相关论文: The Toughness of Kneser Graphs

Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes…

In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are input graphs with $n$ nodes and maximum…

The irregularity strength of a graph $G$, $s(G)$, is the least $k$ such that there exists a $\{1,2,\ldots,k\}$-weighting of the edges of $G$ attributing distinct weighted degrees to all vertices, or equivalently the least $k$ enabling…

Properties of graphs that can be characterized by the spectrum of the adjacency matrix of the graph have been studied systematically recently. Motivated by the complexity of these properties, we show that there are such properties for which…

Let $\Lambda(T)$ denote the set of leaves in a tree $T$. One natural problem is to look for a spanning tree $T$ of a given graph $G$ such that $\Lambda(T)$ is as large as possible. This problem is called maximum leaf number, and it is a…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…

For a fixed positive integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin proved that the maximum $\lambda_1(G)$ among all outerplanar graphs on $n$…

A recent upper bound by Le and Solomon [STOC '23] has established that every $n$-node graph has a $(1+\varepsilon)(2k-1)$-spanner with lightness $O(\varepsilon^{-1} n^{1/k})$. This bound is optimal up to its dependence on $\varepsilon$; the…

Brouwer's Conjecture states that, for any graph $G$, the sum of the $k$ largest (combinatorial) Laplacian eigenvalues of $G$ is at most $|E(G)| + \binom{k+1}{2}$, $1 \leq k \leq n$. We present several interrelated results establishing…

A connected graph $G$ with at least $2m + 2n + 2$ vertices which contains a perfect matching is $E(m, n)$-{\it extendable}, if for any two sets of disjoint independent edges $M$ and $N$ with $|M| = m$ and $|N|= n$, there is a perfect…

The toughness $\tau(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a cut set of vertices in}~G\}$ for $G\ncong K_n.$ The concept of toughness initially proposed by Chv$\mathrm{\acute{a}}$tal in 1973, which serves as a simple way to measure…

Bollob\'as and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $\lambda^2_1(G)+\lambda^2_2(G)\leq \frac{r-1}{r}\cdot2m$,…

Determining the size of a maximum independent set of a graph $G$, denoted by $\alpha(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $\alpha (G)$ for special classes of…

Random K-out graphs are used in several applications including modeling by sensor networks secured by the random pairwise key predistribution scheme, and payment channel networks. The random K-out graph with $n$ nodes is constructed as…

The $k$-token graph $F_k(G)$ of a graph $G$ on $n$ vertices is the graph whose vertices are the ${n\choose k}$ $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices…

The \emph{$k$-restricted edge-connectivity} of a graph $G$, denoted by $\lambda_k(G)$, is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least $k$ vertices. This graph…

In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the…

The toughness of a graph $G$ is defined as the largest real number $t$ such that for any set $S\subseteq V(G)$ such that $G-S$ is disconnected, $S$ has at least $t$ times more elements than $G-S$ has components (unless $G$ is complete, in…

In 1962, Erd\H{o}s proved that if a graph $G$ with $n$ vertices satisfies $$ e(G)>\max\left\{\binom{n-k}{2}+k^2,\binom{\lceil(n+1)/2\rceil}{2}+\left\lfloor \frac{n-1}{2}\right\rfloor^2\right\}, $$ where the minimum degree $\delta(G)\geq k$…

The Kneser graph ${\rm KG}_{n,k}$ is a graph whose vertex set is the family of all $k$-subsets of $[n]$ and two vertices are adjacent if their corresponding subsets are disjoint. The classical Erd\H{o}s-Ko-Rado theorem determines the…