Let PK 2 (Q(ε)) be the projective Klingenberg plane coordinated by the dual quaternion ring Q(ε) = Q + Qε = {x + yε | x, y ∈ Q} where Q is any quaternion ring. In this paper, we determine the addition and multiplication of the points on the line [0, 1, 0] of PK 2 (Q(ε)) as the image of some collineations of the plane PK 2 (Q(ε)). To do this, we give the collineations S a and L a . Later we show that the addition and multiplication of the nonneighbor points on the line [0, 1, 0] can be obtained as the images under that S a and L a . MSC: 51C05; 51J10; 12E15

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Graphs are essential tools to illustrate relationships in given datasets visually. Therefore, generating graphs from another concept is very useful to understand it comprehensively. This paper will introduce a new yet simple method to obtain a graph from any finite affine plane. Some combinatorial properties of the graphs obtained from finite affine planes using this graph-generating algorithm will be examined. The relations between these combinatorial properties and the order of the affine plane will be investigated. Wiener and Zagreb indices, spectrums, and energies related to affine graphs are determined, and appropriate theorems will be given. Finally, a characterization theorem will be presented related to the degree sequences for the graphs obtained from affine planes.

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This paper deals with a certain class of projective Klingenberg planes over the local ring F[η]/<η^{m}> with F an arbitrary field, known as the plural algebra of order m. In particular addition and multiplication of points on a line is defined geometrically and interpreted algebraically, by using the coordinate ring.