Component factors and binding number conditions in graphs

<abstract><p>Let $ G $ be a graph. For a set $ \mathcal{H} $ of connected graphs, an $ \mathcal{H} $-factor of a graph $ G $ is a spanning subgraph $ H $ of $ G $ such that every component of $ H $ is isomorphic to a member of $ \mathcal{H} $. A graph $ G $ is called an $ (\mathcal{H}, m) $-factor deleted graph if for every $ E'\subseteq E(G) $ with $ |E'| = m $, $ G-E' $ admits an $ \mathcal{H} $-factor. A graph $ G $ is called an $ (\mathcal{H}, n) $-factor critical graph if for every $ N\subseteq V(G) $ with $ |N| = n $, $ G-N $ admits an $ \mathcal{H} $-factor. Let $ m $, $ n $ and $ k $ be three nonnegative integers with $ k\geq2 $, and write $ \mathcal{F} = \{P_2, C_3, P_5, \mathcal{T}(3)\} $ and $ \mathcal{H} = \{K_{1, 1}, K_{1, 2}, \cdots, K_{1, k}, \mathcal{T}(2k+1)\} $, where $ \mathcal{T}(3) $ and $ \mathcal{T}(2k+1) $ are two special families of trees. In this article, we verify that (i) a $ (2m+1) $-connected graph $ G $ is an $ (\mathcal{F}, m) $-factor deleted graph if its binding number $ bind(G)\geq\frac{4m+2}{2m+3} $; (ii) an $ (n+2) $-connected graph $ G $ is an $ (\mathcal{F}, n) $-factor critical graph if its binding number $ bind(G)\geq\frac{2+n}{3} $; (iii) a $ (2m+1) $-connected graph $ G $ is an $ (\mathcal{H}, m) $-factor deleted graph if its binding number $ bind(G)\geq\frac{2}{2k-1} $; (iv) an $ (n+2) $-connected graph $ G $ is an $ (\mathcal{H}, n) $-factor critical graph if its binding number $ bind(G)\geq\frac{2+n}{2k+1} $.</p></abstract>

A sufficient condition for the existence of a k-factor excluding a given r-factor

Let G be a graph, and let k, r be nonnegative integers with k ≥ 2. A k-factor of G is a spanning subgraph F of G such that dF(x) = k for each x ∈ V (G), where dF(x) denotes the degree of x in F. For S ⊆ V (G), NG(S) = ∪x∊SNG(x). The binding number of G is defined by bind$\begin{array}{} (G) = {\rm{min }}\{ \frac{{|{N_G}(S)|}}{{|S|}}:\emptyset \ne S \subset V(G),{N_G}(S) \ne V(G)\} \end{array}$. In this paper, we obtain a binding number and neighborhood condition for a graph to have a k-factor excluding a given r-factor. This result is an extension of the previous results.

Binding Number and Wheel Related Graphs

The binding number of a graph G is defined to be the minimum of [Formula: see text] taken over all nonempty [Formula: see text] such that [Formula: see text]. Binding number, one indicator to better understand graph, is an important characteristic quantity of a graph. In this paper, the relationships between the binding number and some other graph vulnerability parameters, namely the toughness, integrity, rupture degree and scattering number, are established. Exact values for the binding numbers of wheel related graphs namely gear, helm, sunflower and friendship graph are obtained.

Determination of rate constant, binding constant, and binding number by fluorescence measurements of Gd3N@C80(OH)20 in D2O

ABSTRACT Today MRI imagining techniques are capable of discerning between abnormal and normal complex tissues by providing contrasting images of these tissues. One drawback of using MRI imagining is its low sensitivity. However, this sensitivity can be greatly enhanced by introducing contrasting agents who can provide a new pathway for water molecules to significantly relax faster and hence generate the desired “contrast” between healthy and unhealthy tissues. We report the first ever recorded fluorescence emission spectrum of Gd3N@C80(OH)20; where -(OH)20 is the average number of hydroxyl groups. Our emission data indicates that the H2O- Gd3N@C80(OH)20 interactions lead to fluorescence quenching via a static quenching mechanism. The binding constant, Kb, on the other hand, was found to be of the same magnitude as interactions between human serum albumin and small organic acid but quite different, several orders of magnitude smaller, than protein nanoparticle complexes. Interestingly, the binding number, n, was found to be approximately 0.5, which in cases like this, is rounded to a whole number of one. The data also indicated an extremely fast rate constant on the order of 1012 L mol-1 s-1 which is outside of the diffusion-control regime. These results are presented within this report.