We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

BackgroundSpecific dietary and other lifestyle behaviors may affect the success of the straightforward-sounding strategy “eat less and exercise more” for preventing long-term weight gain. MethodsWe performed prospective investigations involving three separate cohorts that included 120,877 U.S. women and men who were free of chronic diseases and not obese at baseline, with follow-up periods from 1986 to 2006, 1991 to 2003, and 1986 to 2006. The relationships between changes in lifestyle factors and weight change were evaluated at 4-year intervals, with multivariable adjustments made for age, baseline body-mass index for each period, and all lifestyle factors simultaneously. Cohort-specific and sex-specific results were similar and were pooled with the use of an inverse-variance–weighted meta-analysis. ResultsWithin each 4-year period, participants gained an average of 3.35 lb (5th to 95th percentile, −4.1 to 12.4). On the basis of increased daily servings of individual dietary components, 4-year weight cha...

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Changes in Diet and Lifestyle and Long-Term Weight Gain in Women and Men

Background Nasal carriers of Staphylococcus aureus are at increased risk for health care–associated infections with this organism. Decolonization of nasal and extranasal sites on hospital admission may reduce this risk. Methods In a randomized, double-blind, placebo-controlled, multicenter trial, we assessed whether rapid identification of S. aureus nasal carriers by means of a real-time polymerase-chain-reaction (PCR) assay, followed by treatment with mupirocin nasal ointment and chlorhexidine soap, reduces the risk of hospital-associated S. aureus infection. Results From October 2005 through June 2007, a total of 6771 patients were screened on admission. A total of 1270 nasal swabs from 1251 patients were positive for S. aureus. We enrolled 917 of these patients in the intention-to-treat analysis, of whom 808 (88.1%) underwent a surgical procedure. All the S. aureus strains identified on PCR assay were susceptible to methicillin and mupirocin. The rate of S. aureus infection was 3.4% (17 of 504 patients) in the mupirocin–chlorhexidine group, as compared with 7.7% (32 of 413 patients) in the placebo group (relative risk of infection, 0.42; 95% confidence interval [CI], 0.23 to 0.75). The effect of mupirocin–chlorhexidine treatment was most pronounced for deep surgical-site infections (relative risk, 0.21; 95% CI, 0.07 to 0.62). There was no significant difference in all-cause in-hospital mortality between the two groups. The time to the onset of nosocomial infection was shorter in the placebo group than in the mupirocin–chlorhexidine group (P = 0.005). Conclusions The number of surgical-site S. aureus infections acquired in the hospital can be reduced by rapid screening and decolonizing of nasal carriers of S. aureus on admission. (Current Controlled Trials number, ISRCTN56186788.)

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Preventing Surgical-Site Infections in Nasal Carriers of Staphylococcus aureus

The understanding of nature and nurture within developmental science has evolved with alternating ascendance of one or the other as primary explanations for individual differences in life course trajectories of success or failure. A dialectical perspective emphasizing the interconnectedness of individual and context is suggested to interpret the evolution of developmental science in similar terms to those necessary to explain the development of individual children. A unified theory of development is proposed to integrate personal change, context, regulation, and representational models of development.

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A Unified Theory of Development: A Dialectic Integration of Nature and Nurture

Emotion has a substantial influence on the cognitive processes in humans, including perception, attention, learning, memory, reasoning, and problem solving. Emotion has a particularly strong influence on attention, especially modulating the selectivity of attention as well as motivating action and behavior. This attentional and executive control is intimately linked to learning processes, as intrinsically limited attentional capacities are better focused on relevant information. Emotion also facilitates encoding and helps retrieval of information efficiently. However, the effects of emotion on learning and memory are not always univalent, as studies have reported that emotion either enhances or impairs learning and long-term memory (LTM) retention, depending on a range of factors. Recent neuroimaging findings have indicated that the amygdala and prefrontal cortex cooperate with the medial temporal lobe in an integrated manner that affords (i) the amygdala modulating memory consolidation; (ii) the prefrontal cortex mediating memory encoding and formation; and (iii) the hippocampus for successful learning and LTM retention. We also review the nested hierarchies of circular emotional control and cognitive regulation (bottom-up and top-down influences) within the brain to achieve optimal integration of emotional and cognitive processing. This review highlights a basic evolutionary approach to emotion to understand the effects of emotion on learning and memory and the functional roles played by various brain regions and their mutual interactions in relation to emotional processing. We also summarize the current state of knowledge on the impact of emotion on memory and map implications for educational settings. In addition to elucidating the memory-enhancing effects of emotion, neuroimaging findings extend our understanding of emotional influences on learning and memory processes; this knowledge may be useful for the design of effective educational curricula to provide a conducive learning environment for both traditional "live" learning in classrooms and "virtual" learning through online-based educational technologies.

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The Influences of Emotion on Learning and Memory.

More than ten years ago in 2008, a new kind of graph coloring appeared in graph theory, which is the {\it rainbow connection coloring} of graphs, and then followed by some other new concepts of graph colorings, such as {\it proper connection coloring, monochromatic connection coloring, and conflict-free connection coloring} of graphs. In about ten years of our consistent study, we found that these new concepts of graph colorings are actually quite different from the classic graph colorings. These {\it colored connection colorings} of graphs are brand-new colorings and they need to take care of global structural properties (for example, connectivity) of a graph under the colorings; while the traditional colorings of graphs are colorings under which only local structural properties (adjacent vertices or edges) of a graph are taken care of. Both classic colorings and the new colored connection colorings can produce the so-called chromatic numbers. We call the colored connection numbers the {\it global chromatic numbers}, and the classic or traditional chromatic numbers the {\it local chromatic numbers}. This paper intends to clarify the difference between the colored connection colorings and the traditional colorings, and finally to propose the new concepts of global colorings under which global structural properties of the colored graph are kept, and the global chromatic numbers.

Graph colorings under global structural conditions.

An edge-colored graph G is conflict-free connected if every two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is the smallest number of colors needed in order to make G conflict-free connected. For a graph G,  let C(G) be the subgraph of G induced by its set of cut-edges. In this paper, we first show that, if G is a connected non-complete graph G of order $$n\ge 9$$
 with C(G) being a linear forest and with the minimum degree $$\delta (G)\ge \max \{3, \frac{n-4}{5}\}$$
 , then $$cfc(G)=2$$
 . The bound on the minimum degree is best possible. Next, we prove that, if G is a connected non-complete graph of order $$n\ge 33$$
 with C(G) being a linear forest and with $$d(x)+d(y)\ge \frac{2n-9}{5}$$
 for each pair of two nonadjacent vertices x, y of V(G), then $$cfc(G)=2$$
 . Both bounds, on the order n and the degree sum, are tight. Moreover, we prove several results concerning relations between degree conditions on G and the number of cut edges in G.

Graphs with Conflict-Free Connection Number Two

The $(k,\ell)$-rainbow index $rx_{k, \ell}(G)$ of a graph $G$ was introduced by Chartrand et. al. For the complete graph $K_n$ of order $n\geq 6$, they showed that $rx_{3,\ell}(K_n)=3$ for $\ell=1,2$. Furthermore, they conjectured that for every positive integer $\ell$, there exists a positive integer $N$ such that $rx_{3,\ell}(K_{n})=3$ for every integer $n \geq N$. More generally, they conjectured that for every pair of positive integers $k$ and $\ell$ with $k\geq 3$, there exists a positive integer $N$ such that $rx_{k,\ell}(K_{n})=k$ for every integer $n \geq N$. This paper is to give solutions to these conjectures.

Solutions to conjectures on the $(k,\ell)$-rainbow index of complete graphs

For a simple digraph G of order n with vertex set {v1, v2, . . . , vn}, let d+i and d − i denote the out-degree and in-degree of a vertex vi in G, respectively. Let D (G) = diag(d+1 , d + 2 , . . . , d + n ) and D−(G) = diag(d1 , d − 2 , . . . , d − n ). In this paper we introduce SL(G) = D(G)−S(G) to be a new kind of skew Laplacian matrix of G, where D(G) = D+(G)−D−(G) and S(G) is the skew-adjacency matrix of G, and from which we define the skew Laplacian energy SLE(G) of G as the sum of the norms of all the eigenvalues of SL(G). Some lower and upper bounds of the new skew Laplacian energy are derived and the digraphs attaining these bounds are also determined.

https://journals.ui.ac.ir/article_2833_dc29c895a9ecdf11e850c68593ab72de.pdf

New skew Laplacian energy of simple digraphs

Let S(G � ) be the skew-adjacency matrix of an oriented graph G � . The skew energy of Gis defined as the sum of all singular values of its skew- adjacency matrix S(G � ). In this paper, we first deduce an integral formula for the skew energy of an oriented graph. Then we determine all oriented graphs with minimal skew energy among all connected oriented graphs on n vertices with m (n ≤ m < 2(n − 2)) arcs, which is an analogy to the conjecture for the energy of undirected graphs proposed by Caporossi et al. (G. Caporossi, D. Cvetkovic, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.).

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Xueliang Li

Papers

Graph colorings under global structural conditions.

Graphs with Conflict-Free Connection Number Two

Solutions to conjectures on the $(k,\ell)$-rainbow index of complete graphs

New skew Laplacian energy of simple digraphs

On oriented graphs with minimal skew energy