ON RECIPROCALS LEAP INDICES OF GRAPHS

In the field of chemical graph theory, topological indices are calculated based on the molecular graph of a chemical compound. Topological indices are used in the development of Quantitative structure Activity/Propoerty Relations. To study the physico-chemical properties of molecules most commonly used are the Zagreb indices. In this paper, we introduce reciprocals leap indices as a modified version of leap Zagreb indices. The exact values of reciprocals leap indices of some well-known classes of graphs are calculated. Lower and upper bounds on the reciprocals leap indices of graphs are established. The relationship between reciprocals leap indices and leap Zagreb indices are presented.


Introduction
In last decade, graph theory has found a considerable use in the mathematical chemistry. In this area we can apply tools of graph theory to model the chemical phenomenon mathematically. This theory contributes a prominent in chemical science. A chemical structure of molecules can be represent by molecular graph, where vertices represent the atoms and edges represent the bonds between them. The graph theory based structure descriptors can be determined by considering graph vertices and edges. A simply arithmetic operators are carried out to get numerical indices. Topological indices are used in the development of Quantitative Structure Activity/Property Relations (QSAR/QSPR). A graph is a collection of points and lines connecting a subset of them. The points and lines of a graph are also called vertices and edges of the graph, respectively. The vertex and edge sets of a graph G are denoted by V (G) and E(G), respectively. Let |V (G)| = n and |E(G)| = m, if two vertices u and v of the graph G are adjacent, then the edge connecting them will be denoted by uv. If u, v ∈ V (G) then the distance d G (u, v) between u and v is defined as the length of a shortest path in G connecting them. The diameter of a connected graph G is the length of any longest geodesic, denoted by diam(G). In a graph G, the degree (first degree) of a vertex v, denoted d(v), is the number of first neighbors (the number of edges incident with v) and the second degree of v, denoted wheel W 1,n and a friendship graphs are defined as W 1,n = K 1 + C n and F n = K 1 + n−1 2 K 2 , respactively. A graph G is called F -free graph if no induced subgraph of G is isomorphic to F . In this paper, we only conceder with a simple connected graphs. Any undefined term or notation in this paper can be found in ( [2] [7]).
One of the oldest and most commonly used to study the physico-chemical properties of molecules topological index are Zagreb indices introduced by Gutman and Trinajstic on based degree of vertices of G. The first and second Zagreb indices of a graph G are defined as: The quantity M 1 (G) was first time considered in 1972 [4], whereas M 2 (G) in 1975 [5]. For more information on Zagreb and beyond topological indices, readers are referred to the survey [6], and the references therein.
In 2017, Naji et al. [9] have introduced a new distance-degree-based topological indices conceived depending on the second degrees of vertices, and are so-called leap Zagreb indices of a graph G and are defined as: The leap Zagreb indices have several chemical applications. Surprisingly, the first leap Zagreb index has very good correlation with physical properties of chemical compound, like bolling point, entropy, DHVAP, HVAP and eccentric factor [1]. Consequently, the new class of graphs, that so called leap graphs was defined and studied in [10], and was defined as, A graph G is said to be a leap graph, if and only if for every vertex The inverse degree index of a graph the first was introduced in 2005 [11], and was defined by The inverse degree has attracted attention through a conjecture generating computer Graffiti [3]. The modified first Zagreb index m M 1 (G) was first introduced in [8], and defined as Motivated by the inverse degree index, we herewith define the inverse second degree index of a graph, as following Note that,we added one to the second degree of a vertex, because there are infinity graph with some vertex whose d(v) = n − 1 and so d 2 (v) = 0. Thus, the ID 2 (G) defined here is well-defined for every graph. We need the following fundamental results to prove our main results.
Equality holds if and only if G is a (C 3 , C 4 ) free.
Equality holds if and only if G having diameter of most two.

Reciprocals Leap Indices of Graphs
In this section, we present the definitions of first, second and third reciprocals leap indices of a graph and explore how we can calculation them for a connected simple graph.
For a connected graph G, the first, second and third reciprocals leap indices are defined as: .
To illustrate these invariants, we compute them for a graph G shown in Figure 1.

Reciprocals Leap Indices for some Families of Graphs
In this section, we establish the formulaes of the exact values of reciprocals leap indices for some wellknown graph classes.
For a positive integer n ≥ 2, (1) For the complete graph K n , • RL 3 (K n ) = n (n−1) . (2) For the path P n , (4) For the complete bipartite graph K r,s , 1 ≤ r ≤ s ≤ n − 1, (5) For the star graph K 1,n−1 , (6) For the wheel graph, W 1,n , n ≥ 3, Proposition 3.2. Let G be a connected k-regular C 3 , C 4 -free graph. Then Proof. The proof is immediately consequence of Lemma 1.4 and the definitions of reciprocals leap indices of graphs.

Bounds on Reciprocals Leap Indices of graphs
In this section, we present some upper and lower bounds on reciprocals leap indices of a graph, in term of number of vertices, number of edges, minimum (maximum) degree, Inverse degree and second inverse degree indices of a graph.
Theorem 4.1. Let G be a connected graph with n ≥ 2 vertices. Then, Equality holds if and only if G is a complete graph.
Proof. Let G be a connected graph with n ≥ 2 vertices. Since for every v ∈ V (G), d 2 (v) ≥ 0, which led to To prove the equality, we assume, on the contrary, that G = K n , for n ≥ 2. Then there are at least two which is a contradiction.
Conversely, if G is a complete graph, then it is immediate the result follow from Proposition 3.1. Theorem 4.2. Let G be a connected graph with m ≥ 1 edegs. Then, Equality holds if and only if G is a complete graph.
Proof. The proof is similar to the proof of Theorem 4.1, then we omit it.
Equality holds if and only if G = K 2 .
Proof. Let G be a connected graph with n ≥ 2 vertices. Since d(v) ≥ 1 and and  Theorem 4.4. Let G be a connected graph with n vertices. Then, The lower bound attains on K 1,n−1 and K n , whereas the upper bound attains on K n .
Proof. Let G be a connected graph with n vertices. Since for every vertex v ∈ V (G), δ ≤ d(v) ≤ ∆ and Theorem 4.5. For any connected graph G with n vertices, Equality holds if and only if G is a regular graph with diameter at most two.
Proof. Since for any connected graph d(v) ≥ δ, for every v ∈ V (G) and by using Lemma Suppose the equality RL 1 (G) = n (n−δ) 2 is holding. Then by Lemma 1.3, equality if and only if diam(G) ≤ 2 and G is δ-regular graph. Therefore, equality RL 1 (G) = n (n−δ) 2 hold if and only if G is a regular graph with diam(G) ≤ 2.
The following results immediately follow from the facts that, δ(G) ≥ 1 for any connected graph G and for every vertex v ∈ V (G) and if G is a k-regular graph, then δ(G) = k for every vertex v ∈ V (G).
Equality holds if and only if diam(G) ≤ 2.
Theorem 4.6. For any connected graph G, Equality holds if and only if G is a regular graph with diameter at most two.
Proof. The proof is similar to the proof of Theorem 4.5.
Corollary 4.6. For any connecte k-regular graph, Equality holds if and only if G having diameter at most two. .
Equality holds if and only if G is regular graph with diameter at most two.
Proof. Let G be a connected graph with minimum and maximum degrees 1 ≤ δ ≤ ∆. Then by Lemma 1.3, . This complete the proof.
Equality holds if and only if G is regular graph with diameter at most two. Equality holds if and only if diam(G) ≤ 2.
Theorem 4.8. For any connected graph G, Equality holds if and only if G is a complete.
Proof. Since d 2 (v) is a positive integer number for every v ∈ V (G). Then Equality holds if and only if G = K n .
Proof. Let G be a connected graph. Since d 2 (v) + 1 ≥ 1, for every v ∈ V (G). Then Conversely, it is easy to check that if G = K n , then RL 3 (G) = ID(G).
Theorem 4.10. Let G be a connected graph of order n and with maximum degree ∆ ≥ 1, The equality holds if and only if G is regular.
Conversely, let G be a k-regular graph. Then Theorem 4.11. Let G be a connected graph of order n. Then The equality holds in (1), (2) and (3) if and only if G is a complete.
Proof. Let G be a connected graph with n vertices, m edges and minimum (maximum) degrees δ ≥ 1 ( ∆ ≤ n − 1). We prove an inequality for RL 1 (G), and the proof of the inequalities (2) and (3) are similar.
Assume the function f (x) = x − ln x − 1. Easy calculating gives us that f (x) ≥ 0, for every positive real number. Thus, for every v ∈ V (G), we get By taking the summation on two sides of the inequality over the vertex set of the graph, we get By using the fact, for every v ∈ V (G), d 2 (v) + 1 ≤ n − δ, we get,  Then Equalities holds in (1), (2) and (3) if and only if G is a regular.
Proof. Let G be a connected leap graph with n ≥ 4 vertex, minimum degree δ ≥ 1 and maximum degree ∆ ≤ n − 1. We prove part (1)only, and the proofs of (2) and (3) are similar. Since G is a leap graph and by taken the summation over vertex set of G, we get v∈V (G) Suppose the equality holds in the lower bound. Then d(v) + 1 = ∆ + 1, for every v ∈ V (G). That mean G is a ∆-regular graph. Similarly, for the upper bound, which let G is δ-regular graph.
Conversely, let G be a k-regular leap graph. Then ∆ = δ = k, and hence

Relationship Between Reciprocals Leap Indices and Leap Zagreb Indices of graphs
In this section, we investigate the relationship between reciprocals leap indices and leap Zagreb indices of graphs. Also, the relation between first and third reciprocals leap indices of a graph are presented.
Theorem 5.1. For any connected graph G with n vertices and m edges. Therefore, Since To prove the inequality (2) . By taken the summation over edge set of graph G, we get The proof of an equality in (2) is very similar to the proof of part (1).

Now, by put
and hence, by taken the summation over vertex set of G, The proof of equality in (3) is similar to the proof of part (1).
Theorem 5.2. Let G be a connected graph of order n and size m. Then The equality holds in (1) and (2), if and only if G = K n , whereas in (3) if and only if G = K 2 .
Proof. Let G be a connected graph of order n and size m. Cauchy-Schwartz state, for every real numbers If we put a i = 1 d2(v)+1 and b i = d 2 (v) + 1, for i = 1, 2, ..., n, and for every vertex v ∈ V (G), then and b i = (d 2 (u) + 1)(d 2 (v) + 1) for every edge uv ∈ E(G), we get Then for every v ∈ V (G), Theorem 5.3. Let G be a connected graph of order n. Then Equality holds if and only if G is an n 2 -regular graph.
Proof. Let G be a connected graph of order n. Put in Cauchy Schwartz inequality. Then Therefore  Theorem 5.4. Let G be a connected graph of order n. Then . Equality holds in (1), (2) and (3) if and only if G is regular graph with diam(G) ≤ 2.
Proof. Let G be a connected graph of order n, size m and δ ≥ 1, ∆ ≤ n − 1. Diaz-Metcalf inequality stat that if a i = 0 and b i for i = 0, 1, 2, ..., n satisfy t ≤ bi ai ≤ T , then Equality holds if and only if b i = ta i or b i = T a i , for i = 1, 2, ..., n.
Conversely, let G be a k-regular graph with diam(G) ≤ 2. Then by Lemma 1.2, d 2 (v) + 1 = n − k, for every v ∈ V (G) and hence, Therefore, the equality holds.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.