Related papers: Spectral extremal graphs without intersecting tria…

Let $G$ be a simple graph of order $n$ with degree sequence $(d_1,d_2,\cdots,d_n)$. For an integer $p>1$, let $e_p(G)=\sum_{i=1}^n d^{p}_i$ and let $ex_p(n,H)$ be the maximum value of $e_p(G)$ among all graphs with $n$ vertices that do not…

A graph $G$ is called $H$-free, if it does not contain $H$ as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Tur\'{a}n type problem: what is the maximum spectral radius of an $H$-free graph of order $n$? In this paper, we…

The Tur\'an problem asks for the largest number of edges in an $n$-vertex graph not containing a fixed forbidden subgraph $F$. We construct a new family of graphs not containing $K_{s,t}$, for $t= C^s$, with $\Omega(n^{2-1/s})$ edges…

A graph is color-critical if it contains an edge whose deletion reduces its chromatic number. This class of graphs, including cliques and odd cycles, plays a central role in extremal graph theory. In this paper, following an influential…

It is known that the spectral radius of a digraph with k edges is \le \sqrt{k}, and that this inequality is strict except when k is a perfect square. For k=m^2 + \ell, \ell fixed, m large, Friedland showed that the optimal digraph is…

A graph is said to be \textit{$H$-minor free} if it does not contain $H$ as a minor. In this paper, we characteristic the unique extremal graph with maximal $Q$-index among all $n$-vertex $K_{1,t}$-minor free graphs ($t\ge3$).

In this paper, we prove that if $G$ is a $K_{2,t}$-minor free graph of order $n\geq t^2+4t+1$ with $t\geq 3$, the signless Laplacian spectral radius $q(G)\leq \frac{1}{2}(n+2t-2+\sqrt{(n-2t+2)^2+8t-8}\ )$ with equality if and only if…

This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs…

Graph theory on surfaces extends classical graph structures to topological surfaces, providing a theoretical foundation for characterizing the embedding properties of complex networks in constrained spaces. The study of bounding the…

A stability result due to Ren, Wang, Wang and Yang [SIAM J. Discrete Math. 38 (2024)] shows that if $3\le r \le 2k$ and $n\ge 318 (r-2)^2k$, and $G$ is a $C_{2k+1}$-free graph on $n$ vertices with $e(G)\ge \lfloor {(n-r+1)^2}/{4}\rfloor +{r…

Given graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality, the friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all bijections $\sigma:V(X)\to V(Y)$, where two bijections…

For every positive integer $t$ we construct a finite family of triple systems ${\mathcal M}_t$, determine its Tur\'{a}n number, and show that there are $t$ extremal ${\mathcal M}_t$-free configurations that are far from each other in…

A central theme in extremal combinatorics is the study of the maximum number of edges in an $r$-uniform hypergraph ($r$-graph) with matching number at most $s$ (the Erd\H{o}s Matching Conjecture) or with pairwise intersection at least $t$…

A set of cycles is called independent if no two of them have a common vertex. Let $S_{n, 2k-1}$ be the complete split graph, which is the join of a clique of size $2k-1$ with an independent set of size $n-2k+1$. In 1962, Erd\H{o}s and…

A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\geq0$ is an integer. Zhai and Lin [J. Graph Theory 102 (2023) 502-520] proved that for $n\geq\frac{13}{2}r$, if $G$ is a $B_{r+1}$-free graph of order $n$, then…

Let $t\geq3$ and $G$ be a graph of order $n,$ with no $K_{2,t}$ minor. If $n>400t^{6}$, then the spectral radius $\mu\left( G\right) $ satisfies \[ \mu\left( G\right) \leq\frac{t-1}{2}+\sqrt{n+\frac{t^{2}-2t-3}{4}}, \] with equality if and…

A connected nontrivial graph $G$ is {\it matching covered} if every edge of $G$ is contained in some perfect matching of $G$. A matching covered graph $G$ is {\it minimal} if $G-e$ is not matching covered for each edge $e$ of $G$. A graph…

Let $\rho(G)$ be the spectral radius of a graph $G$ with $m$ edges. Let $S_{m-k+1}^{k}$ be the graph obtained from $K_{1,m-k}$ by adding $k$ disjoint edges within its independent set. Nosal's theorem states that if $\rho(G)>\sqrt{m}$, then…

We resolve a problem posed by Guiduli (1996) on the spectral radius of graphs satisfying the Hereditarily Bounded Property $P_{t,r}$, which requires that every subgraph $H$ with $|V(H)| \geq t$ satisfies $|E(H)| \leq t|V(H)| + r$. For an…

We study the topic of "extremal" planar graphs, defining $\mathrm{ex_{_{\mathcal{P}}}}(n,H)$ to be the maximum number of edges possible in a planar graph on $n$ vertices that does not contain a given graph $H$ as a subgraph. In…