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It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No 112 examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In this paper, we prove that digraphs of degree three and diameter k ! 3 which miss the Moore bound by one do not exist. ß

It is known that Moore digraphs of degree d > 1 and diameter k > 1 do not exist (see 1201 or IS]). Furthermore, for degree 2, it is shown that for k 2 3 there are no digraphs of order "close" to, i.e., one less than, Moore bound [181. In this paper, w e shall consider digraphs of diameter k, degree 3 and number of vertices one less than Moore bound. W e give a necessary condition for the existence of such digraphs and, using this condition, w e deduce that such digraphs do not exist for infinitely many values of the diameter.

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This study is a quasi-experimental nonrandomized pretest-posttest control group design. The experiment group is treated by APOS theory instruction (APOS), that implements four characteristics of APOS theory, (1) mathematical knowledge was constructed through mental construction: actions, processes, objects, and organizing these in schemas, (2) using computer, (3) using cooperative learning groups, and (4) using ACE teaching cycle (activities, class discussion, and exercise). The control group is treated by conventional/traditional mathematics instruction (TRAD). The main purpose of this study is to analyze about achievement in proof. 180 students from two different universities (two classes at the Department of Mathematics UNAND and two classes at the Department of Mathematics Education UNP PADANG) were engaged as the research subjects. Based on the result of data analysis, the main result of this study is that the proof ability of students' in the APOS group is significantly better than student in TRAD group, so it is strongly suggested to apply APOS theory in Abstract Algebra course.