A new neighborhood condition for graphs to be fractional (k,m)-deleted graphs

AbstractLet G be a graph of order n, and let k≥2 and m≥0 be two integers. Let h:E(G)→[0,1] be a function. If ∑e∋xh(e)=k holds for each x∈V(G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh={e∈E(G):h(e)>0}. A graph G is called a fractional (k,m)-deleted graph if there exists a fractional k-factor G[Fh] of G with indicator function h such that h(e)=0 for any e∈E(H), where H is any subgraph of G with m edges. In this paper, it is proved that G is a fractional (k,m)-deleted graph if δ(G)≥k+2m, n≥8k2+4k−8+8m(k+1)+4m−2k+m−1 and ∣NG(x)∪NG(y)∣≥n2 for any two nonadjacent vertices x and y of G such that NG(x)∩NG(y)≠0̸. Furthermore, it is shown that the result in this paper is best possible in some sense

Signless Laplacian spectral radius for a k-extendable graph

Let $k$ and $n$ be two nonnegative integers with $n\equiv0$ (mod 2), and let $G$ be a graph of order $n$ with a 1-factor. Then $G$ is said to be $k$-extendable for $0\leq k\leq\frac{n-2}{2}$ if every matching in $G$ of size $k$ can be extended to a 1-factor. In this paper, we first establish a lower bound on the signless Laplacian spectral radius of $G$ to ensure that $G$ is $k$-extendable. Then we create some extremal graphs to claim that all the bounds derived in this article are sharp.Comment: 11 page

Spanning k-trees and distance spectral radius in graphs

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 $D(G)$ denotes the distance matrix of $G$. In this paper, we verify a upper bound for $\lambda_1(D(G))$ in a connected graph $G$ to guarantee the existence of a spanning $k$-tree in $G$.Comment: 11 page

Sufficient conditions for fractional [a,b]-deleted graphs

Let $a$ and $b$ be two positive integers with $a\leq b$, and let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. Let $h:E(G)\rightarrow[0,1]$ be a function. If $a\leq\sum\limits_{e\in E_G(v)}{h(e)}\leq b$ holds for every $v\in V(G)$, then the subgraph of $G$ with vertex set $V(G)$ and edge set $F_h$, denoted by $G[F_h]$, is called a fractional $[a,b]$-factor of $G$ with indicator function $h$, where $E_G(v)$ denotes the set of edges incident with $v$ in $G$ and $F_h=\{e\in E(G):h(e)>0\}$. A graph $G$ is defined as a fractional $[a,b]$-deleted graph if for any $e\in E(G)$, $G-e$ contains a fractional $[a,b]$-factor. The size, spectral radius and signless Laplacian spectral radius of $G$ are denoted by $e(G)$, $\rho(G)$ and $q(G)$, respectively. In this paper, we establish a lower bound on the size, spectral radius and signless Laplacian spectral radius of a graph $G$ to guarantee that $G$ is a fractional $[a,b]$-deleted graph.Comment: 1

Two sufficient conditions for graphs to admit path factors

Let $\mathcal{A}$ be a set of connected graphs. Then a spanning subgraph $A$ of $G$ is called an $\mathcal{A}$-factor if each component of $A$ is isomorphic to some member of $\mathcal{A}$. Especially, when every graph in $\mathcal{A}$ is a path, $A$ is a path factor. For a positive integer $d\geq2$, we write $\mathcal{P}_{\geq d}=\{P_i|i\geq d\}$. Then a $\mathcal{P}_{\geq d}$-factor means a path factor in which every component admits at least $d$ vertices. A graph $G$ is called a $(\mathcal{P}_{\geq d},m)$-factor deleted graph if $G-E'$ admits a $\mathcal{P}_{\geq d}$-factor for any $E'\subseteq E(G)$ with $|E'|=m$. A graph $G$ is called a $(\mathcal{P}_{\geq d},k)$-factor critical graph if $G-Q$ has a $\mathcal{P}_{\geq d}$-factor for any $Q\subseteq V(G)$ with $|Q|=k$. In this paper, we present two degree conditions for graphs to be $(\mathcal{P}_{\geq3},m)$-factor deleted graphs and $(\mathcal{P}_{\geq3},k)$-factor critical graphs. Furthermore, we show that the two results are best possible in some sense