The sum number of the cocktail party graph

A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a finite number of isolated vertices to it will always yield a sum graph, and the sum number oe(G) of G is the smallest number of isolated vertices that will achieve this result. A labelling that realizes G + K oe(G) as a sum graph is said to be optimal. In this paper we consider G = H m;n , the complete n-partite graph on n 2 sets of m 2 nonadjacent vertices. We give an optimal labelling to show that oe(H 2;n ) = 4n \Gamma 5, and in the general case we give constructive proofs that oe(H m;n ) 2 \Omega\Gamma mn) and oe(H m;n ) 2 O(mn 2 ). We conjecture that oe(H m;n ) is asymptotically greater than mn, the cardinality of the vertex set; if so, then H m;n is the first known graph with this property. We also provide for the first time an optimal labelling of the complete bipatite graph Kmn whose smallest label is 1

Limit-feeding a high-concentrate diet may alter nutrient absorption

Feeding newly arrived cattle is commonly characterized by a few days of feeding longstemmed hay followed by a series of step-up diets, wherein concentrate levels are increased to promote ruminal adaptation to a highconcentrate finishing diet. This is done to give the rumen microbes time to adjust to larger amounts of readily fermentable starches in cereal grains. Rumen epithelial adaptation may be achievable by limit-feeding a finishing diet, with gradual increases in feed intake, until the cattle are on full feed. If this can be achieved without causing ruminal disorders and days off feed, then the cost of feeding cattle can be reduced. By limit-feeding, the higher roughage step-up diets are replaced with a single high-concentrate diet. The cost of grain is less than that of roughage, and there are decreased costs in terms of storage space, waste disposal (due to decreased manure production), and mixing and hauling of rations. The purpose of this experiment was to examine the effects of a traditional step-up program versus a limit-fed finishing diet in terms of dry matter intake, acetate to propionate ratio, and ruminal dilution rate. Diet effects on volatile fatty acid concentration and absorption were also examined by using valerate as a marker for volatile fatty acid absorption

Comparison of three fresh cow feeding programs

We evaluated the impact on performance of top dressing a based total mixed ration (TMR) with long-stem alfalfa hay with or without additional dry-rolled corn to the lactating cow diet during the first 5 days postpartum. The three dietary treatments and numbers of cows assigned to each diet were: 1) total mixed ration (TMR; n = 19); 2) TMR + long-stem alfalfa hay (TMR + A; n= 20); and 3) TMR + long-stemmed alfalfa hay + dry-rolled corn (TMR + A + C; n = 20). Top dressing the lactating TMR with long-stem alfalfa hay with or without dry-rolled corn did not reduce the incidence of metabolic disorders in early lactating cows. Six cows, two on each diet, were treated for displaced abomasums. Cows consuming only the TMR lost slightly more body weight during the first 30 days after calving compared to cows fed the other diets. Milk and energy corrected milk (ECM) yields were similar among diets. Fat, protein, and urea nitrogen content in milk were not different among dietary treatments. Lactose content in milk was greater for cows consuming TMR + A than those consuming TMR or TMR + A + C. Concentrations of glucose and urea nitrogen in plasma were not affected by treatment during the initial 5 days of lactation. Concentrations of glucose and urea nitrogen on days 2 and 3 were less for multiparous cows consuming TMR than for multiparous cows consuming TMR + A. Rumen contractions during the first 5 days of lactation were not different among diets. Top dressing the lactating TMR with long-stem alfalfa hay with or without dry-rolled corn was not beneficial in this study. On a dry matter basis, the lactating TMR contained 22% chopped alfalfa hay, 10% corn silage, 20% wet corn gluten feed, 9% whole fuzzy cottonseed, 7.1% expeller soybean meal, 27.4% ground shelled corn, 1.2% molasses, 1.3% Menhaden fishmeal, and 2.0% mineral-vitamin premix. Cows fed diets containing corn silage as the predominant fiber source may respond differently.; Dairy Day, 2003, Kansas State University, Manhattan, KS, 2003

Responses of lactating holstein cows to low-pressure soaking or high-pressure misting during heat stress

Lactating dairy cattle were used to evaluate three different cooling systems. Eight cows were arranged in a replicated Latinsquare design and assigned to each of four treatments. Treatments were control, lowpressure soaking (LPS), high-pressure misting with 1.7 gallons per minute of water (HP-1.7), or high-pressure misting with 3.4 gallons per minute of water (HP-3.4). Cows were allowed to become heat stressed in a free-stall facility, and then were moved to a tie-stall barn for 2 hours of observations during four hot and humid afternoons. Respiration rates declined when heat abatement systems were used. Respiration rates at the end of the observation period were reduced by 20, 36, and 48% for HP-1.7, HP-3.4, and LPS, respectively. Rearudder skin surface temperature was reduced at a faster rate under the HP-4 treatment than with LPS, but the two treatments did not differ in final rear-udder skin surface temperature or vaginal temperature. The HP-3.4 treatment used the greatest amount of water during the 2-hour testing period. The result was a combination of air-cooling and soaking. Results indicated that a combination of air cooling and soaking may result in faster reduction of surface temperature. When only air cooling was used (HP-1.7), heat stress was reduced, but it was less effective than either LPS or HP-3.4. Use of a low-pressure soaking system is superior to high-pressure misting unless cattle become soaked by the high-pressure system.; Dairy Day, 2004, Kansas State University, Manhattan, KS, 2004

Responses of lactating holstein cows to differing levels and direction of supplemental airflow

Seven heat-stressed, lactating Holstein cows were exposed to six different cooling systems to evaluate the effects of air velocity and direction of airflow. Cows were arranged in a 7 x 7 Latin-square design. Six cooling treatments were compared with a control. Supplemental airflow was provided by axial flow at one of three velocities: 500, 750, or 900 cubic feet per minute (CFM). Airflow was either from the front to rear (FRT) or from the right side (SIDE) of the cow. Combined cooling treatments were FRT-500, FRT- 750, FRT-900, SIDE-500, SIDE-750, or SIDE-900. All cooling systems used a lowpressure soaking system that operated 1 minute every 5 minutes. Respiration rates, rearudder skin surface temperature, and vaginal temperature were measured and recorded during 2 hours of treatment during seven hot and humid afternoons. Cooling systems reduced respiration rate, rear-udder skin surface temperature, and vaginal temperature. When airflow was 750 or 900 CFM, no differences were observed among treatments. When airflow was 500 CFM, rate of decline of rearudder skin surface temperature and vaginal temperature were reduced, compared with those of other treatments. These results indicate that there was no advantage to increasing airflow more than 750 CFM when using a low-pressure soaking system that wets the cattle every 5 minutes. Differences due to airflow direction were only observed when airflow was reduced to 500 CFM. At 500 CFM, airflow from head to tail was not as effective as from the side. Current recommendations of 750 CFM of airflow directed at the side of the cow are effective in reducing heat stress of lactating dairy cattle.; Dairy Day, 2004, Kansas State University, Manhattan, KS, 2004

Comparative value of full-fat corn germ, whole cottonseed and tallow as energy sources for lactating dairy cows

We used 24 multiparous Holstein cows in 4 x 4 Latin square design to evaluate full-fat corn germ as a replacement for whole cottonseed and tallow in total mixed diets for lactating dairy cows. Experimental diets on a dry matter basis were: 1) control 3.5% fat; 2) whole cottonseed 5.1% fat; 3) tallow 5.1% fat; 4) full-fat corn germ 5.1% fat. Diets were fed as total mixed rations typical of that fed on commercial dairy operations. Cottonseed meal and cottonseed hulls were included in the control, tallow, and full-fat corn germ diets to balance for fiber and protein fractions equal to those in the whole cottonseed diet. Dry matter intake, milk production, and energy corrected milk did not differ among the diets. Milk from cows fed full-fat corn germ contained less fat than milk from cows fed whole cottonseed but was similar to that of milk from cows fed control or tallow diets. Milk protein percentage was lower for cows fed full-fat corn germ than those fed control, but similar to cows fed whole cottonseed or tallow. Percentage milk lactose did not differ among dietary treatments. Cows fed WCS produced more pounds of milk fat than cows fed full-fat corn germ or tallow, but protein and lactose yield did not differ among the diets. Cows fed whole cottonseed produced milk more efficiently than cows fed control, tallow, or full-fat corn germ. Unexpectedly, efficiency of energy corrected milk production was not improved by tallow and tallow did not depress dry matter intake. Somatic cell count did not differ among experimental diets. Urea nitrogen concentration was lower in milk from cows fed full-fat corn germ and tallow than those fed whole cottonseed. All diets led to gains in body weight. The handling and storage characteristics of full-fat corn germ enhances its desirability as a feedstuff for dairy cattle. Full-fat corn germ supported milk production as well as whole cottonseed but not milk fat percentage or fat yield at the level fed in our diets. Additional studies need to be conducted to determine the most advantageous amount to feed full-fat corn germ and clarify the mechanisms by which it depresses milk fat production.; Dairy Day, 2002, Kansas State University, Manhattan, KS, 2002

Effect of soaking and misting on respiration rate, body surface temperature, and body temperature of heat stressed dairy cattle

Reducing heat stress is a key issue for dairy producers. Use of feedline soaking and supplemental airflow effectively reduces heat stress and increases milk production and profitability. High-pressure misting allows water to evaporate in the air, reduces air temperature, and increases relative humidity. Misting also soaks the skin of cattle, resulting in additional cooling as water evaporates from skin surfaces, similar to the cooling effect of feedline soaking. Impact of soaking frequency (5-, 10-, or 15-minute intervals) was compared to continuous high-pressure misting. Cows cooled with either system had lower respiration rates, body surface temperatures, and internal body temperatures than controls. Soaking cattle every 5 minutes or 5-minute soaking plus high-pressure misting produced similar body temperatures, but lower (P\u3c0.01) than those when soaking occurred every 10 or 15 minutes. Skin surface temperatures from the thurl, shoulder, and rear udder were less when cattle were cooled with high-pressure misting. Cattle cooled with high-pressure misting became soaked, thus the cooling effect is the combination of cooler air and water evaporation from the skin. These results indicate that either frequent soaking (every 5 minutes) or continuous high-pressure misting that soaks the skin could be equally effective in reducing heat stress in dairy cattle.; Dairy Day, 2003, Kansas State University, Manhattan, KS, 2003

The sum number of a disjoint union of graphs

In this paper we consider the disjoint union of graphs as sum graphs. We provide an upper bound on the sum number of a disjoint union of graphs and provide an application for the exclusive sum number of a graph. We conclude with some open problems

Labelling wheels for minimum sum number

A simple undirected graph G is called a sum graph if there exists a labelling L of the vertices of G into distinct positive integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label L(w) = L(u) +L(v). It is obvious that every sum graph has at least one isolated vertex, namely the vertex with the largest label. The sum number oe(H) of a connected graph H is the least number r of isolated vertices K r such that G = H+K r is a sum graph. It is clear that if H is of size m, then oe(H) m. Recently Hartsfield and Smyth showed that for wheels W n of order n+1 and size m = 2n, oe(W n ) 2 Theta(m); that is, that the sum number is of the same order of magnitude as the size of the graph. In this paper we refine these results to show that for even n 4, oe(W n ) = n=2 + 2, while for odd n 5 we disprove a conjecture of Hartsfield and Smyth by showing that oe(W n ) = n. Labellings are given that achieve these minima

On optimal summable graphs

For a graph G , let σ ( G ) and δ ( G ) denote, respectively, its sum number and minimum degree. Trivially, σ ( G ) ≥ δ ( G ) . A nontrivial connected graph G is called a k -optimum summable graph , where k ≥ 1 , if σ ( G ) = δ ( G ) = k . In this paper, we show that if G is a k -optimum summable graph of order n , k ≥ 3 , then (1) n ≥ 2 k ; (2) the complete bipartite graph K k,n − k is not a spanning subgraph of G . We also describe new families of k -optimum summable graphs for k ≥ 1