Stability for large forbidden subgraphs

We extend the classical stability theorem of Erdos and Simonovits for forbidden graphs of logarithmic order.Comment: Some polishing. Updated reference

Large joints in graphs

We show that if G is a graph of sufficiently large order n containing as many r-cliques as the r-partite Turan graph of order n; then for some C>0 G has more than Cn^(r-1) (r+1)-cliques sharing a common edge unless G is isomorphic to the the r-partite Turan graph of order n. This structural result generalizes a previous result that has been useful in extremal graph theory.Comment: 9 page

An extension of Tur\'an's Theorem, uniqueness and stability

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.Comment: 12 pages, 1 figure; outline of the proof added and other referee's comments incorporate

An extension of Turán's theorem, uniqueness and stability

We determine the maximum number of edges of an n -vertex graph G with the property that none of its r -cliques intersects a fixed set M⊂V(G) . For (r−1)|M|≥n , the (r−1) -partite Turán graph turns out to be the unique extremal graph. For (r−1)|M|<n , there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results

Spectral extremal graphs for edge blow-up of star forests

The edge blow-up of a graph $G$, denoted by $G^{p+1}$, is obtained by replacing each edge of $G$ with a clique of order $p+1$, where the new vertices of the cliques are all distinct. Yuan [J. Comb. Theory, Ser. B, 152 (2022) 379-398] determined the range of the Tur\'{a}n numbers for edge blow-up of all bipartite graphs and the exact Tur\'{a}n numbers for edge blow-up of all non-bipartite graphs. In this paper we prove that the graphs with the maximum spectral radius in an $n$-vertex graph without any copy of edge blow-up of star forests are the extremal graphs for edge blow-up of star forests when $n$ is sufficiently large.Comment: 22. arXiv admin note: text overlap with arXiv:2208.0655

Extremal problems for disjoint graphs

For a simple graph $F$, let $\mathrm{EX}(n, F)$ and $\mathrm{EX_{sp}}(n,F)$ be the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph $F$, respectively. Let $F$ be a graph with $\mathrm{ex}(n,F)=e(T_{n,r})+O(1)$. In this paper, we show that $\mathrm{EX_{sp}}(n,kF)\subseteq \mathrm{EX}(n,kF)$ for sufficiently large $n$. This generalizes a result of Wang, Kang and Xue [J. Comb. Theory, Ser. B, 159(2023) 20-41]. We also determine the extremal graphs of $kF$ in term of the extremal graphs of $F$.Comment: 23 pages. arXiv admin note: text overlap with arXiv:2306.1674

Spectral extremal results on edge blow-up of graphs

The edge blow-up $F^{p+1}$ of a graph $F$ for an integer $p\geq 2$ is obtained by replacing each edge in $F$ with a $K_{p+1}$ containing the edge, where the new vertices of $K_{p+1}$ are all distinct. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and maximum spectral radius of an $F$-free graph of order $n$, respectively. In this paper, we determine the range of $spex(n,F^{p+1})$ when $F$ is bipartite and the exact value of $spex(n,F^{p+1})$ when $F$ is non-bipartite for sufficiently large $n$, which are the spectral versions of Tur\'{a}n's problems on $ex(n,F^{p+1})$ solved by Yuan [J. Combin. Theory Ser. B 152 (2022) 379--398]. This generalizes several previous results on $F^{p+1}$ for $F$ being a matching, or a star. Additionally, we also give some other interesting results on $F^{p+1}$ for $F$ being a path, a cycle, or a complete graph. To obtain the aforementioned spectral results, we utilize a combination of the spectral version of the Stability Lemma and structural analyses. These approaches and tools give a new exploration of spectral extremal problems on non-bipartite graphs

Spectral extremal problem on tt copies of ℓ\ell-cycle

Denote by $tC_\ell$ the disjoint union of $t$ cycles of length $\ell$. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and spectral radius over all $n$-vertex $F$-free graphs, respectively. In this paper, we shall pay attention to the study of both $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any integers $t,\ell$ and $n\ge f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This generalizes the result on $ex(n,tC_3)$ of Erd\H{o}s [Arch. Math. 13 (1962) 222--227] as well as the research on $ex(n,C_{2\ell+1})$ of F\"{u}redi and Gunderson [Combin. Probab. Comput. 24 (2015) 641--645]. On the other hand, we focus on the spectral Tur\'{a}n-type function $spex(n,tC_{\ell})$, and determine the extremal graph for any fixed $t,\ell$ and large enough $n$. Our results not only extend some classic spectral extremal results on triangles, quadrilaterals and general odd cycles due to Nikiforov, but also develop the famous spectral even cycle conjecture proposed by Nikiforov (2010) and confirmed by Cioab\u{a}, Desai and Tait (2022).Comment: 25 pages, one figur