This study analyzed the structure of onion marketing in Yola North Local Government Area of Adamawa State, Nigeria. Data were collected from Jimeta Modern market, Old market and Yola by-pass market based on the existing sampling frame. Descriptive statistics, Gini index and Marketing efficiency were the analytical tools employed. The results revealed that majority (92.5%) of the respondents were male, 62.5% were married with an average family size of 4 persons per household and 52.5% had some level of formal education with a mean marketing experience of 10 years. The average sales recorded per month for wholesalers and retailers were N2,888,000.00 and N372,237.50 respectively, while their respective net incomes per month were N234,610.00 and N35,743.73 respectively. The value of Gini coefficient for wholesalers and retailers were 0.47 and 0.52 respectively, an indication of inequality in earnings among the marketers due to high market concentration resulting to poor market structure. The results further revealed that onion marketing is inefficient in the study area with marketing efficiency scores of 8.82% and 10.62% for wholesalers and retailers respectively. The Return on Investment (ROI) for wholesalers and retailers were N0.081 and N0.096 respectively with retailers having relatively higher ROI. Lack of credit facilities (80%), high cost of transportation (73%) and security challenges (42.5%) were identified as the major problems affecting onion marketers in the area. The study recommends that government should expand its anchor borrower scheme program to include marketers in order to enable them expand their business and improve marketing efficiency.

This paper studies various results on vertex colorings of simple connected graphs, chromatic number, chromatic polynomials and some Algebraic properties of chromatic polynomials. Results were obtained on the roots of chromatic polynomials of simple connected graphs based on Read's conjecture. The chromatic number of every graph is the minimum number of colors to properly color the graph. Chromatic polynomial of a graph is a polynomial in integer and the leading coefficient of chromatic polynomial of a graph of order n and size m is always 1, whose coefficient alternate in sign. Through the application of famous graph theorem (the hand shaking lemma) by whiskey which states that: "the order of a graph twice its size". Hence, every graph has a chromatic polynomial but not all polynomials are chromatic. For example, the polynomial λ 5 − 11 λ 4 + 14 λ 3 − 6 λ 2 + 2 λ is a polynomial for a graph on five vertices and eleven edges which does not exists. Because the maximum number size for a graph of order five is ten. The paper equally gave some practical applications of Vertex coloring in real life situations such as scheduling, allocation of channels to television and radio stations, separation of chemicals and traffic light signals.

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The study determined allocative efficiency among rainfed rice farmers in Ardo-kola and Jalingo local government area of Taraba State, Nigeria. The objectives of the paper were to determined allocative efficiency of rice farmers in the study area. Purposive and simple random sampling techniques were used to collect data from 180 rice farmers through a structured questionnaire. Both descriptive statistics and stochastic frontier approach were used to analyze data collected. Diagnostic statistic, gamma (0.856) and sigma squared (0.191) were statistically significant at P≤ 0.01. probability level. The result further revealed that cost of seed and cost of fertilizer were significant at P≤ 0.01, while cost of family labour and cost of agrochemicals were significant at P≤ 0.05. The mean allocative efficiency (AE) is 0.89. The study also revealed that education, age and extension contact and gender increases allocative efficiency of rice farmers in the study area. The study recommends diversification of income, especially to rice farmers with large household size. However, farmers should be encouraged with improved facilities either from government and non-governmental agencies.