In this paper, we first introduce a new graph

Semigroups are one of the types of algebra to which the methods of universal algebra are applied. During the last three decades, graph theory has established itself as an important mathematical tool in a wide variety of subjects. One of the several ways to study the algebraic structures in mathematics is to consider the relations between graph theory and semigroup theory known as algebraic graph theory. Cayley graphs of semigroups have been extensively studied, and many interesting results have been obtained (see [

When a new structure appears, it is necessary to investigate the properties. For example, in [

Some basic preliminaries, useful notations, and valuable mathematical terminologies needed in what follows are prescribed. Note that a graph

Recall that the girth of a graph

Rees matrix semigroups were first introduced by Rees [

Let us write

Here, every

The diagram for the semigroup

We note that index sets mentioned in

The vertices are all elements of

If

One may address three major problems related to graph theory when a new structure emerges:

An example of the graph

In this section, by considering the graph

If

In the graph of the semigroup

For the semigroup

By considering the definition of

The degree of a vertex

For the semigroup

Let us consider the vertex

The degree sequence is the list of degree of all the vertices of the graph. This sequence is denoted by

Let

The main idea of the proof is to consider whether the degree of each vertex is the same. In accordance with definition of the graph

A subset

For the semigroup

Let us consider the graph

Graph coloring is an assignment of labels, called colors, to the vertices of a graph such that no two adjacent vertices share the same color. The chromatic number

Let us give the following lemma known as perfect graph theorem of Lovasz [

A graph is perfect if and only if its complement is perfect.

The chromatic number of

It is well known that the vertex

The vertex

The types of

The types of

Then,

The clique number of

Now, let us consider the complete subgraph

Then,

By keeping in our minds the definition of perfect graphs [

The graph

By using Lemma

A complement of the graph

We recall that any graph

A graph is perfect if and only if it is Berge.

The graph

As a final note, we may refer [

In this section, we give some Zagreb indices of the graph

A graph invariant is a number related to a graph which is a structural invariant. The well-known graph invariants are the Zagreb indices. Two of the most important Zagreb indices are called first and second Zagreb indices denoted by

They were first defined by Gutman and Trinajstić in [

In [

(see [

Both the first Zagreb index and the second Zagreb index give greater weights to the inner vertices and edges and smaller weights to outer vertices and edges, which opposes intuitive reasoning. Hence, they were amended as follows [

The hyper-Zagreb index, defined as

(see [

It is seen that

Let us remind a well-known result called by the

Now, let us give the results related to Zagreb indices for the graph

A number of vertex for the graph

It is known that there are four different types of vertices in this graph. A type of the vertex

A number of edge for the graph

By equation (

Then, a number of edge for the graph

Let

Let us start with the proof of the first Zagreb index. The graph of

For second Zagreb index, we clearly mention the definition of second Zagreb index for graph in this section. Firstly, we must consider adjacent vertices of the graph, which is as follows.

The first multiplicative Zagreb index and second multiplicative Zagreb index are

We give the number of vertex and degree of the graph

Second multiplicative Zagreb index is

The first Zagreb coindex and second Zagreb coindex are

We know that the numbers of vertex and edge for the graph

Similarly, by Lemma

The first modified Zagreb index and second modified Zagreb index are

The definition of modified Zagreb index is to insert inverse values of the vertex-degrees into

The forgotten index is

We know that

For the graph of

The hyper-Zagreb index and hyper-Zagreb coindex are

We obtain forgotten index and second Zagreb index of the graph

The most important aspect of thinking graph on a new algebraic structure is that the graph reflects new results for both algebraic structure and graph. When a new graph appears, it is important to study some graph properties, namely, diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, and clique number (in Section

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

The author declares that there are no conflicts of interest.