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.

In 1990, Gutman and Mizoguchi conjectured that all roots of the β-polynomial β(G,C,x) of a graph G are real. Since then, there has been some literature intending to solve this conjecture. However, in all existing literature, only classes of graphs were found to show that the conjecture is true; for example, monocyclic graphs, bicyclic graphs, graphs such that no two circuits share a common edge, graphs without 3-matchings, etc, supporting the conjecture in some sense. Yet, no complete solution has been given. In this paper, we show that the conjecture is true for all graphs, and therefore completely solve this conjecture.

http://www.cs.cityu.edu.hk/~lwang/research/jmc03a.pdf

A Complete Solution to a Conjecture on the β-Polynomials of Graphs

Let D be a digraph with skew-adjacency matrix S(D). The skew energy of D is dened as the sum of the norms of all eigenvalues of S(D). Two digraphs are said to be skew equienergetic if their skew energies are equal. We establish an expression for the characteristic polynomial of the skew adjacency matrix of the join of two digraphs, and for the respective skew energy, and thereby construct non-cospectral, skew equienergetic digraphs on n vertices, for all n � 6. Thus we arrive at the solution of some open problems proposed in (X. Li, H. Lian, A survey on the skew energy of oriented graphs, arXiv:1304.5707).

http://toc.ui.ac.ir/article_9372_1d921f94d58d62e8f06d43db2dc426d5.pdf

Skew equienergetic digraphs

The general σ all-ones problem for trees

‎Let $D$ be a digraph with skew-adjacency matrix $S(D)$‎. ‎Then the‎ ‎skew energy of $D$ is defined as the sum of the norms of all‎ ‎eigenvalues of $S(D)$‎. ‎Denote by $mathcal{O}_n$ the class of‎ ‎digraphs of order $n$ with no even cycles‎, ‎and by‎ ‎$mathcal{O}_{n,m}$ the class of digraphs in $mathcal{O}_n$ with‎ ‎$m$ arcs‎. ‎In this paper‎, ‎we first give the minimal skew energy‎ ‎digraphs in $mathcal{O}_n$ and $mathcal{O}_{n,m}$ with $n-1leq‎ ‎mleq frac{3}{2}(n-1)$‎. ‎Then we determine the maximal skew energy‎ ‎digraphs in $mathcal{O}_{n,n}$ and $mathcal{O}_{n,n+1}$‎, ‎and in‎ ‎the latter case we assume that $n$ is even‎.

https://toc.ui.ac.ir/article_4059_ed37714018ee8fbea9af32b692471f46.pdf

Extremal skew energy of digraphs with no even cycles

Abstract A path in an edge-colored graph G is called a rainbow path if no two edges on the path have the same color. The graph G is called rainbow connected if between every pair of distinct vertices of G, there is a rainbow path. Recently, Johnson et al. considered this concept with the additional requirement that the coloring of G is proper. The proper rainbow connection number of G, denoted by prc(G), is the minimum number of colors needed to properly color the edges of G so that G is rainbow connected. Similarly, the proper strong rainbow connection number of G, denoted by psrc(G), is the minimum number of colors needed to properly color the edges of G such that for any two distinct vertices of G, there is a rainbow geodesic (shortest path) connecting them. In this paper, we characterize those graphs with proper rainbow connection numbers equal to the size or within 1 of the size. Moreover, we completely solve a question proposed by Johnson et al. by proving that if G = Kp1Kpn, where n≥ 1, and p1, . . . , pn>1 are integers, then prc(G) = psrc(G) = χ′(G), where χ′(G) denotes the chromatic index of G. Finally, we investigate some su cient conditions for a graph G to satisfy prc(G) = rc(G), and make some slightly positive progress by using a relation between rc(G) and the girth of the graph.

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Xueliang Li

Papers

A Complete Solution to a Conjecture on the β-Polynomials of Graphs

Skew equienergetic digraphs

The general σ all-ones problem for trees

Extremal skew energy of digraphs with no even cycles

On proper (strong) rainbow connection of graphs