Related papers: Spectral radius and spanning trees of graphs

For any integer $k\geq 2$, a spanning $k$-ended tree is a spanning tree with at most $k$ leaves. In this paper, we provide a tight spectral radius condition for the existence of a spanning $k$-ended tree in $t$-connected graphs, which…

Let $G$ be a connected graph of order $n$. A spanning $k$-tree of $G$ is a spanning tree with the maximum degree at most $k$, and a spanning $k$-ended-tree of $G$ is a spanning tree at most $k$ leaves, where $k\geq2$ is an integer. This…

In this paper, we study some spanning trees with bounded degree and leaf degree from eigenvalues. For any integer $k\geq2$, a $k$-tree is a spanning tree in which every vertex has degree no more than $k$. Let $T$ be a spanning tree of a…

A $k$-ended tree is a tree with at most $k$ leaves. In this note, we give a simple proof for the following theorem. Let $G$ be a connected graph and $k$ be an integer ($k\geq 2$). Let $S$ be a vertex subset of $G$ such that $\alpha_{G}(S)…

For any integer $k\geq1,$ a graph $G$ has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper we prove a sufficient condition in terms of the number of $r$-cliques to guarantee the existence of a $k$-factor in a graph…

A $k$-tree is a spanning tree in which every vertex has degree at most $k$. In this paper, we provide a sufficient condition for the existence of a $k$-tree in a connected graph with fixed order in terms of the adjacency spectral radius and…

A tree is called k-ended tree if it has at most k leaves, where a leaf is a vertex of degree one. In this paper we prove that every 3-regular connected graph with n vertices such that n is greater than 8 has spanning sub tree with at most…

Let $k$, $d$ be a positive integer, $G$ be a connected graph of order $n$, $T$ be a tree. The leaf distance of a tree is defined as the minimum distance between any two leaves. For $v\in V(T)$, the leaf degree of $v$ in $T$ is the number of…

A tree with at most $k$ leaves is called a $k$-ended tree. A spanning 2-ended tree is a Hamilton path. A Hamilton cycle can be considered as a spanning 1-ended tree. The earliest result concerning spanning trees with few leaves states that…

We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\over 4}(s-2)+2$ leaves. Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively…

In 1998, Broersma and Tuinstra [J. Graph Theory \textbf{29} (1998), 227-237] proved that if $G$ is a connected graph satisfying $\sigma_2(G) \geq |G|-k+1$ then $G$ has a spanning $k-$ended tree. They also gave an example to show that the…

Let $k\geq2$ be an integer. A $k$-tree is a tree with maximum degree at most $k$. In this paper, we give a closure result on spanning $k$-trees of graphs with given minimum degree. Let $\delta\geq1$ be an integer, and $G$ be a connected…

A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. For a graph $G$, let $\sigma_2$ be the minimum degree sum of two nonadjacent vertices in $G$. We consider tree…

We prove that every connected graph with $s$ vertices of degree~1 and 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${1\over 3}t +{1\over 4}s+{3\over 2}$ leaves. We present infinite series of graphs showing that…

Let $k\geq2$ be an integer. A tree $T$ is called a $k$-tree if $d_T(v)\leq k$ for each $v\in V(T)$, that is, the maximum degree of a $k$-tree is at most $k$. Let $\lambda_1(D(G))$ denote the distance spectral radius in $G$, where $D(G)$…

Chv\'{a}tal and Erd\"{o}s [Discrete Math. 2 (1972) 111-113] stated that, for an $m$-connected graph $G$, if its independence number $\alpha(G)\leq m-1$, then $G$ is Hamilton-connected. Note that $k$-leaf-connectedness is a natural…

We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for…

The $k$-expansion of a graph $G$ is the $k$-uniform hypergraph obtained from $G$ by adding $k-2$ new vertices to every edge. We determine, for all $k > d \geq 1$, asymptotically optimal $d$-degree conditions that ensure the existence of all…

Let $T$ be a tree. A vertex of degree one is a \emph{leaf} of $T$ and a vertex of degree at least three is a \emph{branch vertex} of $T$. A graph is said to be \emph{$K_{1,4}$-free} if it does not contain $K_{1,4}$ as an induced subgraph.…

The toughness of a graph $G$, denoted by $\tau(G)$, is defined by $\tau(G)=$min $\{\frac{|S|}{c(G-S)}:S\subseteq V(G)$ and $c(G-S)\geq2\}$. A graph $G$ is said to be $\tau$-tough if $\tau(G)\geq \tau$. Let $k\geq2$ be an integer. A tree $T$…