Related papers: New proofs of stability theorems on spectral graph…

The following sharpening of Tur\'an's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there…

In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from `strong stability' forms of the corresponding (pure) extremal results. These results hold for…

The Erd\H{o}s-Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erd\H{o}s-Simonovits type stability theorem in multi-partite graphs. Different from the Erd\H{o}s-Simonovits stability…

We extend the classical stability theorem of Erdos and Simonovits in two directions: first, we allow the order of the forbidden graph to grow as log of order of the host graph, and second, our extremal condition is on the spectral radius of…

We study the extremal problem that relates the spectral radius $\lambda (G)$ of an $F$-free graph $G$ with its number of edges. Firstly, we prove that for any graph $F$ with chromatic number $\chi (F)=r+1\ge 3$, if $G$ is an $F$-free graph…

In this paper, we extend and refine previous Tur\'an-type results on graphs with a given circumference. Let $W_{n,k,c}$ be the graph obtained from a clique $K_{c-k+1}$ by adding $n-(c-k+1)$ isolated vertices each joined to the same $k$…

In 2002, Nikiforov proved that for an $n$-vertex graph $G$ with clique number $\omega$ and edge number $m$, the spectral radius $\lambda(G)$ satisfies $\lambda (G) \leq \sqrt{2(1 - 1/\omega) m}$, which confirmed a conjecture implicitly…

The classical stability theorem of Erd\H{o}s and Simonovits states that, for any fixed graph with chromatic number $k+1 \ge 3$, the following holds: every $n$-vertex graph that is $H$-free and has within $o(n^2)$ of the maximal possible…

Confirming a conjecture of Elphick and Edwards and strengthening a spectral theorem of Wilf, Nikiforov proved that for any $K_{r+1}$-free graph $G$, $\lambda(G)^2 \leq 2 (1 - 1/r) m$, where $\lambda(G)$ is the spectral radius of $G$, and…

We present new short proofs to both the exact and the stability results of two extremal problems. The first one is the extension of Tur\'{a}n's theorem in hypergraphs, which was firstly studied by Mubayi $\cite{MU06}$. The second one is…

The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order $n$…

We study the behaviour of $K_{r+1}$-free graphs $G$ of almost extremal size, that is, typically, $e(G)=ex(n,K_{r+1})-O(n)$. We show that such graphs must have a large amount of 'symmetry', in particular that all but very few vertices of $G$…

The well-known Tur\'{a}n 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 Tur\'{a}n graph $T_{n,r}$. A graph $F$ is called color-critical if it…

Keevash, Lenz and Mubayi developed a general criterion for hypergraph spectral extremal problems in their seminal work (SIAM J. Discrete Math., 2014). Their framework shows that extremal results on the $\alpha$-spectral radius (for $\alpha…

We present a sufficient condition for the stability property of extremal graph problems that can be solved via Zykov's symmetrisation. Our criterion is stated in terms of an analytic limit version of the problem. We show that, for example,…

A classical result of Nosal asserts that every $m$-edge graph with spectral radius $\lambda (G)> \sqrt{m}$ contains a triangle. A celebrated extension of Nikiforov [35] states that if $G$ is an $m$-edge graph with $\lambda (G)> \sqrt{(1-…

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability Theorem of Erd\H{o}s and Simonovits shows that if a $K_{r+1}$-free graph with $n$ vertices has close to the maximal $t_r(n)$ edges, then it is close…

We present a method which provides a unified framework for most stability theorems that have been proved in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph problems to the simpler question of…

We investigate two conjectured spectral graph theoretic strengthenings of Tur\'an's theorem. Let $\mu_1 \ge \ldots \ge \mu_n$ denote the eigenvalues of a graph $G$ with $n$ vertices, $m$ edges and clique number $\omega(G)$. The concise…

A fundamental result in extremal graph theory is attributed to Mantel's theorem, which states that every graph on $n$ vertices with more than $\lfloor n^2/4 \rfloor$ edges must contain a triangle. Lov\'{a}sz and Simonovits (1975) provided a…