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

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

Introduction and editorial overview — In search of new sources of growth: what China should do next?

[Extract] Welcome to this special issue of Singapore Economic Review on “In Search of New Sources of Growth: What China Should Do Next”! In the past three decades, remarkable achievements have been made in China’sdevelopment, with economic growth averaged at around 9% per annum. Rapid economic growth benefits ordinary Chinese, as can be observed that more and more Chinese travel overseas, for sight-seeing, shopping and even hunting for real estate assets. Coupled with these remarkable achievements are a number of challenges that are exerting an increasingly significant constraint on China’s road ahead, such as environmental pollutions, income inequality and regional disparities. For example, the widely spread smog in China’s major cities is likely to affect people’s health in a negative way

Toughness, isolated toughness and path factors in graphs

http://dx.doi.org/10.1017/S000497271200033

The Existence Of P_≥3-Factor Covered Graphs

A spanning subgraph F of a graph G is called a P≥3-factor of G if every component of F is a path of order at least 3. A graph G is called a P≥3-factor covered graph if G has a P≥3-factor including e for any e ∈ E(G). In this paper, we obtain three sufficient conditions for graphs to be P≥3-factor covered graphs. Furthermore, it is shown that the results are sharp

Independence number and connectivity for fractional (

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 ⊆ V(G) with |U| = k, which is first defined by (Zhou, Xu and Sun, Inf. Process. Lett. 152 (2019) 105838). Furthermore, they derived a degree condition for a graph to be a fractional (a, b, k)-critical covered graph. In this paper, we gain an independence number and connectivity condition for a graph to be a fractional (a, b, k)-critical covered graph and verify that G is a fractional (a, b, k)-critical covered graph if k(G) ≥ max {2b(a+1)(b+1)+4bk+5/4b,(a+1)2(G)+4bk+5/4b}