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.

In this section, we start with some definitions and fundamental notions regarding affine planes from [

An affine plane

Any two distinct points lie on a unique line

For each point

There exists a set of three noncollinear points.

The lines

Condition (A2) is also called as

If

Let us suppose

If

Each line is parallel to

Each line

There are

In [

The lines can be grouped into three sets of parallel lines:

When a finite affine plane is given, there is a number

Let us now recall some fundamental notions of graph theory from [

Let

A graph

A graph

Obviously,

Affine plane of order 2.

The spectrum of a finite graph

The eigenvalues represented as

Let

A topological index related to a graph is a real number that must be a structural invariant. The topological indices are important for numerical relationships with the structure.

Let

The distance

Two of the most useful topological graph indices are the

Affine planes are one of the fundamental examples of incidence geometry. Considering the interpretation of graphs to social sciences and their application to network technologies, with the regularity and parallelism classes of affine planes, it is an interesting issue to obtain graphs from affine planes as one of the transitionalities that can yield results that are most logical and suitable for real-life practices.

We are going to be in an investigation for the answer to the following question: what if someone perceives the lines of a geometric structure as an element of graph theory and what could be obtained from this? To do so, we are introducing a new perspective. In this study, we work only with finite affine planes.

In this part, a method will be presented to obtain a graph from a given finite affine plane. We note that obtaining a graph

(1) Let

Throughout this paper

If we try to obtain a graph from a finite affine plane

Let us consider a line

For a line

In the following theorem, we obtain a graph from a finite affine plane

If

For any line

Let

When

When

In other cases, it is obvious from the construction of

Throughout this work, we are going to use the method introduced by Theorem

We should point out that we have already developed a new and different algorithm to create a new graph generating system from various incidence structures.

If

We know that if graph

Firstly, let us assume

Secondly, let us assume

Since

For a graph

From Theorem

If

We found that

If there were no condition

A graph obtained from an affine plane

Now, we will examine how affine graphs are obtained when the affine plane of order 2 and the affine plane of order 3 is taken as

We are going to investigate the graph obtained from the smallest affine plane given in Example

In this situation, as a result of Corollary

Thus,

Adjacency matrix for

The Wiener index of

Zagreb indices of

Let us take the affine plane of order 3 with points and lines given, respectively,

With the method given earlier, we know that

The vertex and edge sets of the graph

Thus,

The Wiener index of

Zagreb indices of

Affine plane of order 3.

Corresponding triangle for the line

The

Let us take the affine plane of order 4 with points and lines given, respectively:

With the method given earlier, we know that

Thus,

The Wiener index of

Zagreb indices of

When we obtain the affine graph for the affine plane of order 5, the affine graph of order 5, we calculate it as 25 vertex and 150 edges, and it is 12-regular. The adjacency matrix for this graph is given below.

The Wiener index for affine graph of order 5, namely,

Zagreb indices of

Also, it has spectrum

The results regarding the spectra and energies of affine graphs are consistent with the lower and upper boundaries, given for regular graphs in [

The

Affine graphs of order

Now, we give a theorem and a corollary for the characterization of the graphs that are obtained from affine planes.

Affine graphs of order

Let

We know that, in affine planes of order

The point

Therefore, every vertex in

Let

Proof can be done by following simple calculations.

Let

Proof can be done by the following simple calculations.

With this study, we start giving some relations between the incidence structures and graph theory from our new perspective. It is easily seen that there are some differences and similarities between the affine graphs of order

There are also some open problems in this subject that we are constantly studying on and which we think we are going to be able to examine and answer in the future. Some of them are in the following paragraphs.

We know that it is possible to obtain

We know that an affine plane is obtained when a line, with the points, is thrown away from a projective plane. Under which conditions can we find an affine graph of order

We know that affine planes of order

How can we determine whether a given graph is an affine graph?

Is there a relation between the isomorphism of geometric structures and associated graphs?

How can we determine the chromatic index of affine graphs? Is there a relationship between the parallel groups of affine planes and the chromatic index of corresponding affine graphs?

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.