Judul: Hubungan Graf, Ruang Kebebasan, dan Dual Graf Pada Struktur Cycle dan Struktur Cutset
Abstrak: Ruang kebebasan adalah suatu ruang yang pembentukannya berdasarkan konsep sifat bebas linear dan konsep graf. Ruang kebebasan terdiri dari koleksi himpunan bebas-himpunan bebas yang merupakan himpunan bagian dari suatu himpunan berhingga, dan himpunan berhingga itu sendiri secara umum dapat dikatakan mirip dengan himpunan sisi pada graf. Himpunan bagian yang bukan himpunan bebas disebut himpunan takbebas. Koleksi himpunan bebas-himpunan bebas disebut struktur kebebasan. Himpunan bebas maksimal disebut basis dan himpunan takbebas minimal disebut sirkuit. Dari sebuah graf terhubungkan dapat diperoleh dua buah struktur kebebasan yaitu struktur cycle dan struktur cutset. Struktur cycle adalah himpunan sisi yang tidak memuat cycle, sedangkan struktur cutset adalah himpunan sisi yang tidak memuat cutset. Koleksi basis dari struktur cicle merupakan spanning Tree-spanning Tree pada graf, sedangkan koleksi basis dari struktur culset "merupakan komplemenkomplemen dari spanning Iree. Dengan melihat basis, hubungan graf dan fuang kebebasan dapat dinyatakan sebagai berikut, struktur cycle dan struktur culset merupakan dual struktur kebebasan.
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Judul: Cycle Hamilton pada Graf Lengkap, Graf Regular, dan Graf 2-Connected 4-Regular Berorder Kurang dari Sepuluh.
Abstrak: Suatu graf disebut graf Hamilton apabila graf tersebut memuat cycle Hamilton, yaitu cycle yang melewati setiap vertex tepat satu kali. Menentukan keberadaan cycle Hamilton pada suatu graf dapat menggunakan syarat cukup yang berupa teorema-teorema yang telah diperkenalkan sejak tahun 1950-an. Beberapa syarat cukup yang dibahas dalam karya ilmiah ini adalah syarat cukup yang berhubungan dengan graf lengkap dan graf regular. ,,,
Keyword: graf Hamilton, graf regular, graf lengkap, graf 2-connected 4-regular.

Judul: Utilization of hydrolized duckweed (Lemna minor) meal as a soybean replacement in Nile tilapia (Oreochromis niloticus) diet,
Abstrak: Tilapia is a fast growing fish widely cultured in many tropical and sub tropical countries. The recent production figures reported our global production estimates for 2015 is 5,576.8 metric tons. In developing countries the major challenges facing aquaculture industry is high cost of production associated with fish feeds. Feed represents 40-70% of operating costs for aquaculture operations, increasing expenses and potential declining supplies soybean meal have led scientists to search for less expensive but compatible alternative protein sources for fish feed; most of which are based on animal or plant protein sources. The objectives of study was to determine the incubation period for hydrolysis of duckweed and to evaluate the use of hydrolized duckweed as aprotein source for the growth performance of Nile tilapia Three experiments were conducted to determine the composition and digestibility of hydrolyzed duckweed meal by sheep rumen liquor as the replacement of soybean meal in the diet of Nile tilapia. The first trial examined the effectivity of sheep rumen liquor enzyme to decrease crude fiber contents of (DWM), the added volume of sheep rumen liquor was 100 ml/kg DWM and then it was incubated for 0, 12, 24 hr . The second trial was to evaluate the apparent digestibility of hydrolyzed DWM for Nile Tilapia, apparent digestibility were determined using chromic oxide indicator added to both reference diet and test diets. The feed ingredients used in the trial were hydrolyzed DWM (HDWM) and unhydrolyzed DWM (DWM). Ten fish with an average weight of 18±0.36g were used in the trial and reared in the aquarium of size 30×40×50cm for 15 days; feces were collected from three replicates using a fecal collection column attached to the aquarium. Third trial was to evaluate the growth performance of Tilapia fed test diet with hydrolyzed Duckweed meal (HDWM). The experimental design was set according to completely randomized design with four treatments and three replicates of HDWM inclusion levels that is 0%, 15%, 30% and 45%. Nile tilapia fish with initial body weight of 18±0.36g was kept in 12 units aquarium size of 30×40×50cm at stocking density of 10fish/aquarium, the fish were fed two times a day at 08:00am and 16:00pm to apparent satiation for a period of 60 days. The results indicated that duckweed meal (DWM) hydrolyzed with 100 mL/kg and incubated for 24 hrs showed the lowest crude fiber contents of 13.68% compared to other treatments. Total digestibility of hydrolyzed duckweed (HDWM) meal was 87.3% compared with unhydrolyzed duckweed (DWM) which was 66.8%. Total final weight (TFW), survival rate (SR), specific growth rate (SGR), feed conversional rate (FCR) and protein retention (PR) shows growth performance at 15% inclusion. The results of this study showed that 100mL/kg sheep rumen liquor at incubation period of 24 hr can be suitable for hydrolysis of duckweed to reduce crude fiber also hydrolyzed duckweed can substitute soybean meal at an incubation rate of 15% in the diet of Nile tilapia and had the same growth and FCR with control diet
Keyword: Digestibility, Duckweed meal, Growth performance, Sheep rumen