On Almost Distance-Regular Graphs

2010 Mathematics Subject Classification: 05E30, 05C50;distance-regular graph;walk-regular graph;eigenvalues;predistance polynomial

Dual concepts of almost distance-regularity and the spectral excess theorem

Generally speaking, `almost distance-regular' graphs share some, but not necessarily all, of the regularity properties that characterize distance-regular graphs. In this paper we propose two new dual concepts of almost distance-regularity, thus giving a better understanding of the properties of distance-regular graphs. More precisely, we characterize $m$-partially distance-regular graphs and $j$-punctually eigenspace distance-regular graphs by using their spectra. Our results can also be seen as a generalization of the so-called spectral excess theorem for distance-regular graphs, and they lead to a dual version of it

On almost distance-regular graphs

Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study `almost distance-regular graphs'. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called $m$-walk-regularity. Another studied concept is that of $m$-partial distance-regularity or, informally, distance-regularity up to distance $m$. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of $(\ell,m)$-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem

Lack of effect of pravastatin on cerebral blood flow or parenchymal volume loss in elderly at risk for vascular disease

<p><b>Background and Purpose:</b> Ageing is associated with a decline in cerebral blood flow. Animal studies have shown that cholesterol-lowering therapy with statins might preserve cerebral blood flow (CBF). We examined the effect of 40 mg pravastatin on the decline in CBF and brain volume in a subset of elderly subjects participating in the PROspective Study of Pravastatin in the Elderly at Risk (PROSPER) trial.</p> <p><b>Methods:</b> Randomization was not stratified according to whether or not subjects participated in the MRI substudy. In 391 men (n=226) and women (n=165) aged 70 to 82 years (mean±SD, 75±3.2), we measured total CBF (in mL/min) at baseline and after a mean±SD follow-up of 33±1.4 months with a gradient-echo phase-contrast MRI technique. Total CBF was defined as the summed flows in both internal carotid and vertebral arteries. Parenchymal volume (whole brain) was segmented with the use of in-house–developed semiautomatic software.</p> <p><b>Results:</b> Total CBF significantly declined in the placebo-allocated group, from 521±83 to 504±92 mL/min (P=0.0036) and in the pravastatin-allocated group from 520±94 to 506±92 mL/min (P=0.018). This decline was not significantly different between treatment groups (P=0.56). There was also a significant reduction in brain volume over time (P<0.001), which was not different between the treatment groups (P=0.47). When expressed per unit of parenchymal volume, the decline in CBF over time was no longer statistically significant.</p> <p><b>Conclusions:</b> Elderly people at risk for cerebral vascular disease had a significant decline in CBF with increasing age that was explained by a concomitant reduction in brain volume. Treatment with 40 mg pravastatin daily had no beneficial effect on total CBF.</p&gt

Strong coupling in massive gravity by direct calculation

We consider four-dimensional massive gravity with the Fierz-Pauli mass term. The analysis of the scalar sector has revealed recently that this theory becomes strongly coupled above the energy scale \Lambda = (M_{Pl}^2 m^4)^{1/5} where m is the mass of the graviton. We confirm this scale by explicit calculations of the four-graviton scattering amplitude and of the loop correction to the interaction between conserved sources.Comment: 9 pages, 3 figures, some clarifications adde