A Short Note On Hyper Zagreb Index

abstract: In this paper, we present and analyze the upper and lower bounds on the Hyper-Zagreb index χ(G) of graph G in terms of the number of vertices (n), number of edges (m), maximum degree (∆), minimum degree (δ) and the inverse degree (ID(G)). In addition, a counter example on the upper bound of the second Zagreb index for Theorems 2.2 and 2.4 from [20] is provided. Finally, we present the lower and upper bounds on χ(G) + χ(G), where G denotes the complement of G.


Introduction
Let G be a simple graph with the vertex set V (G) and the edge set E(G).As usual, we denote the degree of a vertex by d i = d(v i ) for i = 1, 2, . . ., n such that d 1 ≥ d 2 ≥ • • • ≥ d n , with the maximum, second maximum and the minimum vertex degree of G are denoted by ∆ = ∆(G), ∆ 2 = ∆ 2 (G) and δ = δ(G) respectively.G denotes the complement of G, with the same vertex set such that two vertices u and v are adjacent in G if and only if they are not adjacent in G.A line graph L(G) obtained from G in which V (L(G)) = E(G), where two vertices of L(G) are adjacent if and only if they are adjacent edges of G.
In 1972, the first and second Zagreb indices are introduced by Gutman and Trinajstić [13,14] and are defined as In 1987, the inverse degree first attracted attention through conjectures of the computer program Graffiti [11].The inverse degree of a graph G with no isolated vertices are defined as In 2005, Li and Zheng [15] introduced the generalized version of the first Zagreb index.For α ∈ R and G be any graph which satisfies the important identity (1.1) In 2010, Ashrafi, Došlić and Hamzeha introduced the concept of sum of nonadjacent vertex degree pairs of the graph G, known as first and second Zagreb coindices [2] and are defined as In 2013, Shirdel, Rezapour, and Sayadi [16] defined the Hyper-Zagreb index as In 2015, Fortula and Gutman [12,13] introduced the forgotten topological index and for α = 2 in (1.1) turns it as a very special case formula, defined by As usual P n , K 1,n−1 , C n , K n denotes the path, star, cycle and complete graphs on n vertices respectively.The wheel graph W n is join of the graphs C n−1 and K 1 .Bidegreed graph is a graph whose vertices have exactly two vertex degrees ∆ and δ.The Helm graph H n is obtained from W n by adjoining a pendant edge at each vertex of the cycle.Let G and H be any graph.Then σ G (H) represents the total number of distinct subgraphs of the graph G which are isomorphic to H.The tensor product of the two simple graphs G and H are denoted by G × H, whose vertex set is V (G) × V (H) in which (g 1 , h 1 ) and (g 2 , h 2 ) are adjacent whenever g 1 g 2 is an edge in G and h 1 h 2 is an edge in H.
For computational purposes, we use the software GraphTea [1] considering various phases of testing.GraphTea is a graph visualization software designed specifically to visualize and explore graph algorithms and topological indices interactively.

Upper bounds for χ 2 (G)
An equivalent formula for the Hyper-Zagreb index was already in use, pertaining to the first and second Zagreb index.In 2010, Zhou and Trinajstić [21] proposed the general sum-connectivity index defined as (2.1) 1) turns the Hyper-Zagreb index as its special case.At first we give the identity for the Hyper-Zagreb index.
Lemma 2.1.Let G be any simple graph, then Proof.By the definition of the general sum-connectivity index and using (1.1), we get Thus, by using M 2 1 (G), M 3 1 (G) and M 1 2 (G) from [4], we complete the proof.✷ It is easy to see that, an upper bound for either M 1 2 (G) or M 3 1 (G) suits for χ 2 (G).In the preparations of presenting the upper bounds for χ 2 (G) through the existing upper bounds for the second Zagreb index, we encountered the following upper bounds Theorem 2.2.[20] For a simple connected graph G, Remark 2.4.Counterexamples for the above two theorems.For any edge need not be true for all graphs.For K 1,3 , u∈V (G) d(u) = 6, and for uv∈E(G) d(u) we have the following combinations 3, 5, 7, 9. Therefore Inequality (2.4) is not true in general.In addition, for the helm H 3 (See Figure 1) with ∆ = 4 and second Zagreb index is 96, but the 2∆m is 72.In analogy, Inequality (2.5) is also not true in general.By considering L(H 3 ) with the first Zagreb coindex is 126 and ∆n denote the total number of combinations of sum of the vertices u, v in G and is represented as For any simple graph G with δ ≥ 2 then, it is easy to see that d(u) By adding over all the edges, we have , but this inequality is mentioned in the Theorem 2.4 of [20] in reverse order, which leads to the counterexample in Figure 1.
Note that forgotten topological index [12] has only few lower bounds.At first, we give an upper bound for M 3 1 (G) which leads to the upper bound for χ 2 (G).
Theorem 2.5.Let G be any simple graph with no isolated vertices.Then equality if and only if G is regular or bidegreed graph.
Proof.Let a, A ∈ R and x i , y i be two sequences with the property ay i ≤ x i ≤ Ay i for i = 1, 2, . . ., n and w i be any sequence of positive real numbers, it holds w i (Ay i − x i ) (x i − ay i ) ≥ 0. Since w i is a positive sequence, choose Substituting the above inequality into (2.1)completes the proof and the equality holds if and only if G is regular.✷ Theorem 2.6.Let G be any simple graph with n vertices and m edges.Then equality if and only if G is regular or bidegreed graph.
Proof.The proof follows by using similar arguments as in the proof of Theorem 2.5 with setting m i = d(v i ) and n i = 1.✷ Remark 2.7.The upper bounds (2.6) and (2.8) are incomparable.For the graphs H 3 and L(H 3 ) depicted in Figure 1, (2.6) is better than (2.8) and for the graphs ) is better than (2.6), as shown in the next table Zhou and Trinajstić [21] obtained the following lower bound for χ 2 (G).
Theorem 3.1.[21] Let G be a simple graph G with m ≥ 1 edges.Then equality holds if and only if d(u) + d(v) is a constant for any edge uv.
which completes the proof, and the equality holds if and only if G is regular.✷ Theorem 3.3.Let G be a simple graph with no isolated vertices.Then equality holds if and only if G is regular.
Proof.Consider w 1 , w 2 , . . ., w n be the non-negative weights, then we have the weighted version of Cauchy-Schwartz inequality Since w i is non-negative, we assume that w .
By combining the above inequality with (2.1), we complete the proof and the equality holds if and only if G is regular.✷ Theorem 3.4.Let G be a simple graph with n vertices and m edges, then  In [21], the following lower and upper bound for χ 2 (G)+χ 2 (G) was established: By using Theorems 2.5 and 3.4, we deduce a finer bound for χ 2 (G) + χ 2 (G).

Figure 1 :
Figure 1: The Helm H 3 and its Line graph L(H 3 ).

Theorem 3 . 2 .
Let G be a simple graph with n vertices and m edges, thenχ 2 (G) ≥ 4M 1 2 (G) (3.2)equality holds if and only if G is regular.Proof.For any two non-negative real numbers a, b we have 1 4 (a + b) 2 ≥ ab.Thus, by fixing a = d(u) and b = d(v) for uv ∈ E(G), then adding over all the edges of G