• Communications of the Korean Mathematical Society
• /
• v.17 no.2
• /
• pp.363-370
• /
• 2002

We show that G(x) = $e^{(x}$(1-x))/ -1 is the exponential generating function for the labeled digraphs whose weak components are transitive tournaments and derive both a recursive formula and an explicit formula for the number of them on n vertices. Moreover, we investigate the asymptotic behavior for the coefficients of G(x) using Hayman's method.d.

• Journal of Life Science
• /
• v.15 no.3 s.70
• /
• pp.365-373
• /
• 2005

The quantitative inheritance of some yield characters in Gosyium spp was carried out by means of a $10\times10$ diallel cross. In this study, 45 combinations of $F_1\;and\;F_2$ generations were genetically analyzed through 10 different cultivars diallel cross population of cotton (Gosyium spp) at an experimental field. The results of Vr-Wr graph analysis of six characters such as number of boll, boll weight, lint weight per boll, 100 seeds weight, fiber fineness and fiber length in those combinations by the Hayman's method were as follow: 1. The significant difference was observed from the genetic variance of all the examined characters. 2. On based the Vr-Wr graphical analysis, $F_1$ showed a complete dominance in all the experimental characters except boll weight, lint weight per boll and fiber fineness, but the dominance degree and gene arrangement of $F_2$ were somewhat different from those of $F_1$.

• Korean Journal of Agricultural Science
• /
• v.4 no.2
• /
• pp.254-282
• /
• 1977

To obtain information on the inheritance of the quantitative characters related with the vegetative and reproductive growth of rice, the $F_1$ seeds were obtained in 1974 from the all possible combinations of the diallel crosses among five leading rice varieties : Nongbaek, Tongil, Palgueng, Mangyeong and Gimmaze. The $F_1$'s including reciprocals and parents were grown under the standard cultivation method at Chungnam Provincial Office of Rural Development in 1975. The arrangement of experimental plots was randomized block design with 3 replications and 12 characters were used for the analysis. Analytical procedure for genetic components was followed the Griffing's and Hayman's methods and the results obtained are summarized as follows. 1. In all $F_1$'s of Tongil crosses, the longer duration to heading was due to dominant effect of Tongil and each $F_1$ showed high heterosis in delaying the heading time. It was assumed that non-allelic gene action besides dominant gene effect might be involed in days to heading character. However, in all $F_1$'s from the crosses among parents excluding Tongil the shorter duration was due to dominant gene action and the degree of dominance was partial, since dominance effects were not greater than the additive effect. The non-allelic gene interaction was not significant. Considering the results mentioned above, it was regarded that there were two kinds of Significantly different genetic systems in the days to heading. 2. The rate of heterosis was significantly different depending upon the parents used in the crosses. For instance, the $F_1$'s from Togil cross showed high rate of heterosis in longer culm. Compared to short culm, longer culm was due to recesive gene action and short culm was due to recesive gene action. The dominant gene effect was greater than the additive gene effect in culm length. The narrow sense of heretability was very low and the maternal effects as well as reciprocal effects were significantly recognized. 3. The lenght of the of the uppermost internode of each $F_1$ plant was a little lorger than these of respective parental means or same as those of parents having long internodes, indicating partial dominance in the direction of lengthening the uppermost internodes. The additive gene effects on the uppermost internode was greater than the dominance gene effect. The narrow as well as broad sense of heritabilities for the character of the uppermost internode were very high. There were significant maternal and reciprocal effect in the uppermost internode. 4. The gene action for the flag leaf angle was rather dominance in a way of getting narrower angle. However, in the Palgueng combinations, heterosis of $F_1$ was observed in both narrow and wide angles of the flag leaf. The dominant effects were greater than the additive effects on the flag leaf angle. There were observed also a great deal of non-allelic gene interacticn on the inheritance of the flag leaf angle. 5. Even though the dominant gene action on the length and width of flag leaf was effective in increasing the length or width of the flag leaf, there were found various degrees of hetercsis depending upon the cross combination. Over-dominant gene effect were observed in the inheritance of length of the flag leaf, while additive gene effects was found in the inheritance of the width of the flag leaf. High degree of heretabilities, either narrow or broad sense, were found in both length and width of the flag leaf. No maternal and reciprocal effect were found in both characters. 6. When Tongil was used as one parent in the cross, the length of panicle of $F_1$'s was remarkedly longer than that of parents. In other cross comination, the length of panicle of $F_1$'s was close to the parental mean values. Rather greater dominent gene effect than additive gene effect was observed in the inheritance of panicle length and the dominant gene was effective in increasing the panicle length. 7. The effect of dominant genes was effective in increasing the number of panicles. The degree of heterosis was largely dependent on the cross combination. The effect of dominant gene in the inheritance of panicle number was a little greater than that of additive genes, and the inheritance of panicle number was assumed to be due to complete dominant gene effects. Significantly high maternal and reciprocal effects were found in the character studied. 8. There were minus and plus values of heterosis in the kernel number per panicle depending upon the cross combination. The mean dominant effect was effective in increasing the kernel number per panicle, the degree of dominant effect varied with cross combination. The dominant gene effect and non-allelic gene interaction were found in the inheritance of the kernel number per panicle. 9. Genetic studies were impossible for the maturing ratio, because of environmental effects such as hazards delaying heads. The dominant gene effect was responsible for improving the maturing ratio in all the cross combinations excluding Tongil 10. The heavier 1000 grain weight was due to dominant gene effects. The additive gene effects were greater than the dominant gene effect in the 1000 grain weight, indicating that partial dominance was responsible for increasing the 1000 grain weight. The heritabilites, either narrow or broad sense of, were high for the grain weight and maternal or reciprocal effects were not recognized. 11. When Tongil was used as parent, the straw weight was showing high heterosis in the direction of increasing the weight. But in other crosses, the straw weight of $F_1$'s was lower than those of parental mean values. The direction of dominant gene effect was plus or minus depending upon the cross combinations. The degree of dominance was also depending on the cross combination, and apparently high nonallelic gene interaction was observed.

• KOREAN JOURNAL OF CROP SCIENCE
• /
• v.28 no.2
• /
• pp.216-226
• /
• 1983

Genetic studies on the $F_2$ generation of a set of half diallel crosses involving six cowpea varieties were conducted. by the randomized block design with three replications to determine combining ability, gene action and the relationships between parents and their $F_2$ hybrids. The 12 agronomic characters namely, days to flowering, days from flowering to maturity, days to maturity, diameter of stem, length of internode, number of branches per plant, length of pod, number of pods per plant, number of grains per pod, number of grains per plant, 100 grain weight and grain weight per plot were observed, and the $F_2$ generation of this diallel set of crosses was analysed for each character according to the method by Jinks and Hayman. The results obtained are summarized as follows: 1. Vr-Wr graphical analyses; The following seven characters, days to flowering, number of branches per plant, length of pod, number of pods per plant, number of grains per plant, 100 grain weight and grain weight per plot appeared to be partially dominant, and over dominance was found for days from flowering to maturity, days to maturity, length of internode and number of grains per pod. But diameter of stem indicated partial dominance near complete dominance. 2. Estimates of genetic variance components; In the degree of dominance,. eight characters, that is, days to flowering, days from flowering to maturity, days to maturity, length of internode, number of pods per plant, number of grains per pod, number of grains per plant and grain weight per plot were expressed larger than 1. And the characters, days from flowering to maturity, number of branches per plant and number of grains per plant as the degree of mean dominance ($H_1$/D) were found to be negative value over other characters. On the other hand, apprent asymmetry of dominance-recessive allele ($H_2$ /$4H_1$) produced comparatively estimates with lower value on days from flowering to maturity, length of internode, number of branches per plant and number of grains per pod. 3. Analyses of combining ability; Mean square value of GCA(general combining ability) appeared to be more important than those of SCA (specific combining ability) for most characters, and among them, grain weight per plot showed the highest mean square value in GCA and SCA. 4. Effect of combining ability; Variety 178 was expressed as the highest GCA effects in five characters (days to flowering days to maturity, number of pods per plant, number of grains per plant and grain weight per plot). SCA effects were differed from parents, characters and crosses, but crosses between TVu 1857 $\times$ TVu 2885 and TVu 2702 $\times$ J78 were shown to be highly with SCA effects on yield.