On the bounds of degree-based topological indices of the Cartesian product of F-sum of connected graphs.

Topological indices are the mathematical tools that correlate the chemical structure with various physical properties, chemical reactivity or biological activity numerically. A topological index is a function having a set of graphs as its domain and a set of real numbers as its range. In QSAR/QSPR study, a prediction about the bioactivity of chemical compounds is made on the basis of physico-chemical properties and topological indices such as Zagreb, Randić and multiple Zagreb indices. In this paper, we determine the lower and upper bounds of Zagreb indices, the atom-bond connectivity (ABC) index, multiple Zagreb indices, the geometric-arithmetic (GA) index, the forgotten topological index and the Narumi-Katayama index for the Cartesian product of F-sum of connected graphs by using combinatorial inequalities.

Introduction and preliminary results

We consider G as a simple, connected and finite graph with a vertex set , an edge set , the order of and the size of . An edge with end vertices and is denoted by . The number of edges having u as an end vertex is called the degree of u in G and is denoted by . The minimum and maximum degrees of graph G are denoted by and , respectively. and are used for path and cycle with order n, respectively.

The branch of chemistry in which we discuss and predict the chemical structure by using mathematical tools without referring to quantum mechanics is called mathematical chemistry [1, 2]. The branch of mathematical chemistry which applies graph theory to mathematical modeling of chemical phenomena is known as chemical graph theory [2]. This theory has a remarkable role in the development of chemical sciences.

The Zagreb indices are the first degree-based structure descriptors [3, 4]. The terms , and first appeared in the topological formula for total π-energy of conjugated molecules that was derived in 1972 by Gutman and Trinajstić [3]. Ten years later, Balaban et al. included and among topological indices and named them ‘Zagreb group indices’ [5], which was later on abbreviated to ‘Zagreb indices’, and now and are called the first and second Zagreb indices. Afterwards these indices were used as branching indices [6]. Later on, the Zagreb indices found applications in QSPR and QSAR studies [1, 7]. These indices have been used to study molecular complexity, chirality, ZE-isomorphism and hetero-systems. Chemical applications and mathematical properties of Zagreb indices can be studied from [8-10].

Narumi and Katayama studied the degree product of a graph G for the first time in 1984. The Narumi-Katayama index proposed by Narumi and Katayama [11] is defined as follows: Estrada et al. [12] introduced the atom-bond connectivity index defined as follows: It has been applied up till now to study the stability of alkanes and the strain energy of cycloalkanes [12, 13]. The ABC-index can be used for modeling thermodynamic properties of organic chemical compounds. The ABC-index happens to be the only topological index for which theoretical, quantum-theory-based, foundation and justification have been found.

The first geometric-arithmetic connectivity index or simply geometric-arithmetic (GA) index of a connected graph G was introduced by Vukičević et al. in 2009 [14] and is defined as follows:

The augmented Zagreb index proposed by Furtula et al. in 2010 [15] is defined as follows: This graph invariant is a valuable predictive index in the study of the heat of formation in octanes and heptanes [15].

The first multiple Zagreb index was introduced by Ghorbani and Azimi in 2012 [16] and defined as follows: Clearly, the first multiple Zagreb index is the square of Narumi-Katayama index.

The third Zagreb index was introduced by Shirdel in 2013 [17] and defined as follows:

Furtula and Gutman showed that the term has a very promising application potential [18]. They called it the forgotten topological index or shortly the F-index, and it is defined as follows: They proved that the linear combination yields a highly accurate mathematical model of certain physico-chemical properties of alkanes [18].

Clearly, this index is a combination of second and third Zagreb indices, i.e., The Cartesian product is an important method to construct a bigger graph and plays an important role in the design and analysis of networks [19]. The Cartesian product of the graphs G and H, denoted by , is a graph with a vertex set and whenever [ and ] or [ and ].

Now we state distinct properties of the Cartesian product of graphs in form of the following lemma.

Lemma 1

Let and be graphs of orders , and sizes , , respectively. Then we have:

and ,

.

For a connected graph G, define four related graphs , , and as follows:

is the graph obtained by inserting an additional vertex in each edge of G, i.e., replacing each edge of G by a path of length 2. The graph is also known as a subdivision graph of G.

is the graph obtained by adding a new vertex corresponding to each edge of G, then joining each new vertex to the end vertices of the corresponding edge.

is the graph obtained by inserting a new vertex into each edge of G, then joining with edges those pairs of new vertices on adjacent edges of G.

has as its vertices, the edges and vertices of G. Adjacency in is defined as adjacency or incidence for the corresponding elements of G. The graph is called the total graph of G.

The four operations, , , and on a graph G are illustrated in Figure 1.

Figure 1

The graphs , , , and .

Eliasi and Taeri [20] introduced four new operations that are based on , , , as follows.

Let F be one of the symbols S, R, Q or T. The F-sum, denoted by , of graphs G and H having orders and , respectively, is a graph with the set of vertices and if and only if [ and ] or [ and ], where . consists of copies of the graph , and we label these copies by vertices of H. The vertices in each copy have two types, the vertices in (black vertices) and the vertices in (white vertices). Now we join only black vertices with the same name in in which their corresponding labels are adjacent in H. The graphs are shown in Figure 2.

Figure 2

The graphs , , and .

Eliasi and Taeri [20] computed the expression for the Wiener index of four graph operations which are based on these graphs , , , and in terms of and . Deng et al. [21] computed the first and second Zagreb indices for the graph operations , , and . Akhter and Imran computed bounds for the general sum-connectivity index of F-sums of graphs [22]. Some explicit computing formulas for different topological indices of some important graphs can be found in [7, 23–27].

To avoid computational complications, it is important to express the formulas for the product of F-sum of graphs in terms of their factor graphs. So, we presented bounds for the first Zagreb index, the ABC-index, the third Zagreb index, the augmented Zagreb index, the F-index, the first multiple Zagreb index and the GA-index for the Cartesian product of F-sum of graphs in form of its factor graphs.

Main results and discussions

This section is meant for determination of bounds for the first Zagreb, the third Zagreb, the augmented Zagreb, the first multiple Zagreb, ABC and GA indices of the Cartesian product of F-sum of graphs in terms of their factor graphs. Bounds for the F-index and the Narumi-Katayama index are also discussed. The following lemmas are useful for determination of these bounds.

In the following lemma, we compute the size of F-sum of graphs for .

Lemma 2

If , then the size of G is , where , , and .

Proof

We know that is a subdivision of , therefore the size of is .

Hence . □

In the following lemma, we compute the size of F-sum of graphs for .

Lemma 3

If , then the size of G is , where , , and .

By using combinations, the size of .

Hence . □

In the following lemma, we compute the size of F-sum of graphs for .

Lemma 4

If , then the size of G is , where , , and .

We know that the size of is equal to three times the size of , therefore the size of is .

Hence . □

In the following lemma, we compute the size of F-sum of graphs for .

Lemma 5

If , then the size of G is , where , , and .

By using combinations, the size of .

Hence . □

Let , , , be simple, connected graphs such that , , , , , , and .

In the following theorem the lower and upper bounds for the first Zagreb, the third Zagreb, the atom-bond connectivity (ABC), the augmented Zagreb, the first multiple Zagreb and geometric-arithmetic (GA) indices of the Cartesian product of F-sum of graphs in terms of their factor graphs for are determined.

Theorem 1

Let and , then where , and .

,

,

,

,

,

,

Let G and H be the graphs with vertex sets and , respectively. Then

(a) By definition,

By using Lemma 1, part (b), we obtain Since, for any vertex , and , therefore, by using these facts, we obtain which implies the inequality

By using inequality (2) in equation (1), we obtain Since , , , , and , therefore we obtain

By using similar arguments with , we obtain Hence part (a) of the theorem is proved by substituting in inequalities (3) and (4).

(b) By definition, By using Lemma 1, part (b), we obtain Since, for any vertex , and , therefore, by using these facts, we obtain which implies the inequality By using inequalities (2) and (6) in equation (5), we obtain Since , , , , and , therefore we get By using similar arguments with , we obtain Hence from inequalities (7) and (8), part (b) of the theorem is proved.

(c) By definition, By using inequality (2) in equation (9) and adopting the same procedure as in part (a) of this theorem, and Inequalities (10) and (11) complete the proof of part (c) of the theorem.

(d) By definition, By using inequalities (2) and (6) in equation (12) and adopting the same procedure as in parts (a) and (b) of this theorem, we obtain and Inequalities (13) and (14) complete the proof of part (d) of the theorem.

(e) By definition,

By using inequality (2) in equation (15) and adopting the same procedure as in part (i) of this theorem, and Hence from inequalities (16) and (17), part (e) of the theorem is proved.

(f) By definition,

By using inequalities (2) and (6) in equation (18) and adopting the same procedure as in part (b) of this theorem, and Hence from inequalities (19) and (20), part (f) of the theorem is proved. □

We determine the lower and upper bounds for the F-index and the Narumi-Katayama index of the Cartesian product of F-sum of graphs in terms of their factor graphs for .

Corollary 1

Let and , then where , and .

,

,

(a) By using the relation , in Theorem 1, we obtain the required result.

(b) By using the relation , in Theorem 1, we obtain the required result. □

In the following theorem the lower and upper bounds for the first Zagreb, the third Zagreb, the atom-bond connectivity (ABC), the augmented Zagreb, the first multiple Zagreb and geometric-arithmetic (GA) indices of the Cartesian product of F-sum of graphs in terms of their factor graphs for are determined.

Theorem 2

Let and , then where , and .

,

,

,

,

,

,

Let G and H be the graphs with vertex sets and , respectively. The proof is similar to that of Theorem 1 using Lemma (3). □

Corollary 2

Let and , then where , and .

,

,

(a) By using the relation , in Theorem 2, we obtain the required result.

(b) By using the relation , in Theorem 2, we obtain the required result. □

In the following theorem we determine the lower and upper bounds for the first Zagreb, ABC, the third Zagreb, the augmented Zagreb, the first multiple Zagreb and GA indices of the Cartesian product of F-sum of graphs in terms of their factor graphs for .

Theorem 3

Let and , then where , and .

,

.

,

,

,

,

Let G and H be the graphs with vertex sets and , respectively. The proof is similar to that of Theorem 1 using Lemma (4). □

We determine the lower and upper bounds for the F-index and the Narumi-Katayama index of the Cartesian product of F-sum of graphs in terms of their factor graphs for .

Corollary 3

Let and , then where , and .

,

,

(a) Using the relation , in Theorem 3, we get the required result.

(b) Using the relation , in Theorem 3, we get the required result. □

In the following theorem we determine the lower and upper bounds for the first Zagreb, ABC, the third Zagreb, the augmented Zagreb, the first multiple Zagreb and GA indices of the Cartesian product of F-sum of graphs in terms of their factor graphs for .

Theorem 4

Let and , then where , and .

,

,

,

,

,

,

Let G and H be the graphs with vertex sets and , respectively. The proof is similar to that of Theorem 1 using Lemma (5). □

We determine the lower and upper bounds for the F-index and the Narumi-Katayama index of the Cartesian product of F-sum of graphs in terms of their factor graphs for .

Corollary 4

Let and , then where , and .

,

,

(a) Using the relation , in Theorem 4, we get the required result.

(b) Using the relation , in Theorem 4, we get the required result. □

In the following theorem we determine the lower and upper bounds for the first Zagreb, ABC, the third Zagreb, the augmented Zagreb, the first multiple Zagreb and GA indices of the Cartesian product of F-sum of graphs in terms of their factor graphs for and .

Theorem 5

Let and , then where , and .

,

,

,

,

,

,

Let G and H be the graphs with vertex sets and , respectively. The proof is similar to that of 1 with and . □

We determine the lower and upper bounds for the F-index and the Narumi-Katayama index of the Cartesian product of F-sum of graphs in terms of their factor graphs for and .

Corollary 5

Let and , then where , and .

,

,

(a) Using the relation , in Theorem 5, we get the required result.

(b) Using the relation , in Theorem 5, we get the required result. □