Structure of Cubic Lehman Matrices

A pair $(A,B)$ of square $(0,1)$-matrices is called a \emph{Lehman pair} if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called \emph{Lehman matrices}. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman matrices with $k=-1$ are essentially equivalent to \emph{partitionable graphs} (also known as $(\alpha,\omega)$-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, we view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite graph, focusing in particular on the case where the graph is cubic. From this perspective, we identify two constructions that generate cubic Lehman graphs from smaller Lehman graphs. The most prolific of these constructions involves repeatedly replacing suitable pairs of edges with a particular $6$-vertex subgraph that we call a $3$-rung ladder segment. Two decades ago, L\"{u}tolf \& Margot initiated a computational study of mni matrices and constructed a catalogue containing (among other things) a listing of all cubic Lehman matrices with $k =1$ of order up to $17 \times 17$. We verify their catalogue (which has just one omission), and extend the computational results to $20 \times 20$ matrices. Of the $908$ cubic Lehman matrices (with $k=1$) of order up to $20 \times 20$, only two do not arise from our $3$-rung ladder construction. However these exceptions can be derived from our second construction, and so our two constructions cover all known cubic Lehman matrices with $k=1$

On regular induced subgraphs of generalized polygons

The cage problem asks for the smallest number $c(k,g)$ of vertices in a $k$-regular graph of girth $g$ and graphs meeting this bound are known as cages. While cages are known to exist for all integers $k \ge 2$ and $g \ge 3$, the exact value of $c(k, g)$ is known only for some small values of $k, g$ and three infinite families where $g \in \{6, 8, 12\}$ and $k - 1$ is a prime power. These infinite families come from the incidence graphs of generalized polygons. Some of the best known upper bounds on $c(k,g)$ for $g \in \{6, 8, 12\}$ have been obtained by constructing small regular induced subgraphs of these cages. In this paper, we first use the Expander Mixing Lemma to give a general lower bound on the size of an induced $k$-regular subgraph of a regular bipartite graph in terms of the second largest eigenvalue of the host graph. We use this bound to show that the known construction of $(k,6)$-graphs using Baer subplanes of the Desarguesian projective plane is the best possible. For generalized quadrangles and hexagons, our bounds are new. In particular, we improve the known lower bound on the size of a $q$-regular induced subgraphs of the classical generalized quadrangle $\mathsf{Q}(4,q)$ and show that the known constructions are asymptotically sharp. For prime powers $q$, we also improve the known upper bounds on $c(q,8)$ and $c(q,12)$ by giving new geometric constructions of $q$-regular induced subgraphs in the symplectic generalized quadrangle $\mathsf{W}(3,q)$ and the split Cayley hexagon $\mathsf{H}(q)$, respectively. Our constructions show that $$c(q,8) \le 2(q^3 - q\sqrt{q} - q)$$ for $q$ an even power of a prime, and $$c(q, 12) \le 2(q^5 - 3q^3)$$ for all prime powers $q$. For $q \in \{3,4,5\}$ we also give a computer classification of all $q$-regular induced subgraphs of the classical generalized quadrangles of order $q$.Comment: Published version, proof of Lemma 5.3 simplified, for computer code see previous versio

A simplified search strategy for identifying randomised controlled trials for systematic reviews of health care interventions : a comparison with more exhaustive strategies

Background It is generally believed that exhaustive searches of bibliographic databases are needed for systematic reviews of health care interventions. The CENTRAL database of controlled trials (RCTs) has been built up by exhaustive searching. The CONSORT statement aims to encourage better reporting, and hence indexing, of RCTs. Our aim was to assess whether developments in the CENTRAL database, and the CONSORT statement, mean that a simplified RCT search strategy for identifying RCTs now suffices for systematic reviews of health care interventions. Methods RCTs used in the Cochrane reviews were identified. A brief RCT search strategy (BRSS), consisting of a search of CENTRAL, and then for variants of the word random across all fields (random$.af.) in MEDLINE and EMBASE, was devised and run. Any trials included in the meta-analyses, but missed by the BRSS, were identified. The meta-analyses were then re-run, with and without the missed RCTs, and the differences quantified. The proportion of trials with variants of the word random in the title or abstract was calculated for each year. The number of RCTs retrieved by searching with "random$.af." was compared to the highly sensitive search strategy (HSSS). Results The BRSS had a sensitivity of 94%. It found all journal RCTs in 47 of the 57 reviews. The missing RCTs made some significant differences to a small proportion of the total outcomes in only five reviews, but no important differences in conclusions resulted. In the post-CONSORT years, 1997–2003, the percentage of RCTs with random in the title or abstract was 85%, a mean increase of 17% compared to the seven years pre-CONSORT (95% CI, 8.3% to 25.9%). The search using random$.af. reduced the MEDLINE retrieval by 84%, compared to the HSSS, thereby reducing the workload of checking retrievals. Conclusion A brief RCT search strategy is now sufficient to locate RCTs for systematic reviews in most cases. Exhaustive searching is no longer cost-effective, because in effect it has already been done for CENTRAL

First national survey of practitioners with early years’ professional status

The first national survey of practitioners who have achieved Early Years Professional Status (EYPS) set out to ascertain: • more detailed demographic information about their backgrounds and experience • their views on their ability to carry out their role since gaining EYPS • information about career trajectories including their intentions to change setting, role or career • an overview of their professional development activities and plans • an assessment of the impact of obtaining EYPS on professional identity • their views on the difficulty of achieving change in their settings. This survey is part of a three year longitudinal study investigating the role and impact of early years professionals (EYPs) in their working environments (settings) and also investigating practitioners’ personal career development and aspirations. There are two main parts to the study: • a survey of all EYPs, asking about their career development needs and aspirations • case studies in 30 settings across the country, looking at how EYPs have an impact on the quality of education and care available to children. The survey, with slight modifications, will be repeated in year three of the study. The intention was to make the survey accessible to all who have achieved EYPS, with the aim of generating responses from approximately 10-15 per cent of respondents. The survey went live between January and February 2010 and by the close of the survey some 1,045 completed questionnaires had been generated, representing nearly 30 per cent of the total number of practitioners with EYPS. This sample was broadly representative of the total population of practitioners with EYPS based on gender, ethnicity, geographical distribution and the pathway they had followed to achieve EYPS