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.

Let $G$ be a simple graph with vertex set $V(G) = {v_1‎, ‎v_2,ldots‎, ‎v_n}$ and $d_i$ the degree of its vertex $v_i$‎, ‎$i = 1‎, ‎2‎, ‎dots‎, ‎n$‎. ‎Inspired by the Randic matrix and the general Randic‎ ‎index of a graph‎, ‎we introduce the concept of general Randi'c‎ ‎matrix $textbf{R}_alpha$ of $G$‎, ‎which is defined by‎ $(textbf{R}_alpha)_{i,j}=(d_id_j)^alpha$ if $v_i$ and $v_j$ are‎ ‎adjacent‎, ‎and zero otherwise‎. ‎Similarly‎, ‎the general Randic‎ ‎eigenvalues are the eigenvalues of the general Randic} matrix‎, ‎the greatest general Randic eigenvalue is the general Randic‎ ‎spectral radius of $G$‎, ‎and the general Randic energy is the sum‎ ‎of the absolute values of the general Randic eigenvalues‎. ‎In ‎this paper‎, ‎we prove some properties of the general Randi'c matrix‎ ‎and obtain lower and upper bounds for general Randic energy‎, ‎also‎, ‎we get some lower bounds for general Randic spectral‎ ‎radius of a connected graph‎. ‎Moreover‎, ‎we give a new sharp upper‎ ‎bound for the general Randic energy when $alpha=-1/2$‎.

https://toc.ui.ac.ir/article_5451_12d35e7ada9a784103a3b287d094dee6.pdf

General Randic matrix and general Randic energy

The concept of k-connectivity $$\kappa '_{k}(G)$$źkź(G) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity $$\kappa _{k}(G)$$źk(G), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if $$\kappa '_k(G)\ne n-k+1$$źkź(G)źn-k+1 where $$k\ge 3$$kź3, then $$0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1$$0≤źkź(G)-źk(G)≤n-k-1; otherwise, $$-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k$$-źk2ź+1≤źkź(G)-źk(G)≤n-k. Moreover, all of these bounds are sharp. Some specific study is focused for the case $$k=3$$k=3. As results, we characterize the graphs with $$\kappa '_3(G)=\kappa _3(G)=t$$ź3ź(G)=ź3(G)=t for $$t\in \{1, n-3, n-2\}$$tź{1,n-3,n-2}, and give a necessary condition for $$\kappa '_3(G)=\kappa _3(G)$$ź3ź(G)=ź3(G) by showing that for a connected graph G of order n and size m, if $$\kappa '_3(G)=\kappa _3(G)=t$$ź3ź(G)=ź3(G)=t where $$1\le t\le n-3$$1≤t≤n-3, then $$m\le {n-2\atopwithdelims ()2}+2t$$m≤n-22+2t. Moreover, the unique extremal graph is given for the equality to hold.

On the difference of two generalized connectivities of a graph

Hansen et‎. ‎al.‎, ‎using the AutoGraphiX software‎ ‎package‎, ‎conjectured that the Szeged index $Sz(G)$ and the‎
‎Wiener index $W(G)$ of a connected bipartite graph $G$ with $n geq‎ ‎4$ vertices and $m geq n$ edges‎, ‎obeys the relation‎ ‎$Sz(G)-W(G) geq 4n-8$‎. ‎Moreover‎, ‎this bound would be the best possible‎. ‎This paper offers a proof to this conjecture‎.

http://toc.ui.ac.ir/article_2450_71f471bc2fe6a994e640ca1d23353d16.pdf

Xueliang Li

Papers

General Randic matrix and general Randic energy

On the difference of two generalized connectivities of a graph

On a relation between Szeged and Wiener indices of bipartite graphs

On the ABC spectra radius of unicyclic graphs

Tight upper bound of the rainbow vertex-connection number for 2-connected graphs