Related papers: Spectral radius and $[a,b]$-factors in graphs

A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers all vertices of $G$. The leaf distance of $T$ is defined as the minimum of distances between any two leaves of $T$. A fractional matching of a graph $G$…

A graph of order $n$ is said to be \emph{$k$-factor-critical} ($0\leq k <n$) if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not…

Let $\mu(G)$ denote the spectral radius of a graph $G$. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erd\H{o}s-S\'os Conjecture that any tree of order $t$ is contained in a graph of…

We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree…

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte's reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the…

The study of the existence of hamiltonian cycles in a graph is a classic problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chv\'{a}tal's conjecture from another perspective: what is the spectral…

Let $G$ be a graph and $T$ be a spanning tree of $G$. We use $Q(G)=D(G)+A(G)$ to denote the signless Laplacian matrix of $G$, where $D(G)$ is the diagonal degree matrix of $G$ and $A(G)$ is the adjacency matrix of $G$. The signless…

A graph $G$ is a fractional $(a,b,k)$-critical covered graph if $G-U$ is a fractional $[a,b]$-covered graph for every $U\subseteq V(G)$ with $|U|=k$, which is first defined by Zhou, Xu and Sun (S. Zhou, Y. Xu, Z. Sun, Degree conditions for…

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right] $, write $A_{\alpha}\left( G\right) $ for the matrix \[ A_{\alpha}\left( G\right)…

Let ${\rm EX}(n,H)$ and ${\rm SPEX}(n,H)$ denote the families of $n$-vertex $H$-free graphs with the maximum size and the maximum spectral radius, respectively. A graph $H$ is said to be spectral-consistent if ${\rm SPEX}(n,H)\subseteq {\rm…

Let $G$ be a connected uniform hypergraphs with maximum degree $\Delta$, spectral radius $\lambda$ and minimum H-eigenvalue $\mu$. In this paper, we give some lower bounds for $\Delta-\lambda$, which extend the result of [S.M. Cioab\u{a},…

An $H_n$-factor of a graph $G$ is defined to be a spanning subgraph $F$ of $G$ such that each vertex has degree belonging to the set $\{1,3,5,...,2n-1,2n\}$ in $F$. In this paper, we investigate $H_n$-factors of graphs by using Lov\'asz's…

We prove that, for any $t\ge 3$, there exists a constant $c=c(t)>0$ such that any $d$-regular $n$-vertex graph with the second largest eigenvalue in absolute value~$\lambda$ satisfying $\lambda\le c d^{t-1}/n^{t-2}$ contains vertex-disjoint…

An $n$-vertex graph is said to to be $(p,\beta)$-bijumbled if for any vertex sets $A,B\subseteq V(G)$, we have \[e(A,B)=p|A||B|\pm \beta \sqrt{|A||B|}.\] We prove that for any $3\leq r\in \mathbb{N}$ and $c>0$ there exists an…

A long-standing conjecture asserts that there exists a constant $c>0$ such that every graph of order $n$ without isolated vertices contains an induced subgraph of order at least $cn$ with all degrees odd. Scott (1992) proved that every…

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and…

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\rho(G)$ be the spectral radius of a graph $G$. Let $\theta(1,p,q)$ denote the theta graph, which is obtained by connecting two distinct vertices with three internally…

In Gao's previous work, the authors determined several graph degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if $b=f(x)=g(x)=a$ for all vertices $x$…

Let $k,l$ be two positive integers. An $S_{k,l}$ is a graph obtained from disjoint $K_{1,k}$ and $K_{1,l}$ by adding an edge between the $k$-degree vertex in $K_{1,k}$ and the $l$-degree vertex in $K_{1,l}$. An {\em $S_{k,l}$-free} graph is…