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Recently, Ning and Zhai (2023) proved that every $n$-vertex graph $G$ with $\lambda (G) \ge \sqrt{\lfloor n^2/4\rfloor}$ has at least $\lfloor n/2\rfloor -1$ triangles, unless $G=K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}$. The aim of this paper is two-fold. Using the supersaturation-stability method, we prove a stability variant of Ning-Zhai's result by showing that such a graph $G$ contains at least $n-3$ triangles if no vertex is in all triangles of $G$. This result could also be viewed as a spectral version of a result of Xiao and Katona (2021). The second part concerns with the spectral supersaturation for the bowtie, which consists of two triangles sharing a common vertex. A theorem of Erd\H{o}s, F\"{u}redi, Gould and Gunderson (1995) says that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor +1$ edges contains a bowtie. For graphs of given order, the spectral supersaturation problem has not been considered for substructures that are not color-critical. In this paper, we give the first such theorem by counting the number of bowties. Let $K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}^{+2}$ be the graph obtained from $K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}$ by embedding two disjoint edges into the vertex part of size $\lceil \frac{n}{2} \rceil$. Our result shows that every graph $G$ with $n\ge 8.8 \times 10^6$ vertices and $\lambda (G)\ge \lambda (K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}^{+2})$ contains at least $\lfloor \frac{n}{2} \rfloor$ bowties, and $K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}^{+2}$ is the unique spectral extremal graph. This gives a spectral correspondence of a theorem of Kang, Makai and Pikhurko (2020). The method used in our paper provides a probable way to establish the spectral counting results for other graphs, even for non-color-critical graphs., Comment: 32 pages. Some spectral extremal graph problems are proposed in the end. Any suggestions are welcome. arXiv admin note: text overlap with arXiv:2406.13176

A well-known theorem of Mantel states that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor $ edges contains a triangle. An interesting problem in extremal graph theory studies the minimum number of edges contained in triangles among graphs with a prescribed number of vertices and edges. Erd\H{o}s, Faudree and Rousseau (1992) showed that a graph on $n$ vertices with more than $\lfloor n^2/4\rfloor $ edges contains at least $2\lfloor n/2\rfloor +1$ edges in triangles. Such edges are called triangular edges. In this paper, we present a spectral version of the result of Erd\H{o}s, Faudree and Rousseau. Using the supersaturation-stability and the spectral technique, we prove that every $n$-vertex graph $G$ with $\lambda (G) \ge \sqrt{\lfloor n^2/4\rfloor}$ contains at least $2 \lfloor {n}/{2} \rfloor -1$ triangular edges, unless $G$ is a balanced complete bipartite graph. The method in our paper has some interesting applications. Firstly, the supersaturation-stability can be used to revisit a conjecture of Erd\H{o}s concerning with the booksize of a graph, which was initially proved by Edwards (unpublished), and independently by Khad\v{z}iivanov and Nikiforov (1979). Secondly, our method can improve the bound on the order $n$ of a graph by dropping the condition on $n$ being sufficiently large, which is obtained from the triangle removal lemma. Thirdly, the supersaturation-stability can be applied to deal with the spectral extremal graph problems on counting triangles, which was recently studied by Ning and Zhai (2023)., Comment: 30 pages. At the end of the paper, we proposed many spectral extremal graph problems for readers. Any comments are welcome

A classical result of Erd\H{o}s and Rademacher (1955) indicates a supersaturation phenomenon. It says that if $G$ is a graph on $n$ vertices with at least $\lfloor {n^2}/{4} \rfloor +1$ edges, then $G$ contains at least $\lfloor {n}/{2}\rfloor$ triangles. We prove a spectral version of Erd\H{o}s--Rademacher's theorem. Moreover, Mubayi [Adv. Math. 225 (2010)] extends the result of Erd\H{o}s and Rademacher from a triangle to any color-critical graph. It is interesting to study the extension of Mubayi from a spectral perspective. However, it is not apparent to measure the increment on the spectral radius of a graph comparing to the traditional edge version (Mubayi's result). In this paper, we provide a way to measure the increment on the spectral radius of a graph and propose a spectral version on the counting problems for color-critical graphs., Comment: 27 pages, 5 figures. Any comments and suggestions are welcome

A well-known result of Nosal states that a graph $G$ with $m$ edges and $\lambda(G) > \sqrt{m}$ contains a triangle. Nikiforov [Combin. Probab. Comput. 11 (2002)] extended this result to cliques by showing that if $\lambda (G) > \sqrt{2m(1-1/r)}$, then $G$ contains a copy of $K_{r+1}$. Let $C_k^+$ be the graph obtained from a cycle $C_k$ by adding an edge to two vertices with distance two, and let $F_k$ be the friendship graph consisting of $k$ triangles that share a common vertex. Recently, Zhai, Lin and Shu [European J. Combin. 95 (2021)], Sun, Li and Wei [Discrete Math. 346 (2023)], and Li, Lu and Peng [Discrete Math. 346 (2023)] proved that if $ m\ge 8$ and $\lambda (G) \ge \frac{1}{2} (1+\sqrt{4m-3})$, then $G$ contains a copy of $C_5,C_5^+$ and $F_2$, respectively, unless $G=K_2\vee \frac{m-1}{2}K_1$. In this paper, we give a unified extension by showing that such a graph contains a copy of $V_5$, where $V_5=K_1\vee P_4$ is the join of a vertex and a path on four vertices. Our result extends the aforementioned results since $C_5,C_5^+$ and $F_2$ are proper subgraphs of $V_5$. In addition, we prove that if $m\ge 33$ and $\lambda (G) \ge 1+ \sqrt{m-2}$, then $G$ contains a copy of $F_3$, unless $G=K_3\vee \frac{m-3}{3}K_1$. This confirms a conjecture on the friendship graph $F_k$ in the case $k=3$. Finally, we conclude some spectral extremal graph problems concerning the large fan graphs and wheel graphs., Comment: 21 pages,2 figures

An $(n, k_1, \dots, k_t)$-cross intersecting system is a set of non-empty pairwise cross-intersecting families $\mathcal{F}_1\subset{[n]\choose k_1}, \mathcal{F}_2\subset{[n]\choose k_2}, \dots, \mathcal{F}_t\subset{[n]\choose k_t}$ with $t\geq 2$ and $k_1\geq k_2\geq \cdots \geq k_t$. If an $(n, k_1, \dots, k_t)$-cross intersecting system contains at least two families which are cross intersecting freely and at least two families which are cross intersecting but not freely, then we say that the cross intersecting system is of mixed type. All previous studies are on non-mixed type, i.e, under the condition that $n \ge k_1+k_2$. In this paper, we study for the first interesting mixed type, an $(n, k_1, \dots, k_t)$-cross intersecting system with $k_1+k_3\leq n

Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. We call $t$ families $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ pairwise cross-intersecting families if $\mathcal{A}_i$ and $\mathcal{A}_j$ are cross-intersecting when $1\le i

Erd\H{o}s and Simonovits asked the following question: For an integer $r\geq 2$ and a family of non-bipartite graphs $\mathcal{H}$, what is the tight bound of $\alpha$ such that any $\mathcal{H}$-free $n$-vertex graph with minimum degree at least $\alpha n$ has chromatic number at most $r$? We answer this question for $r=2$ and any family of odd cycles. Let $l\le k$ and $n\ge 1000k^{8}$ be positive integers. Let ${\mathcal C}$ be a family of odd cycles, $C_{2l+1}$ be the shortest odd cycle not in ${\mathcal C}$, and $C_{2k+1}$ be the longest odd cycle in ${\mathcal C}$. Let $BC_{2l+1}(n)$ denote the graph obtained by taking $2l+1$ vertex-disjoint copies of $K_{\frac{n}{2(2l+1)},\frac{n}{2(2l+1)}}$ and selecting a vertex in each of them such that these vertices form a cycle of length $2l+1$. Let $C_{2k+3}({n \over 2k+3})$ denote the balanced blow up of $C_{2k+3}$ with $n$ vertices. Note that both $BC_{2l+1}(n)$ and $C_{2k+3}({n \over 2k+3})$ are $n$-vertex ${\mathcal C}$-free non-bipartite graphs. We show that if $G$ is an $n$-vertex ${\mathcal C}$-free graph with $\delta(G)>\max\{ \frac{n}{2(2l+1)}, \frac{2}{2k+3}n\}$, then $G$ is bipartite. The bound is tight evident by $BC_{2l+1}(n)$ and $C_{2k+3}({n \over 2k+3})$. Moreover, the only $n$-vertex ${\mathcal C}$-free non-bipartite graph with minimum degree $\max\{ \frac{n}{2(2l+1)}, \frac{2}{2k+3}n\}=\frac{n}{2(2l+1)}$ is $BC_{2l+1}(n)$, and the the only $n$-vertex ${\mathcal C}$-free non-bipartite graph with minimum degree $\max\{ \frac{n}{2(2l+1)}, \frac{2}{2k+3}n\}=\frac{2}{2k+3}n$ is $C_{2k+3}({n \over 2k+3})$. Our result also unifies stability results of Andr\'{a}sfai, Erd\H{o}s and S\'{o}s, H\"{a}ggkvist and Yuan and Peng for large $n$., Comment: 17 pages

For graphs $G_0$, $G_1$ and $G_2$, write $G_0\longmapsto(G_1, G_2)$ if each red-blue-edge-coloring of $G_0$ yields a red $G_1$ or a blue $G_2$. The Ramsey number $r(G_1, G_2)$ is the minimum number $n$ such that the complete graph $K_n\longmapsto(G_1, G_2)$. In [Discrete Math. 312(2012)], Schelp formulated the following question: for which graphs $H$ there is a constant $0c|V(G)|$, $G\longmapsto(H, H)$. In this paper, we prove that for any $m>n$, if $G$ is a balanced bipartite graph of order $2(m+n-1)$ with $\delta(G)>\frac{3}{4}(m+n-1)$, then $G\longmapsto(CM_m, CM_n)$, where $CM_i$ is a matching with $i$ edges contained in a connected component. By Szem\'{e}redi's Regularity Lemma, using a similar idea as introduced by [J. Combin. Theory Ser. B 75(1999)], we show that for every $\eta>0$, there is an integer $N_0>0$ such that for any $N>N_0$ the following holds: Let $\alpha_1>\alpha_2>0$ such that $\alpha_1+\alpha_2=1$. Let $G[X, Y]$ be a balanced bipartite graph on $2(N-1)$ vertices with $\delta(G)\geq(\frac{3}{4}+3\eta)(N-1)$. Then for each red-blue-edge-coloring of $G$, either there exist red even cycles of each length in $\{4, 6, 8, \ldots, (2-3\eta^2)\alpha_1N\}$, or there exist blue even cycles of each length in $\{4, 6, 8, \ldots, (2-3\eta^2)\alpha_2N\}$. Furthermore, the bound $\delta(G)\geq(\frac{3}{4}+3\eta)(N-1)$ is asymptotically tight. Previous studies on Schelp's question on cycles are on diagonal case, we obtain an asymptotic result of Schelp's question for all non-diagonal cases.

A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$, where the equality holds if and only if $G$ is a complete bipartite graph. A well-known spectral conjecture of Bollob\'{a}s and Nikiforov [J. Combin. Theory Ser. B 97 (2007)] asserts that if $G$ is a $K_{r+1}$-free graph with $m$ edges, then $\lambda_1^2(G) + \lambda_2^2(G) \le (1-\frac{1}{r})2m$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] confirmed the conjecture in the case $r=2$. Using this base case, they proved further that $\lambda (G)\le \sqrt{m-1}$ for every non-bipartite triangle-free graph $G$, with equality if and only if $m=5$ and $G=C_5$. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented an improvement by showing $\lambda (G) \le \beta (m)$, where $\beta(m)$ is the largest root of $Z(x):=x^3-x^2-(m-2)x+m-3$. The equality in Zhai--Shu's result holds only if $m$ is odd and $G$ is obtained from the complete bipartite graph $K_{2,\frac{m-1}{2}}$ by subdividing exactly one edge. Motivated by this observation, Zhai and Shu proposed a question to find a sharp bound when $m$ is even. We shall solve this question by using a different method and characterize three kinds of spectral extremal graphs over all triangle-free non-bipartite graphs with even size. Our proof technique is mainly based on applying Cauchy interlacing theorem of eigenvalues of a graph, and with the aid of a triangle counting lemma in terms of both eigenvalues and the size of a graph., Comment: 28 pages. Following reviewer's suggestion, we have changed the original title. arXiv admin note: text overlap with arXiv:2204.09884

Let $F_k$ be the (friendship) graph obtained from $k$ triangles by sharing a common vertex. The $F_k$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioab\u{a}, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)] under the condition that $n$ is sufficiently large. In this paper, we get rid of the condition on $n$ being sufficiently large if $k=2$. The graph $F_2$ is also known as the bowtie. We show that the unique $n$-vertex $F_2$-free spectral extremal graph is the balanced complete bipartite graph adding an edge in the vertex part with smaller size if $n\ge 7$, and the condition $n\ge 7$ is tight. Our result is a spectral generalization of a theorem of Erd\H{o}s, F\"{u}redi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995)], which states that $\mathrm{ex}(n,F_2)=\left\lfloor {n^2}/{4} \right\rfloor +1$. Moreover, we study the spectral extremal problem for $F_k$-free graphs with given number of edges. In particular, we show that the unique $m$-edge $F_2$-free spectral extremal graph is the join of $K_2$ with an independent set of $\frac{m-1}{2}$ vertices if $m\ge 8$, and the condition $m\ge 8$ is tight., Comment: 22 pages, 6 figures. This is the published version

The Lagrangian density of an $r$-uniform hypergraph $H$ is $r!$ multiplying the supremum of the Lagrangians of all $H$-free $r$-uniform hypergraphs. For an $r$-uniform graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$. We say that an $r$-uniform hypergraph $H$ with $t$ vertices is $\lambda$-perfect if $\pi_{\lambda}(H)= r!\lambda{(K_{t-1}^r)}$. A theorem of Motzkin and Straus implies that all $2$-uniform graphs are $\lambda$-perfect. It is interesting to understand what kind of hypergraphs are $\lambda$-perfect. The property `$\lambda$-perfect' is monotone in the sense that an $r$-graph obtained by removing an edge from a $\lambda$-perfect $r$-graph (keep the same vertex set) is $\lambda$-perfect. It's interesting to understand the relation between the number of edges in a hypergraph and the `$\lambda$-perfect' property. We propose that the number of edges in a hypergraph no more than the number of edges in a linear hyperpath would guarantee the `$\lambda$-perfect' property. We show some partial result to support this conjecture. We also give some partial result to support the conjecture that the disjoint union of two $\lambda$-perfect $r$-uniform hypergraph is $\lambda$-perfect. We show that the disjoint union of a $\lambda$-perfect $3$-graph and $S_{2,t}=\{123,124,125,126,...,12(t+2)\}$ is perfect. This result implies the earlier result of Heftz and Keevash, Jiang, Peng and Wu, and several other earlier results., Comment: arXiv admin note: text overlap with arXiv:1810.13077, arXiv:2112.14935

The celebrated Erd\H{o}s-Ko-Rado theorem \cite{EKR1961} states that the maximum intersecting $k$-uniform family on $[n]$ is a full star if $n\ge 2k+1$. Furthermore, Hilton-Milner \cite{HM1967} showed that if an intersecting $k$-uniform family on $[n]$ is not a subfamily of a full star, then its maximum size achieves only on a family isomorphic to $HM(n,k):= \Bigl\{G\in {[n] \choose k}: 1\in G, G\cap [2,k+1] \neq \emptyset \Bigr\} \cup \Bigl\{ [2,k+1] \Bigr\} $ if $n>2k$ and $k\ge 4$, and there is one more possibility in the case of $k=3$. Han and Kohayakawa \cite{HK2017} determined the maximum intersecting $k$-uniform family on $[n]$ which is neither a subfamily of a full star nor a subfamily of the extremal family in Hilton-Milner theorm, and they asked what is the next maximum intersecting $k$-uniform family on $[n]$. Kostochka and Mubayi \cite{KM2016} gave the answer for large enough $n$. In this paper, we are going to get rid of the requirement that $n$ is large enough in the result by Kostochka and Mubayi \cite{KM2016} and answer the question of Han and Kohayakawa \cite{HK2017}., Comment: 28 pages, 1 figure

It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for non-bipartite triangle-free graphs. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented a further improvement. In this paper, we present an alternative method for proving the improvement by Zhai and Shu. Furthermore, the method can allow us to give a refinement on the result of Zhai and Shu for non-bipartite graphs without short odd cycles., Comment: 27 pages, 6 figures. We would like to express sincere thanks to Huiqiu Lin, Bo Ning and Mingqing Zhai for kind discussions, which considerably improves the presentation of the manuscript. The present work can be viewed as the second paper of our previous project arXiv:2204.09194. Any comments and suggestions are welcome

A well-known result in extremal spectral graph theory, due to Nosal and Nikiforov, states that if $G$ is a triangle-free graph on $n$ vertices, then $\lambda (G) \le \lambda (K_{\lfloor \frac{n}{2}\rfloor, \lceil \frac{n}{2} \rceil })$, equality holds if and only if $G=K_{\lfloor \frac{n}{2}\rfloor, \lceil \frac{n}{2} \rceil }$. Nikiforov [Linear Algebra Appl. 427 (2007)] extended this result to $K_{r+1}$-free graphs for every integer $r\ge 2$. This is known as the spectral Tur\'{a}n theorem. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a refinement on this result for non-bipartite triangle-free graphs. In this paper, we provide alternative proofs for the result of Nikiforov and the result of Lin, Ning and Wu. Our proof can allow us to extend the later result to non-$r$-partite $K_{r+1}$-free graphs. Our result refines the theorem of Nikiforov and it also can be viewed as a spectral version of a theorem of Brouwer., Comment: 26 pages, 6 figures. Any comments and suggestions are welcome. This article was completed during a quarantine period due to the COVID-19 pandemic. The authors would like to express their sincere gratitude to all of the volunteers and medical staffs for their kind help and support, which makes our daily life more and more secure

Both the Simonovits stability theorem and the Nikiforov spectral stability theorem are powerful tools for solving exact values of Tur\'{a}n numbers in extremal graph theory. Recently, F\"{u}redi [J. Combin. Theory Ser. B 115 (2015)] provided a concise and contemporary proof of the Simonovits stability theorem. In this note, we present a unified treatment for some extremal graph problems, including short proofs of Nikiforov's spectral stability theorem and the clique stability theorem proved recently by Ma and Qiu [European J. Combin. 84 (2020)]. Moreover, some spectral extremal problems related to the $p$-spectral radius and signless Laplacian radius are also included., Comment: 25 pages. At the end of the paper, we proposed an open problem. Any suggestions and comments are welcome! The first author would like to thank Prof. Lihua Feng, who introduced and encouraged him to the study of fascinating spectral graph theory. Thanks also go to Prof. Vladimir Nikiforov for valuable comments and for pointing out references [51,22]

Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any coloring of the edges of $K_N$ in red or blue yields a red $G$ or a blue $H$. Let $v(G)$ be the number of vertices of $G$ and $\chi(G)$ be the chromatic number of $G$. Let $s(G)$ denote the chromatic surplus of $G$, the cardinality of a minimum color class taken over all proper colorings of $G$ with $\chi(G)$ colors. Burr showed that for a connected graph $G$ and a graph $H$ with $v(G)\geq s(H)$, $R(G,H) \geq (v(G)-1)(\chi(H)-1)+s(H)$. A connected graph $G$ is called $H$-good if $R(G,H)=(v(G)-1)(\chi(H)-1)+s(H)$. In this paper, we mainly confirm the Ramsey number for any tree $T_n$ versus $K_m\cup K_l$. Our result yields that $T_n$ is $K_m\cup K_l$-good.

Given bipartite graphs $H_1$, \dots , $H_k$, the bipartite Ramsey number $br(H_1,\dots, H_k)$ is the minimum integer $N$ such that any $k$-edge-coloring of complete bipartite graph $K_{N, N}$ contains a monochromatic $H_i$ in color $i$ for $1\le i\le k$. There are considerable results on asymptotic values of bipartite Ramsey numbers of cycles. For exact value, Zhang-Sun \cite{Zhangs} determined $br(C_4, C_{2n})$, Zhang-Sun-Wu \cite{Zhangsw} determined $br(C_6, C_{2n})$, and Gholami-Rowshan \cite{GR} determined $br(C_8, C_{2n})$. In this paper, we solve all remaining cases and give the exact values of $br(C_{2n}, C_{2m})$ for all $n\ge m\ge 5$, this answers a question concerned by Buci\'c-Letzter-Sudakov \cite{BLS}, Gholami-Rowshan \cite{GR}, Zhang-Sun \cite{Zhangs}, and Zhang-Sun-Wu \cite{Zhangsw}.

We say that a graph $F$ can be embedded into a graph $G$ if $G$ contains an isomorphic copy of $F$ as a subgraph. Guo and Volkmann \cite{GV} conjectured that if $G$ is a connected graph with at least $n$ vertices and minimum degree at least $n-3$, then any tree with $n$ vertices and maximum degree at most $n-4$ can be embedded into $G$. In this paper, we give a result slightly stronger than this conjecture and obtain a sufficient and necessary condition that a tree with $n$ vertices and maximum degree at most $n-3$ can be embedded into a connected graph G with at least $n$ vertices and minimum degree at least $n-3$. Our result implies that the conjecture of Guo and Volkmann is true with one exception. We also give an application to the Ramsey number of a tree versus a star.

A real number $\alpha\in [0, 1)$ is a jump for an integer $r\ge 2$ if there exists $c>0$ such that no number in $(\alpha , \alpha + c)$ can be the Tur\'an density of a family of $r$-uniform graphs. A classical result of Erd\H os and Stone \cite{ES} implies that that every number in $[0, 1)$ is a jump for $r=2$. Erd\H os \cite{E64} also showed that every number in $[0, r!/r^r)$ is a jump for $r\ge 3$ and asked whether every number in $[0, 1)$ is a jump for $r\ge 3$. Frankl and R\"odl \cite{FR84} gave a negative answer by showing a sequence of non-jumps for every $r\ge 3$. After this, Erd\H os modified the question to be whether $\frac{r!}{r^r}$ is a jump for $r\ge 3$? What's the smallest non-jump? Frankl, Peng, R\"odl and Talbot \cite{FPRT} showed that ${5r!\over 2r^r}$ is a non-jump for $r\ge 3$. Baber and Talbot \cite{BT0} showed that every $\alpha\in[0.2299, 0.2316)\cup [0.2871, \frac{8}{27})$ is a jump for $r=3$. Pikhurko \cite{Pikhurko2} showed that the set of all possible Tur\'an densities of $r$-uniform graphs has cardinality of the continuum for $r\ge 3$. However, whether $\frac{r!}{r^r}$ is a jump for $r\ge 3$ remains open, and $\frac{5r!}{2r^r}$ has remained the known smallest non-jump for $r\ge 3$. In this paper, we give a smaller non-jump by showing that ${54r!\over 25r^r}$ is a non-jump for $r\ge 3$. Furthermore, we give infinitely many irrational non-jumps for every $r\ge 3$.

The {\em Tur\'an 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 {\em Tur\'{a}n density} of $F$ is defined as $\pi(F)=\underset{{n\rightarrow\infty}}{\lim}{ex(n,F) \over {n \choose r }}.$ For graphs, Erd\H{o}s-Stone-Simonovits (\cite{ESi}, \cite{ES}) showed that $\Pi_{\infty}^{(2)}=\Pi_{fin}^{(2)}=\Pi_{1}^{(2)}=\{0, {1 \over 2}, {2 \over 3}, \ldots,{l-1 \over l}, ...\}.$ We know quite few about the Tur\'an density of an $r$-uniform graph for $r\ge 3$. Baber and Talbot \cite{BT}, and Pikhurko \cite{Pikhurko2} showed that there is an irrational number in $\Pi_{3}^{(3)}$ and $\Pi_{fin}^{(3)}$ respectively, disproving a conjecture of Chung and Graham \cite{FG}. Baber and Talbot \cite{BT} asked whether $\Pi_{1}^{(r)}$ contains an irrational number. 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 ${\sqrt 3\over 3}$, consequently, the Tur\'an density of the extension of $F$ is an irrational number, answering the question of Baber and Talbot.

We present an Oppenheim type determinantal inequality for positive definite block matrices. Recently, Lin [Linear Algebra Appl. 452 (2014) 1--6] proved a remarkable extension of Oppenheim type inequality for block matrices, which solved a conjecture of G\"{u}nther and Klotz. There is a requirement that two matrices commute in Lin's result. The motivation of this paper is to obtain another natural and general extension of Oppenheim type inequality for block matrices to get rid of the requirement that two matrices commute., Comment: 12 pages. Any comments and suggestions are welcome. E-mail addresses: ytli0921@hnu.edu.cn (Yongtao Li), ypeng1@hnu.edu.cn (Yuejian Peng, corresponding author). Linear and Multilinear Algebra (2021). Linear and Multilinear Algebra, 2021

A graph $G$ is $k$-edge-Hamiltonian if any collection of vertex-disjoint paths with at most $k$ edges altogether belong to a Hamiltonian cycle in $G$. A graph $G$ is $k$-Hamiltonian if for all $S\subseteq V(G)$ with $|S|\le k$, the subgraph induced by $V(G)\setminus S$ has a Hamiltonian cycle. These two concepts are classical extensions for the usual Hamiltonian graphs. In this paper, we present some spectral sufficient conditions for a graph to be $k$-edge-Hamiltonian and $k$-Hamiltonian in terms of the adjacency spectral radius as well as the signless Laplacian spectral radius. Our results could be viewed as slight extensions of the recent theorems proved by Li and Ning [Linear Multilinear Algebra 64 (2016)], Nikiforov [Czechoslovak Math. J. 66 (2016)] and Li, Liu and Peng [Linear Multilinear Algebra 66 (2018)]. Moreover, we shall prove a stability result for graphs being $k$-Hamiltonian, which could be regarded as a complement of two recent results of F\"{u}redi, Kostochka and Luo [Discrete Math. 340 (2017)] and [Discrete Math. 342 (2019)]., Comment: 21 pages, 1 figure. Any comments and suggestions are welcome. E-mail addresses: ytli0921@hnu.edu.cn (Yongtao Li), ypeng1@hnu.edu.cn (Yuejian Peng, corresponding author). Linear and Multilinear Algebra, 2022

Let $H_{s,t_1,\ldots ,t_k}$ be the graph with $s$ triangles and $k$ odd cycles of lengths $t_1,\ldots ,t_k\ge 5$ intersecting in exactly one common vertex. Recently, Hou, Qiu and Liu [Discrete Math. 341 (2018) 126--137], and Yuan [J. Graph Theory 89 (1) (2018) 26--39] determined independently the maximum number of edges in an $n$-vertex graph that does not contain $H_{s,t_1,\ldots ,t_k}$ as a subgraph. In this paper, we determine the graphs of order $n$ that attain the maximum spectral radius among all graphs containing no $H_{s,t_1,\ldots ,t_k}$ for $n$ large enough., Comment: 25 pages. This is the Journal Version. The problem raised at the end of our paper was recently solved by Chen, Liu and Zhang; see arXiv:2108.03895. The extremal spectral problem involving the intersecting cliques was also solved in another paper; see the joint work arXiv:2108.03587v2. arXiv admin note: text overlap with arXiv:1911.13082 by other authors

For a fixed positive integer $n$ and an $r$-uniform hypergraph $H$, the Tur\'an number $ex(n,H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraph on $n$ vertices, and the Lagrangian density of $H$ is defined as $\pi_{\lambda}(H)=\sup \{r! \lambda(G) : G \;\text{is an}\; H\text{-free} \;r\text{-uniform hypergraph}\}$, where $\lambda(G)$ is the Lagrangian of $G$. For an $r$-uniform hypergraph $H$ on $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$. We say that an $r$-uniform hypergraph $H$ on $t$ vertices is perfect if $\pi_{\lambda}(H)= r!\lambda{(K_{t-1}^r)}$. Let $P_t=\{e_1, e_2, \dots, e_t\}$ be the linear $3$-uniform path of length $t$, that is, $|e_i|=3$, $|e_i \cap e_{i+1}|=1$ and $e_i \cap e_j=\emptyset$ if $|i-j|\ge 2$. We show that $P_3$ and $P_4$ are perfect, this supports a conjecture in \cite{yanpeng} proposing that all $3$-uniform linear hypergraphs are perfect. Applying the results on Lagrangian densities, we determine the Tur\'an numbers of their extensions., Comment: 17 pages, 6 figures. arXiv admin note: text overlap with arXiv:1609.08983; text overlap with arXiv:1510.03461 by other authors

The Lagrangian density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all $F$-free $r$-uniform hypergraphs. For an $r$-graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$. We say that an $r$-unform hypergraph $H$ with $t$ vertices is perfect if $\pi_{\lambda}(H)= r!\lambda{(K_{t-1}^r)}$. A theorem of Motzkin-Straus implies that all $2$-uniform graphs are perfect. It is interesting to explore what kind of hypergraphs are perfect. A hypergraph is linear if any 2 edges have at most 1 vertex in common. We propose the following conjecture: (1) For $r\ge 3$, there exists $n$ such that a linear $r$-unofrm hypergraph with at least $n$ vertices is perfect. (2) For $r\ge 3$, there exists $n$ such that if $G, H$ are perfect $r$-uniform hypergraphs with at least $n$ vertices, then $G\bigsqcup H$ is perfect. Regarding this conjecture, we obtain a partial result: Let $S_{2,t}=\{123,124,125,126,...,12(t+2)\}$. (An earlier result of Sidorenko states that $S_{2,t}$ is perfect \cite{Sidorenko-89}.) Let $H$ be a perfect $3$-graph with $s$ vertices. Then $F=S_{2,t}\bigsqcup H$ is perfect if $s\geq 3$ and $t\geq 3$.

Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such that for any red-blue-green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of at least size $l$ in a component, or a green matching of size at least $m$ in a component. In this paper, we determine the exact value of $r(k,l,m)$ completely. Applying a technique originated by {\L}uczak that applies Szemer\'edi's Regularity Lemma to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component, we obtain the 3-colored asymmetric bipartite Ramsey number of cycles asymptotically., Comment: arXiv admin note: text overlap with arXiv:1803.03689 by other authors

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\ldots,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N,N}$ contains a monochromatic subgraph isomorphic to $H_i$ in color $i$ for some $i$, $1\leq i\leq r$. We show that for $\alpha_1,\alpha_2>0$, $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor})=(\alpha_1+\alpha_2+o(1))n$. We also show that if $r\geq 3, \alpha_1,\alpha_2>0, \alpha_{j+2}\geq [(j+2)!-1]\sum^{j+1}_{i=1} \alpha_i$ for $j=1,2,\ldots,r-2$, then $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor},\ldots,C_{2\lfloor \alpha_r n\rfloor})=(\sum^r_{j=1} \alpha_j+o(1))n.$ For $\xi>0$ and sufficiently large $n$, let $G$ be a bipartite graph with bipartition $\{V_1,V_2\}$, $|V_1|=|V_2|=N$, where $N=(2+8\xi)n$. We prove that if $\delta(G)>(\frac{7}{8}+9\xi)N$, then any $2$-edge coloring of $G$ contains a monochromatic copy of $C_{2n}$., Comment: 19 pages

Frankl and F\"uredi conjectured in 1989 that the maximum Lagrangian of all $r$-uniform hypergraphs of fixed size $m$ is realized by the minimum hypergraph $C_{r,m}$ under the colexicographic order. In this paper, we prove a weaker version of the Frankl and F\"{u}redi's conjecture at $r=3$: there exists an absolute constant $c>0$ such that for any $3$-uniform hypergraph $H$ with $m$ edges, the Lagrangian of $H$ satisfies $\lambda(H)\leq \lambda(C_{3,m+cm^{2/9}})$. In particular, this result implies that the Frankl and F\"{u}redi's conjecture holds for $r=3$ and $m\in [{t-1\choose 3}, {t\choose 3}-(t-2)-ct^{\frac{2}{3}}]$. It improves a recent result of Tyomkyn., Comment: 9 page, 1 figure

An $r$-uniform graph $G$ is dense if and only if every proper subgraph $G'$ of $G$ satisfies $\lambda (G') < \lambda (G)$, where $\lambda (G)$ is the Lagrangian of a hypergraph $G$. In 1980's, Sidorenko showed that $\pi(F)$, the Tur\'an density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all dense $F$-hom-free $r$-uniform hypergraphs. This connection has been applied in estimating Tur\'an density of hypergraphs. When $r=2$, the result of Motzkin and Straus shows that a graph is dense if and only if it is a complete graph. However, when $r\ge 3$, it becomes much harder to estimate the Lagrangians of $r$-uniform hypergraphs and to characterize the structure of all dense $r$-uniform graphs. The main goal of this note is to give some sufficient conditions for $3$-uniform graphs with given substructures to be dense. For example, if $G$ is a $3$-graph with vertex set $[t]$ and $m$ edges containing $[t-1]^{(3)}$, then $G$ is dense if and only if $m \ge {t-1 \choose 3}+{t-2 \choose 2}+1$. We also give sufficient condition condition on the number of edges for a $3$-uniform hypergraph containing a large clique minus $1$ or $2$ edges to be dense., Comment: 23 pages

Let $S^r(n)$ be the $r$-graph on $n$ vertices with parts $A$ and $B$, where the edges consist of all $r$-tuples with $1$ vertex in $A$ and $r-1$ vertices in $B$, and the sizes of $A$ and $B$ are chosen to maximise the number of edges. Let $M_t^r$ be the $r$-graph with $t$ pairwise disjoint edges. Given an $r$-graph $F$ and a positive integer $p\geq |V(F)|$, we define the {\em extension} of $F$, denoted by $H_{p}^{F}$ as follows: Label the vertices of $F$ as $v_1,\dots,v_{|V(F)|}$. Add new vertices $v_{|V(F)|+1},\dots,v_{p}$. For each pair of vertices $v_i,v_j, 1\le i

Given a positive integer $n$ and an $r$-uniform hypergraph (or $r$-graph for short) $F$, the Turan number $ex(n,F)$ of $F$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain $F$ as a subgraph. The extension $H^F $ of $F$ is obtained as follows: For each pair of vertices $v_i,v_j$ in $F$ not contained in an edge of $F$, we add a set $B_{ij}$ of $r-2$ new vertices and the edge $\{v_i,v_j\} \cup B_{ij}$, where the $B_{ij}$ 's are pairwise disjoint over all such pairs $\{i,j\}$. Let $K^r_p$ denote the complete $r$-graph on $p$ vertices. For all sufficiently large $n$, we determine the Turan numbers of the extensions of a $3$-uniform $t$-matching, a $3$-uniform linear star of size $t$, and a $4$-uniform linear star of size $t$, respectively. We also show that the unique extremal hypergraphs are balanced blowups of $K^3_{3t-1}, K^3_{2t}$, and $K^4_{3t}$, respectively. Our results generalize the recent result of Hefetz and Keevash [7]., Comment: 21 pages

There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus in 1965. It is useful in practice if similar results hold for hypergraphs. However the obvious generalization of Motzkin and Straus' result to hypergraphs is false. Frankl and F\"{u}redi conjectured that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$ edges. For $r=2$, Motzkin and Straus' theorem confirms this conjecture. For $r=3$, it is shown by Talbot that this conjecture is true when $m$ is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for $3$-uniform hypergraphs. As an application of this connection, we confirm that Frankl and F\"{u}redi's conjecture holds for bigger ranges of $m$ when $r$=3. We also obtain two weaker versions of Tur\'{a}n type theorem for left-compressed $3$-uniform hypergraphs., Comment: 10pages. arXiv admin note: substantial text overlap with arXiv:1311.1062, arXiv:1311.1409, arXiv:1211.7056, arXiv:1211.7057

Motzkin and Straus established a close connection between the maximum clique problem and a solution (namely graph-Lagrangians) to the maximum value of a class of homogeneous quadratic multilinear functions over the standard simplex of the Euclidean space in 1965. This connection provides a new proof of Tur\'an's theorem. Recently, an extension of Motzkin-Straus theorem was proved for non-uniform hypergraphs whose edges contain 1 or 2 vertices in \cite{PPTZ}. It is interesting if similar results hold for other non-uniform hypergraphs. In this paper, we give some connection between polynomial programming and the clique of non-uniform hypergraphs whose edges contain 1, or 2, and more vertices. Specifically, we obtain some Motzkin-Straus type results in terms of the graph-Lagrangian of non-uniform hypergraphs whose edges contain 1, or 2, and more vertices., Comment: 10pages. arXiv admin note: text overlap with arXiv:1312.3034

In 1965, Motzkin and Straus established a remarkable connection between the order of a maximum clique and the Lagrangian of a graph and provided a new proof of Tur\'an's theorem using the connection. The connection of Lagrangians and Tur\'{a}n densities can be also used to prove the fundamental theorem of Erd\"{o}s-Stone-Simonovits on Tur\'{a}n densities of graphs. Very recently, the study of Tur\'{a}n densities of non-uniform hypergraphs have been motivated by extremal poset problems. In this paper, we attempt to explore the applications of Lagrangian method in determining Tur\'{a}n densities of non-uniform hypergraphs. We first give a definition of the Lagrangian of a non-uniform hypergraph, then give an extension of Motzkin-Straus theorem to non-uniform hypergraphs whose edges contain 1 or 2 vertices. Applying it, we give an extension of Erd\"{o}s-Stone-Simonovits theorem to non-uniform hypergraphs whose edges contain 1 or 2 vertices.

A remarkable connection between the order of a maximum clique and the Graph-Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extension were useful in both combinatorics and optimization. Since then, Graph-Lagrangian has been a useful tool in extremal combinatorics. In this paper, we give a parametrized Graph-Lagrangian for non-uniform hypergraphs and provide several Motzkin-Straus type results for nonuniform hypergraphs which generalize results from [1] and [2]. Another part of the paper concerns a long-standing conjecture of Frankl-F\"uredi on Graph-Lagrangians of hypergraphs. We show the connection between the Graph-Lagrangian of $\{1, r_1, r_2, \cdots, r_l\}$-hypergraphs and $\{ r_1, r_2, \cdots, r_l\}$-hypergraphs. Some of our results provide solutions to the maximum value of a class of polynomial functions over the standard simplex of the Euclidean space., Comment: 24 pages

Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we provide upper bounds on the Lagrangian of a hypergraph containing dense subgraphs when the number of edges of the hypergraph is in certain ranges. These results support a pair of conjectures introduced by Y. Peng and C. Zhao (2012) and extend a result of J. Talbot (2002). \keywords{Cliques of hypergraphs \and Colex ordering \and Lagrangians of hypergraphs \and Polynomial optimization}, Comment: 20pages 5 figures. arXiv admin note: text overlap with arXiv:1211.7056, arXiv:1211.6508, arXiv:1212.2795, arXiv:1311.1062

Frankl and F\"uredi (1989) conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest graph-Lagrangian of all $r$-graphs with $m$ edges. In this paper, we establish some bounds for graph-Lagrangians of some special $r$-graphs that support this conjecture., Comment: 9pages

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extensions were applied in Tur\'{a}n problems of graphs and uniform hypergraphs. Very recently, the study of Tur\'{a}n densities of non-uniform hypergraphs has been motivated by extremal poset problems. In this paper, we give some Motzkin-Straus type results for non-uniform hypergraphs., Comment: 13 pages

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in [7]. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It has been also applied in spectral graph theory. Estimating the Lagrangians of hypergraphs has been successfully applied in the course of studying the Turan densities of several hypergraphs as well. It is useful in practice if Motzkin-Straus type results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus' result to hypergraphs is false. We attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. In this paper, we give some Motzkin-Straus type results for r-uniform hypergraphs. These results generalize and refine a result of Talbot in [19] and a result in [11]., Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:1211.6508, arXiv:1211.7056, arXiv:1211.7057

In 1965, Motzkin and Straus [5] provided a new proof of Turan's theorem based on a continuous characterization of the clique number of a graph using the Lagrangian of a graph. This new proof aroused interests in the study of Lagrangians of r-uniform graphs. The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. Sidorenko and Frankl-Furedi applied Lagrangians of hypergraphs in finding Turan densities of hypergraphs. Frankl and Rodl applied it in disproving Erdos' jumping constant conjecture. In most applications, we need an upper bound for the Lagrangian of a hypergraph. Frankl and Furedi conjectured that the r-uniform graph with m edges formed by taking the first m sets in the colex ordering of $N^(r)$ has the largest Lagrangian of all r-uniform graphs with m edges. Talbot in [14] provided some evidences for Frankl and Furedi's conjecture. In this paper, we prove that the r-uniform graph with m edges formed by taking the first m sets in the colex ordering of $N^(r)$ has the largest Lagrangian of all r-uniform graphs on t vertices with m edges when ${t choose r}-3$ or ${t choose r}-4$. As an implication, we also prove that Frankl and Furedi's conjecture holds for 3-uniform graphs with ${t choose 3}-3$ or ${t choose 3}-4$ edges., Comment: 15 pages; 1 figure. arXiv admin note: substantial text overlap with arXiv:1211.6508, arXiv:1211.7056

The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Lagrangian of a hypergraph. Frankl and Furedi in \cite{FF} conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-graphs with $m$ edges. In this paper, we give some partial results for this conjecture., Comment: 19 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1211.6508

There is a remarkable connection between the maximum clique number and the Lagrangian of a graph given by T. S. Motzkin and E.G. Straus in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we explore evidences that the Lagrangian of a 3-uniform hypergraph is related to the order of its maximum cliques when the number of edges of the hypergraph is in certain range. In particular, we present some results about a conjecture introduced by Y. Peng and C. Zhao (2012) and describe a combinatorial algorithm that can be used to check the validity of the conjecture., Comment: 11 pages

Generalizing the concept of a perfect number, Sloane's sequences of integers A083207 lists the sequence of integers $n$ with the property: the positive factors of $n$ can be partitioned into two disjoint parts so that the sums of the two parts are equal. Following Clark et al., we shall call such integers, Zumkeller numbers. Generalizing this, Clark et al., call a number n a half-Zumkeller number if the positive proper factors of n can be partitioned into two disjoint parts so that the sums of the two parts are equal. An extensive study of properties of Zumkeller numbers, half-Zumkeller numbers and their relation to practical numbers is undertaken in this paper. Clark et al., announced results about Zumkellers numbers and half-Zumkeller numbers and suggested two conjectures. In the present paper we shall settle one of the conjectures, prove the second conjecture in some special cases and prove several results related to the second conjecture. We shall also show that if there is an even Zumkeller number that is not half-Zumkeller it should be bigger than 7233498900., Comment: Generalization of perfect numbers, reationship with practical numbers, contributions to the conjecture that even Zumkeller numbers are half-Zumkeller