Related papers: A general theorem in spectral extremal graph theor…

For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by {ex}$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining {ex}$(K_n,F)$ for a given graph $F$ is a…

Let $\mc{F}$ be a family of graphs. A graph is {\em $\mc{F}$-free} if it contains no copy of a graph in $\mc{F}$ as a subgraph. A cornerstone of extremal graph theory is the study of the {\em Tur\'an number} $ex(n,\mc{F})$, the maximum…

Tait and Tobin [J. Combin. Theory Ser. B 126 (2017) 137--161] determined the unique spectral extremal graph over all outerplanar graphs and the unique spectral extremal graph over all planar graphs when the number of vertices is…

Let $G$ be a simple connected graph of order $n$ and $\partial(G)$ is the spectral radius of the distance matrix $D(G)$ of $G$. The transmission $D_i$ of vertex $i$ is the $i$-th row sum of $D(G)$. Denote by $D_{\max}(G)$ the maximum of…

For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $\delta(H)>0$ and $\Delta(H)$ denote the minimum degree and maximum degree of $H$,…

Let $\mathscr{G}_{n,\beta}$ be the set of graphs of order $n$ with given matching number $\beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue…

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called…

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 a family of $k$-hypergraphs $\mathcal{F}$, $ex(n,\mathcal{F})$ is the maximum number of edges a $k$-hypergraph can have, knowing that said hypergraph has $n$ vertices but contains no copy of any hypergraph from $\mathcal{F}$ as a…

In this paper, we present a sharp upper bound for the spectral radius of an $n$-vertex graph without $F$-minor for sufficient large $n$, where $F$ is obtained from the complete graph $K_r$ by deleting disjointed paths. Furthermore, the…

A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle-free subgraph of $K_n$ is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large $p$, every maximum…

Let $G$ be a $k$-degenerate graph of order $n.$ It is well-known that $G\ $has no more edges than $S_{n,k},$ the join of a complete graph of order $k$ and an independent set of order $n-k.$ In this note it is shown that $S_{n,k}$ is…

One of the earliest results in extremal graph theory, Mantel's theorem, states that the maximum number of edges in a triangle-free graph $G$ on $n$ vertices is $\lfloor n^2/4 \rfloor$. We investigate how this extremal bound is affected when…

In this paper, we study the maximum adjacency spectral radii of graphs of large order that do not contain an even cycle of given length. For $n>k$, let $S_{n,k}$ be the join of a clique on $k$ vertices with an independent set of $n-k$…

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-…

We study the problem of embedding bipartite graphs in Ahlfors-David regular sets of large dimension using results from extremal graph theory. Our main theorem states that any graph satisfying a power-improving bound on the extremal number…

The extremal number of a graph $H$, denoted by $\mbox{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices that does not contain $H$. The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem says that for a complete bipartite graph…

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real $\alpha\in[0,1]$. The largest eigenvalue of $A_\alpha(G)$ is…

Given a hypergraph $\mathcal{H}$ and a graph $G$, we say that $\mathcal{H}$ is a \textit{Berge}-$G$ if there is a bijection between the hyperedges of $\mathcal{H}$ and the edges of $G$ such that each hyperedge contains its image. We denote…

The binding number $b(G)$ of a graph, introduced by Woodall [J. Combin. Theory, Ser. B, 1973], is a central topic of both structural and extremal graph theory. It is closely related to fundamental combinatorial and structural properties of…