Distance spectral conditions for IDID-factor-critical and fractional [a,b][a, b]-factor of graphs

Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A graph is $ID$-factor-critical if for every independent set $I$ of $G$ whose size has the same parity as $|V(G)|$, $G-I$ has a perfect matching. For two positive integers $a$ and $b$ with $a\leq b$, let $h$: $E(G)\rightarrow [0, 1]$ be a function on $E(G)$ satisfying $a\leq\sum _{e\in E_{G}(v_{i})}h(e)\leq b$ for any vertex $v_{i}\in V(G)$. Then the spanning subgraph with edge set $E_{h}$, denoted by $G[E_{h}]$, is called a fractional $[a, b]$-factor of $G$ with indicator function $h$, where $E_{h}=\{e\in E(G)\mid h(e)>0\}$ and $E_{G}(v_{i})=\{e\in E(G)\mid e$ is incident with $v_{i}$ in $G$\}. A graph is defined as a fractional $[a, b]$-deleted graph if for any $e\in E(G)$, $G-e$ contains a fractional $[a, b]$-factor. For any integer $k\geq 1$, a graph has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper, we firstly give a distance spectral radius condition of $G$ to guarantee that $G$ is $ID$-factor-critical. Furthermore, we provide sufficient conditions in terms of distance spectral radius and distance signless Laplacian spectral radius for a graph to contain a fractional $[a, b]$-factor, fractional $[a, b]$-deleted-factor and $k$-factor.Comment: 10 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

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

Summertime partitioning and budget of NOycompounds in the troposphere over Alaska and Canada: ABLE 3B

As part of NASA's Arctic Boundary Layer Expedition 3A and 3B field measurement programs, measurements of NO(x) HNO31, PAN, PPN, and NOy were made in the middle to lower troposphere over Alaska and Canada during the summers of 1988 and 1990. These measurements are used to assess the degree of closure within the reactive odd nitrogen (NxOy) budget through the comparison of the values of NOy measured with a catalytic convertor to the sum of individually measured NOy(i) compounds (i.e., Sigma NOy(i) = NOx + HNO3 + PAN + PPN). Significant differences were observed between the various study regions. In the lower 6 km of the troposphere over Alaska and the Hudson Bay lowlands of Canada a significant traction of the NOy budget (30 to 60 per cent) could not be accounted for by the measured Sigma NOy(i). This deficit in the NOy budget is about 100 to 200 parts per trillion by volume (pptv) in the lower troposphere (0.15 to 3 km) and about 200 to 400 pptv in the middle free troposphere (3 to 6.2 km). Conversely, the NOy budget in the northern Labrador and Quebec regions or Canada is almost totally accounted for within the combined measurement uncertainties of NOy and the various NOy(i) compounds. A substantial portion of the NOx budget's 'missing compounds' appears to be coupled to the photochemical and/or dynamical parameters influencing the tropospheric oxidative potential over these regions. A combination of factors are suggested as the causes for the variability observed in the NOy budget. In addition, the apparent stability of compounds represented by the NOy budget deficit in the lower-attitude range questions the ability of these compounds to participate as reversible reservoirs for "active" odd nitrogen and suggest that some portion of the NOy budget may consist of relatively unreactive nitrogencontaining compounds. Bei der Rationalisierung von Kommissioniersystemen besteht bei vielen Unternehmen noch Nachholbedarf. Dies ergab eine Umfrage des Fraunhofer-Instituts für Materialfluss und Logistik in Dortmund bei ca. 800 Unternehmen. Keins der Unternehmen setzt Kommissionierautomaten ein, die Voraussetzungen für durchgehende Automatisierung fehlen