Search results for "Maximum size"

The prime graph on class sizes of a finite group has a bipartite complement

2020

Abstract Let G be a finite group, and let cs ( G ) denote the set of sizes of the conjugacy classes of G. The prime graph built on cs ( G ) , that we denote by Δ ( G ) , is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cs ( G ) , and two distinct vertices p, q are adjacent if and only if pq divides some number in cs ( G ) . A rephrasing of the main theorem in [8] is that the complement Δ ‾ ( G ) of the graph Δ ( G ) does not contain any cycle of length 3. In this paper we generalize this result, showing that Δ ‾ ( G ) does not contain any cycle of odd length, i.e., it is a bipartite graph. In other words, the vertex set V ( G ) of Δ ( G ) is covered b…

Bounding the number of vertices in the degree graph of a finite group

2020

Abstract Let G be a finite group, and let cd ( G ) denote the set of degrees of the irreducible complex characters of G . The degree graph Δ ( G ) of G is defined as the simple undirected graph whose vertex set V ( G ) consists of the prime divisors of the numbers in cd ( G ) , two distinct vertices p and q being adjacent if and only if pq divides some number in cd ( G ) . In this note, we provide an upper bound on the size of V ( G ) in terms of the clique number ω ( G ) (i.e., the maximum size of a subset of V ( G ) inducing a complete subgraph) of Δ ( G ) . Namely, we show that | V ( G ) | ≤ max { 2 ω ( G ) + 1 , 3 ω ( G ) − 4 } . Examples are given in order to show that the bound is bes…

2013

To estimate the body temperature (BT) of seven dinosaurs Gillooly, Alleen, and Charnov (2006) used an equation that predicts BT from the body mass and maximum growth rate (MGR) with the latter preserved in ontogenetic growth trajectories (BT-equation). The results of these authors evidence inertial homeothermy in Dinosauria and suggest that, due to overheating, the maximum body size in Dinosauria was ultimately limited by BT. In this paper, I revisit this hypothesis of Gillooly, Alleen, and Charnov (2006). I first studied whether BTs derived from the BT-equation of today’s crocodiles, birds and mammals are consistent with core temperatures of animals. Second, I applied the BT-equation to a …

Deterioration patterns in diapausing egg banks of Brachionus (Müller, 1786) rotifer species

2005

Rotifers are cyclical parthenogens that produce sexual diapausing eggs at some stage in their life cycle. These eggs are encysted embryos that remain viable for extended periods in lake and pond sediments, thus acting as an egg bank with many ecological and evolutionary consequences. Despite its importance to rotifer evolution, there are no studies on resting egg deterioration and associated processes in natural environments. In this study, more than 4000 diapausing eggs of species from the Brachionus plicatilis complex, which includes several closely related cryptic species, were collected from different sediment depths in 15 ponds in eastern Spain and were classified according to three fe…