The signless Laplacian spectral Turán problems for color-critical graphs
Jian Zheng, Yongtao Li, Honghai Li
TL;DR
This work extends the Turán-type extremal framework to the signless Laplacian spectral radius. For any color-critical graph $F$ with chromatic number $ chi(F)=r+1\ge 4$ and sufficiently large $n$, the maximum $q(G)$ among $F$-free graphs on $n$ vertices is achieved uniquely by the $r$-partite Turán graph $T_{n,r}$, i.e., $q(G)\le q(T_{n,r})$ with equality iff $G=T_{n,r}$. The authors establish a signless Laplacian analogue of a Keevash–Lenz–Mubayi criterion, enabling a reduction to degree-stable extremal problems and yielding sharp results for generalized books $B_{r,k}$ and even wheels. As an application, they derive a tight bound on the degree-squared sum: for large $n$, $\sum_{v} d^2(v) \le 2\left(1-\frac{1}{r}\right) mn$, with equality for the regular $r$-partite Turán graph. The paper also discusses corollaries, extensions, and conjectures for forbidding other color-critical families, highlighting both the potential and the limitations of the approach in higher chromatic regimes.
Abstract
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)].