Abstract
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.
Highlights
We start with some definitions and fundamental notions regarding affine planes from [1].Definition 1
An affine plane A is an ordered pair (P, L) which we call the elements of P as points and the elements of L as lines, with the following properties: (A1) Any two distinct points lie on a unique line (A2) For each point p not on a line l, there is exactly one line l′ passing through p such that l is parallel to l′
What is the relation between affine and projective graphs which have the same order? For example, how does the existence and absence of the parallel axiom in affine and projective planes affect the graphs that are obtained from these planes?
Summary
Received 24 March 2021; Revised 13 April 2021; Accepted 16 April 2021; Published 4 May 2021. Graphs are essential tools to illustrate relationships in given datasets visually. Erefore, generating graphs from another concept is very useful to understand it comprehensively. Is 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. E 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. A characterization theorem will be presented related to the degree sequences for the graphs obtained from affine planes
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