Abstract. A well-known result in extremal spectral graph theory, known as Nosal’s theorem, states that if [Formula: see text] is a triangle-free graph on [Formula: see text] vertices, then [Formula: see text], equality holds if and only if [Formula: see text]. Nikiforov [ Linear Algebra Appl., 427 (2007), pp. 183–189] extended Nosal’s theorem to [Formula: see text]-free graphs for every integer [Formula: see text]. This is now known as the spectral Turán theorem. Recently, Lin, Ning, and Wu [ Combin. Probab. Comput., 30 (2021), pp. 258–270] proved a refinement on Nosal’s theorem for nonbipartite triangle-free graphs. In this paper, we provide alternative proofs for both the result of Nikiforov and the result of Lin, Ning, and Wu. Moreover, our new proof can allow us to extend the later result to non-[Formula: see text]-partite [Formula: see text]-free graphs. Our result refines the theorem of Nikiforov and it also can be viewed as a spectral version of a theorem of Brouwer.

The Turán number of an $r$-uniform graph $F$, denoted by $ex(n,F)$, is the maximum number of edges in an $F$-free $r$-uniform graph on $n$ vertices. The Turán density of $F$ is defined as $\pi(F)=\underset{{n\rightarrow\infty}}{\lim}{ex(n,F) \over {n \choose r }}.$ Denote $\Pi_{\infty}^{(r)}={ \pi(\cal F): \cal F is a family of r{-uniform graphs}},$ $\Pi_{fin}^{(r)}=\{ \pi(\cal F): \cal F {is a \ finite \ family of} r{{-}uniform graphs}\}$, and $\Pi_{t}^{(r)}=\{\pi(\cal F): \cal F is a family of r{-uniform graphs, and}|\cal F|\le t}.$ For graphs, Erdös and Simonovits [ Studia Sci. Mat. Hungar. 1 (1966), pp. 51--57] and Erdös and Stone [ Bull. Amer. Math. Soc., 52 (1946), pp. 1087--1091] showed that $\Pi_{\infty}^{(2)}=\Pi_{fin}^{(2)}=\Pi_{1}^{(2)}={0, {1 \over 2}, {2 \over 3}, ...,{l-1 \over l}, ...}.$ We know quite little about the Turán density of an $r$-uniform graph for $r\ge 3$. Baber and Talbot [ Electron. J. Combin., 19 (2011)] and Pikhurko [ Israel J. Math., 20 (2014), pp. 415--454] showed that there is an irrational number in $\Pi_{3}^{(3)}$ and $\Pi_{fin}^{(3)}$, respectively, disproving a conjecture of Chung and Graham [ Erdös on Graphs: His Legacy of Unsolved Problems, A. K. Peters, Natick, MA, 1999]. Baber and Talbot [ Electron. J. Combin., 19 (2011)] asked whether $\Pi_{1}^{(r)}$ contains an irrational number. The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. The Lagrangian density of an $r$-uniform graph $F$ is $\pi_{\lambda}(F)=\sup \{r! \lambda(G):G\;is;F-free\}$, where $\lambda(G)$ is the Lagrangian of an $r$-uniform graph $G$. Sidorenko [ Combinatorica, 9 (1989), pp. 207--215] showed that the Lagrangian density of an $r$-uniform hypergraph $F$ is the same as the Turán density of the extension of $F$. In this paper, we show that the Lagrangian density of $F={123, 124, 134, 234, 567}$ (the disjoint union of $K_4^3$ and an edge) is ${\sqrt 3\over 3}$, and consequently, the Turán density of the extension of $F$ is an irrational number, answering the question of Baber and Talbot.