8193810
On the size of graphs with all cycle having distinct length
Abstract Let ƒ(n) be the maximum number of edges in a graph on n vertices in which no two cycles have the same length. In 1975, Erdős raised the question of determining ƒ(n ) (see Bondy and Murty (1976)). In this note, we prove that for n ⩾36·5 t 2 −4·5 t +1 one has ƒ(n)⩾n+9t−1 . We conjecture that lim (ƒ(n)−n)/√n⩽√3 .
7497035
A Lower Bound for the Number of Edges in a Graph Containing No Two Cycles of the Same Length
In 1975, P. Erdős proposed the problem of determining the maximum number $f(n)$ of edges in a simple graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+32t-1$$ for $t=27720r+169 \,\ (r\geq 1)$ and $n\geq {{6911}\over {16}}t^{2}+{{514441}\over {8}}t-{{3309665}\over {16}}$. Consequently, $\liminf _{n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + {2562 \over 6911}}.$
11729796
Ecodynamics and Dissolved Gas Chemistry Routines for Ocean Circulation Models
Abstract Interactions between oceanic nitrate ecology and circulation determine the marine distribution of dissolved, climate relevant trace gases such as dimethyl sulfide (DMS) and carbonyl sulfide (OCS), and a variety of hydrocarbons. Our group is constructing a suite of ecosystem/reaction/transport models, which link nitrate to the chemistry of volatiles near the sea–air interface. In this paper, we describe programs which will be inserted into the high resolution Parallel Ocean Program. Major features of the coding include: (1) ecodynamics represented in seven biological compartments (phytoplankton, zooplankton, bacteria, detritus, nitrate, ammonium and dissolved organic nitrogen). Light limited primary production is computed, along with nitrogen cycling among the bioentities. (2) Photochemistry for the volatile species DMS, OCS, the methyl halides, nonmethane hydrocarbons (NMHC) and ammonia. DMS and the halides are emitted by phytoplankton, while OCS and NMHC are produced by photolysis of dissolved organic material. Ammonia is exuded by animals and bacteria. Removal mechanisms for the gases include consumption by organisms, hydrolysis, chlorination and interfacial transfer. (3) Explicit, efficient and mass conserving numerical solutions for the biological and chemical continuity equations. Production and loss forms are generalized and automated so that they are readily applied to new constituents. Ecology and the chemical transformations are exposed qualitatively to begin, and are then expressed as differential and differencing equations. The structure of the program is described in terms of the major subroutines and their purposes. Results are provided from both one- and three-dimensional sample runs. Computational aspects such as performance and code availability are discussed.