相关论文: On a generalization of the spectral Mantel's theor…

Spectral graph theory studies how the eigenvalues of a graph relate to the structural properties of a graph. In this paper, we solve three open problems in spectral extremal graph theory which generalize the classical Tur\'{a}n-type…

An $r$-graph is a triangle if there exists a positive integer $i \le \lceil r/2 \rceil$ such that it is isomorphic to the following $r$-graph with three edges: \begin{align*} \left\{\{1, \ldots, r\},~\{1, \ldots, i, r+1, \ldots,…

Erd\H{o}s-Stone Theorem is a well-known result in extremal graph theory which determines the asymptotic behaviour of maximum number of edges in an $n$-vertex $H$-free graph. In 2009, Nikiforov gave a spectral version of Erd\H{o}s-Stone…

We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph which does not contain $k+1$ vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in…

In this paper, we study the spectral radius of the clique tensor A(G) associated with a graph G. This tensor is a higher-order extensions of the adjacency matrix of G. A lower bound of the clique number is given via the spectral radius of…

Let $G(n,k)$ be the class of clique trees on $n$ vertices and zero forcing number $k$, where $\left \lfloor \frac{n}{2} \right \rfloor + 1 \le k \le n-1$ and each block is a clique of size at least $3$. In this article, we proved the…

Let $\cal H$ be a family of graphs. The Tur\'an number ${\rm ex}(n,{\cal H})$ is the maximum possible number of edges in an $n$-vertex graph which does not contain any member of $\cal H$ as a subgraph. As a common generalization of…

Let $G$ be a graph with $m$ edges and spectral radius $\lambda_{1}$. Let $bk\left( G\right) $ stand for the maximal number of triangles with a common edge in $G$. In 1970 Nosal proved that if $\lambda_{1}^{2}>m,$ then $G$ contains a…

In 1986, Brualdi and Solheid firstly proposed the problem of determining the maximum spectral radius of graphs in the set $\mathcal{H}_{n,m}$ consisting of all simple connected graphs with $n$ vertices and $m$ edges, which is a very tough…

For a set of graphs $\mathcal{F}$, let $\ex(n,\mathcal{F})$ and $\spex(n,\mathcal{F})$ denote the maximum number of edges and the maximum spectral radius of an $n$-vertex $\mathcal{F}$-free graph, respectively. Nikiforov ({\em LAA}, 2007)…

A sparse version of Mantel's Theorem is that, for sufficiently large $p$, with high probability (w.h.p.), every maximum triangle-free subgraph of $G(n,p)$ is bipartite. DeMarco and Kahn proved this for $p>K \sqrt{\log n/n}$ for some…

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…

The extremal graphs $\mathrm{EX}(n,\mathcal F)$ and spectral extremal graphs $\mathrm{SPEX}(n,\mathcal F)$ are the sets of graphs on $n$ vertices with maximum number of edges and maximum spectral radius, respectively, with no subgraph in…

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 $H$,…

The fundamental theorem of Tur\'{a}n from Extremal Graph Theory determines the exact bound on the number of edges $t_r(n)$ in an $n$-vertex graph that does not contain a clique of size $r+1$. We establish an interesting link between…

Erd\H{o}s and Szekeres's quantitative version of Ramsey's theorem asserts that any complete graph on n vertices that is edge-colored with two colors has a monochromatic clique on at least 1/2log(n) vertices. The famous Erd\H{o}s-Hajnal…

We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint $t$-cliques. The extremal graphs…

We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of $P_2$ and $P_{n-2}$ is the unique graph of maximum spectral radius over all planar…

The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $\Delta(G)$ the maximum degree of a graph $G$ and let $\rho(G)$ be the spectral radius of $G$. In this article we present a lower…

Let $G$ be an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges of $G$ and the number of colors appearing on $E(G)$, respectively. For a vertex $v\in V(G)$, the \emph{color neighborhood} of $v$ is defined as the set…