)]. Consequently, we can determine the signless Laplacian spectral extremal graphs for generalized books and even wheels. As an application, our result gives an upper bound on the degree power of an $F$-free graph. We show that if $n$ is sufficiently large and $G$ is an $F$-free graph on $n$ vertices with $m$ edges, then $\sum_{v\in V(G)} d^2(v) \le 2(1- \frac{1}{r})mn$, with equality if and only if $G$ is a regular Turán graph $T_{n,r}$. This extends a result of Nikiforov and Rousseau [J. Combin. Theory Ser B 92 (2004)].