Related papers: A sharp spectral extremal result for general non-b…

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge…

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The Tur\'an number $ex(n, \mathscr{F})$ is the maximum number of edges in an $n$-vertex…

Erd\H{o}s proved an upper bound on the number of edges in an $n$-vertex non-Hamiltonian graph with given minimum degree and showed sharpness via two members of a particular graph family. F\"{u}redi, Kostochka and Luo showed that these two…

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Tur\'{a}n graph, which is the complete $r$-partite graph on $n$ vertices…

Given an $r$-graph $F$ with $r \ge 2$, let $\mathrm{ex}(n, (t+1) F)$ denote the maximum number of edges in an $n$-vertex $r$-graph with at most $t$ pairwise vertex-disjoint copies of $F$. Extending several old results and complementing…

For an odd integer $k$, let $\mathcal{C}_k = \{C_3,C_5,...,C_k\}$ denote the family of all odd cycles of length at most $k$ and let $\mathcal{C}$ denote the family of all odd cycles. Erd\H{o}s and Simonovits \cite{ESi1} conjectured that for…

In this paper, we characterize the extremal digraphs with the maximal or minimal $\alpha$-spectral radius among some digraph classes such as rose digraphs, generalized theta digraphs and tri-ring digraphs with given size $m$. These digraph…

Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(\Delta,\ldots,\Delta,\delta)$ with…

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\mathcal{G}(m, F)$ denote the family of $F$-free graphs with $m$ edges and without isolated vertices. Let $S_{n,k}$ denote the graph obtained by joining every vertex of…

The spread of a graph $G$ is the difference between the largest and smallest eigenvalues of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{2,t}$-minor. We show that for any $t\geq 2$, there…

Given a family $\mathcal{F}$, a graph is $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. We study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213--230], that is,…

Given a graph $F$, we define $\operatorname{ex}(G_{n,p},F)$ to be the maximum number of edges in an $F$-free subgraph of the random graph $G_{n,p}$. Very little is known about $\operatorname{ex}(G_{n,p},F)$ when $F$ is bipartite, with…

Let $F$ be a graph. A hypergraph is called Berge $F$ if it can be obtained by replacing each edge in $F$ by a hyperedge containing it. Given a family of graphs $\mathcal{F}$, we say that a hypergraph $H$ is Berge $\mathcal{F}$-free if for…

For two integers $r\geq 2$ and $h\geq 0$, the $h$-extra $r$-component connectivity of a graph $G$, denoted by $c\kappa_{r}^{h}$, is defined as the minimum number of vertices whose removal produces a disconnected graph with at least $r$…

Confirming a conjecture posed by Caro, it was shown by Chen and Yu that every graph $G$ with $n$ vertices and at most $2n-4$ edges has a stable cutset, which is a stable set of vertices whose removal disconnects the graph. Le and Pfender…

We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question…

For every integer $k \ge 3$, we determine the extremal structure of an $n$-vertex graph with at most $t$ vertex-disjoint copies of $C_{2k}$ when $n$ is sufficiently large and $t$ lies in the interval…

The generalized Tur\'{a}n number $ex(n,K_s,H)$ is the maximum number of complete graph $K_s$ in an $H$-free graph on $n$ vertices. Let $F_k$ be the friendship graph consisting of $k$ triangles. Erd\H{o}s and S\'os (1976) determined the…

The generalized Tur\'an number $\ex(n,K_s,F)$ denotes the maximum number of copies of $K_s$ in an $n$-vertex $F$-free graph. Let $kF$ denote $k$ disjoint copies of $F$. Gerbner, Methuku and Vizer [DM, 2019, 3130-3141] gave a lower bound for…

Let ${\cal F}$ be a family of graphs. For a graph $G$, the {\em ${\cal F}$-packing number}, denoted $\nu_{{\cal F}}(G)$, is the maximum number of pairwise edge-disjoint elements of ${\cal F}$ in $G$. A function $\psi$ from the set of…