Reformulated Zagreb Indices of Some Derived Graphs

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph.


Introduction
In the fields of mathematical chemistry and chemical graph theory, a topological index is a numerical parameter that is measured based on the molecular graph of a chemical constitution.Topological indices are extensively used in the study of quantitative structure-activity relationships (QSARs) to establish the correlation between different properties of molecules and/or the biological activity with their structure.Topological indices have also been applied in spectral graph theory to measure the resilience and robustness of complex networks [1].
Suppose that Ω denotes the set of all graphs.Then, a function T : Ω → R + is called a topological index if for any pair of isomorphic graphs G and H, we have T(G) = T(H).There are many topological indices that are useful in chemistry, biochemistry, and nanotechnology.
Throughout this paper, we are concerned only with the simple and connected graphs.Let G = (V, E) be such a graph.We use the notations V = V(G) and E = E(G) for the vertex set and edge set of G, respectively.We use the notation d G (i) for the degree of a vertex i in G.The hand shaking lemma says that the sum of the degrees of all the vertices is equal to double the number of edges.Mathematically, For basic definitions and notations, see the book [2].
The first and second Zagreb indices are defined as: Another expression for the first Zagreb index is: These two Zagreb indices defined by Gutman and Trinajstic in 1972 [3] are among the oldest topological indices, having many applications in mathematical chemistry.Zagreb indices are known to be very useful in quantitative structure-property relationships (QSPR) and QSAR [4][5][6].In [7] and [8], the authors have derived the expressions for Zagreb indices of some derived graphs.
Zagreb indices were reformulated by Miličević [9] as: where d G (e) denotes the degree of an edge e = ij in G and defined as the total number of edges incident with e. Mathematically, Here, e ∼ f indicates that the edges e and f are incident.Another expression for the first reformulated Zagreb index is: The first reformulated Zagreb index is closely related to Laplacian eigenvalues [10].Much interest has been shown by researches and scientists in the reformulated Zagreb indices [11][12][13][14][15].
Another vertex-degree-based topological index was found to be useful in the earliest work on Zagreb indices [3,16], but later was totally ignored.Quite recently, some interest has been shown in it [17,18].It is called the forgotten index or simply the F-index and is defined as: In general, for any real number "α", the generalized version of first Zagreb index is defined as: Clearly, the first Zagreb index and F-index are special cases of α M 1 for α = 2 and α = 3, respectively.
Bearing in mind the reformulation of Zagreb indices, here we reformulate the F-index as: We prefer to call it the reformulated forgotten index or simply the reformulated F-index.

Some Derived Graphs
Let, as before, G be a simple and connected graph with vertex set V(G) and edge set E(G).Let |V(G)| = n and |E(G)| = m.Now, we consider the following graphs derived from G: The complement G of G is the graph with the same set of vertices as G, but there is an edge between two vertices of G if and only if there is no edge between these vertices in G.
Line graph: The line graph L = L(G) of G is the graph in which the vertex set is the edge set of G, and there is an edge between two vertices of L if and only if their corresponding edges are incident in G. Thus, |V(L)| = m, and by hand shaking lemma, • Subdivision graph: A subdivision graph of a graph G can be constructed by inserting a vertex on each edge of G, which will change that edge into a path of length two.This graph is denoted as S = S(G).

•
Total graph: The total graph T = T(G) is the union of the vertex-semitotal graph and the edge-semitotal graph.Thus, In [7], one can easily see that M 1 (S) = M 1 + 4m.Thus: In Figure 1, self-explanatory examples of these derived graphs are given for a particular graph.In every derived graph of G (except the paraline graph PL(G)), the vertices corresponding to the vertices of G are denoted by circles and the vertices corresponding to the edges of G are denoted by squares.
The following are some known results on Zagreb indices.
Lemma 1. [7] If G is the complement of G, then: Lemma 2. [7] If G is the complement of G, then: Lemma 3. [17] If G is the complement of G, then: Lemma 4. [15] The reformulated first Zagreb index can be written in terms of the Zagreb indices and F-index as:

Reformulated First Zagreb Index of Some Derived Graphs
In [7] and [8], the authors derived the expressions for Zagreb indices and coindices of those derived graphs, which have been discussed above.In [19], the authors derived the expressions for multiplicative Zagreb indices and coindices of some derived graphs.In this section, we present the expressions for the reformulated first Zagreb index of these derived graphs.
Proof.By the definition of the first reformulated Zagreb index, Using Lemma 1, Lemma 2, and Lemma 3 for fourth, third, and first term, respectively, we have: Proof.By the definition of the reformulated first Zagreb index, Now, by the definition of the line graph, it is clear that: Thus: Using Lemma 4 for term EM 1 , Now, for subdivision graph S, vertex-semitotal graph T 1 , edge-semitotal graph T 2 , and total graph T, we can see that there are two types of vertices in these graphs: first, the vertices corresponding to the vertices of G and, second, the vertices corresponding to the edges of G.We call them α-vertices and β-vertices, respectively.Depending on the nature of end vertices, we can divide the edges of these graphs into three types: 1.
The above idea is very similar to the idea used in [20].
Proof.We can see that for any α-vertex i α of S: and for any β-vertex j β of S: Furthermore, all the edges of S are αβ-edges.Thus: In fact, every α-vertex i α of S is connected with d G (i α ) β-vertices, each of degree two.Hence: Proof.First note that for any α-vertex i α of T 1 : and for any β-vertex j β of T 1 : Furthermore, any edge of T 1 is either an αα-edge or an αβ-edge.Thus: For αβ-edges in the second term, it is clear that every α-vertex i α of T 1 is connected with d G (i α ) β-vertices, each of degree two.Therefore, corresponding to every vertex i α in G, there are d G (i α ) edges in T 1 each of (edge) degree [2d G (i α ) + 2 − 2].Thus: Proof.First note that for any α-vertex i α of T 2 : and for any β-vertex Any edge in T 2 is either an αβ-edge or a ββ-edge.In fact,

1.
Corresponding to every edge i α j α in G, there are two αβ-edges i α x β and x β j α in T 2 such that: and:

ββ-edges are the edges corresponding to the edges of L(G).
Thus: Now, by the definition of the line graph, it is clear that: Thus: Using Lemma 4, Theorem 6.If T = T(G) is the total graph of G, then: Proof.First note that for any α-vertex i α of T: and for any β-vertex j β of T: T has all three types of edges.In fact,

1.
Corresponding to every edge i α j α in G, there is one αα-edge, which is i α j α , and two αβ-edges, which are i α x β and x β j α in T such that: and:

2.
ββ-edges are the edges corresponding to the edges of L(G). Thus: Now, using the facts: and then using Lemma 4, we get: Proof.It can be noted that for any vertex i ∈ V(G), there are d G (i) vertices in PL(G) having the same degree as i such that all these d G (i) vertices are connected with each other.In fact, PL(G) can be obtained from G by replacing every vertex i by K d G (i) .Now, the edges of PL(G) can be divided into two categories: 1.
The edges in K d G (i) , where i ∈ V(G).

2.
Edges corresponding to edges of G.It can be seen that corresponding to every edge in G, there is an edge in PL(G) of the same degree. Thus: Now, in each K d G (i) , there are Solving the first term and using Lemma 4 in the second term, we get:

Reformulated Second Zagreb Index of Some Derived Graphs
In this section, we derive the expressions for the reformulated second Zagreb index of some derived graphs.
Proof.We can divide the pairs of incident edges of S = S(G) into two categories: 1.
For any vertex i ∈ V(G), there are d G (i) edges, each of degree d G (i) in S, and all these edges are incident at i. Therefore, for any vertex i ∈ V(G), the total number of pairs of incident edges lying in this category is Corresponding to every pair of adjacent vertices i and j in G, there is a pair of incident edges of degrees d G (i) and d G (j) in S. Thus: Before going to the next theorem, we prove another lemma here.
Lemma 5.For a graph G, the following equality holds.
Using Lemma 4 and the relation: we have: Proof.By definition We can divide the pairs of incident edges (e, f ) of T 1 = T 1 (G) into three cases: Case 1: When e, f ∈ E(S).Just like Theorem 8, we have two further categories: 1.
For any vertex i ∈ V(G), there are d G (i) edges, each of degree 2d G (i) in S, and all these edges are incident at i. Therefore, for any vertex i ∈ V(G), the total number of pairs of incident edges lying in this category is Corresponding to every pair of adjacent vertices i and j in G, there is a pair of incident edges of degrees 2d G (i) and 2d G (j) in S.
Case 2: When e, f ∈ E(G).For any pair of incident edges e and f in G, e and f are also incident in T 1 .Furthermore, for any edge e = ij in G, This edge is incident with d G (i) edges, each of degree 2d G (i) at i, and d G (j) edges, each of degree 2d G (j) at j. Thus: Using Lemma 5 and the fact: we have: Now, using Lemma 4 and simplifying the above expression, we get: Proposition 1.Let K n be the complete graph on n vertices, then the total number of pairs of incident edges in .
Proof.The total number of pairs of incident edges in Theorem 10.If PL = PL(G) is a paraline graph of G, then: Proof.As we explained in Theorem 7, PL(G) can be obtained from G by replacing every vertex i by K d G (i) .Therefore, we can divide the pairs of incident edges of PL = PL(G) into two cases: Case 1: For any vertex i ∈ V(G), there are d G (i)(d G (i)−1) 2 edges each of degree (2d G (i) − 2) in the corresponding K d G (i) of PL, and all these edges are incident with each other.By Proposition 1, for any vertex i ∈ V(G), the total number of pairs of incident edges in K d G (i) is: Case 2: Corresponding to every edge ij in G, there is an edge in PL(G) of the same degree that is incident with (d G (i) − 1) edges, each of degree (2d G (i) − 2) at i, and (d G (j) − 1) edges, each of degree (2d G (j) − 2) at j.

Conclusions
In this note, we obtained some relations for degrees between a derived graph and its parent graph.Using these relations, the expressions for reformulated Zagreb indices of some derived graphs have been found in terms of some topological indices of the parent graph.Finding expressions of derived graphs like these is an open problem for many other topological indices.