Theory and Applications of Graphs Theory and Applications of Graphs

The augmented Zagreb index (AZI for short) of a graph G , introduced by Furtula et al. in 2010, is defined as AZI


Introduction
Let G(V, E) be a simple graph with |V (G)| = n and |E(G)| = m. The set of neighbors of vertex u is called the open neighbor of u and denoted by N (u). The degree of a vertex u ∈ V (G), denoted by d(u), is d(u) = |N (u)|. The vertex u is called a pendent vertex if d(u) = 1. The maximum and minimum degree of G are denoted by ∆ and δ, respectively. (i, j)-edge denotes an edge whose degrees of terminal vertices are i and j, and let E i,j be the set of edges that all edges are (i, j)-edges, where 1 ≤ i ≤ j ≤ n − 1. A graph is a complete bipartite graph if V (G) can be partitioned into two sets U and V so that uv is an edge of G if and only if u ∈ U and v ∈ V . If |U | = ℓ 1 and |V | = ℓ 2 , then this complete bipartite graph is denoted by K ℓ 1 ,ℓ 2 . A tree T is molecular tree if ∆(T ) ≤ 4. The augmented Zagreb index (AZI for short) of a graph G, introduced by Furtula et al. in [6], is defined as In [6], it has been shown that the prediction power of AZI plays an important role in the study of the heat of formation in octanes and heptanes. In fact, the prediction power of augmented Zagreb index is better than atom-bond connectivity index, we can see [5,8,14].
The tight upper and lower bounds for molecual trees were firstly calculated in [6]. Moreover, in all trees, it has been proven that stars have the minimum AZI value. Unicyclic, bicyclic graphs and tricyclic graphs of fixed order graphs with minimum AZI were shown in [2] and [12]. In [13,9,8,14,7,10,11], some AZI bounds of general graphs, chemical graphs and other graphs were computed.
In a connected graph G, let p be the number of pendent vertices, q be the number of non-pendent edges satisfying the degree of one terminal vertex is exactly 2 and the degree of the other terminal vertex is at least 2, δ 1 be the minimum degree of non-pendent vertices, let |E δ 1 ,δ 1 | = s. Let Φ 1 be the set of connected graphs which has the property Φ 1 = {G|E(G) ⊂ E 1,∆ ∪ E 2,∆ }. Let Φ 2 be the set of connected graphs which has the property where i and j are positive integers. Clearly, i + j ̸ = 2, x ij = x ji and A ij = A ji . Then we can rewrite the augmented Zagreb index of G as follow In [9], Huang et al. obtained the following result, which will be useful.
In [13], Wang et al. obtained the following.
With equality if and only if G is isomorphic to a (1, ∆)-biregular graph or G is isomorphic to a δ 1 -regular graph or G ∈ Φ 1 or G ∈ Φ 2 . 13]). Let G be a connected graph with n ≥ 3 and m ≥ 2. Then The equality holds if and only if G is a ∆-regular graph or G is a (1, δ 1 )-biregular graph.
In Section 2, we improve the results of Theorems 1.1, 1.3 and 1.4 and obtain some new upper and lower bounds for the AZI index of general connected graphs. In Section 3, we give a lower bound of molecular trees. In Section 4, upper and lower bound of connected triangle-free graphs are given.

Results for connected graphs
Let Φ 3 , Φ 4 , Φ 5 and Φ 6 be the sets of connected graphs which have the following properties respectively: We first improve the lower bound in Theorem 1.3.
Theorem 2.1. Let G be a connected graph of order n ≥ 3 and size m ≥ 2. Then The equality holds if and only if G is a (1, ∆)-biregular graph or G ∈ Φ 1 or G ∈ Φ 2 for δ 1 = 2, or G ∈ Φ 3 for δ 1 ≥ 3.
Proof. If δ 1 = 2, then it follows from Theorem 1.2 that

The equality holds if and only if
Then we have the following remark. If m > p + s and δ 1 ≥ 3, then M > N , which means that Theorem 2.1 is better than Theorem1.3.
Corollary 2.2. Let G be a connected graph with order n ≥ 3. Then Proof. Since m ≥ nδ 2 , it follows from Theorem 2.1 that The equality holds if and only if G is isomorphic to δ-regular graph with δ = δ 1 ≥ 2.
Theorem 2.3. Let G be a connected graph with order n ≥ 3 and size m ≥ 2. Then where |E 3,3 | = t. The equality holds if and only if G ∈ Φ 4 .
Proof. From Theorem 1.2, we have Clearly, the equality holds if and only if G ∈ Φ 4 . Let and Then we have the following remark.
Remark 2.2. If m = p + q + t, then Theorems 2.3 and 1.1 have the same lower bound. If m > p + q + t, then M ′ > N ′ , which implies that Theorem 2.3 is more better than Theorem 1.1. Let Then we have the following remark.

Moreover, the equality holds if and only if
If δ 1 ≥ 3, then by Theorem 1.2, we have that With the equality holds if and only if G ∈ Φ 6 . Let Then we have the following remark. The following corollary is immediate. Corollary 2.5. Let G be a connected graph with order n ≥ 3. Then Moreover, the equality holds if and only if G is isomorphic to ∆-regular graph.
Proof. Since m ≤ n∆ 2 , it follows from Theorem 2.4 that the equality holds if and only if G is isomorphic to ∆-regular graph.
Then h(i, j) > 0. The equality holds if and only if T is isomorphic to T ′ .

Results for connected triangle-free graphs
In this section, we calculate the bounds for connected triangle-free graphs. Let G be a connected triangle-free graph. Then it has the property d(v i ) + d(v j ) ≤ n, where d(v i ) and d(v j ) are the degrees of the terminal vertices of edge v i v j ∈ E(G). In [3], Bin et al. showed that the size of a connected triangle-free graph with order n is at most ∆(n − ∆). We use the Mantel's Theorem given by B. Bollobás in [4] to obtian the bounds of the triangle-free graph.
The equality holds if and only if G is the complete bipartite graph K ⌊ n 2 ⌋,⌈ n 2 ⌉ . Proof. Let G be a connected triangle-free graph with order n ≥ 3 and size m. It follows from Theorem 4.1 that n − 1 ≤ m ≤ ⌊ n 2 4 ⌋. Since d(v i ) + d(v j ) ≤ n and 2 ≤ ∆ ≤ n − 1, it follows that d(v i )d(v j ) ≤ (d(v i )+d(v j )) 2 4 ≤ n 2 4 . Then we have The equality holds if and only if G is a K ⌊ n 2 ⌋,⌈ n 2 ⌉ . Corollary 4.3. Let G be a connected triangle-free graph with maximum degree ∆ and order n ≥ 3. Then AZI(G) ≤ ∆(n − ∆) ∆ 6 (2∆ − 2) 3 .
The equality holds if and only if G is a complete bipartite graph K ∆,∆ .
Proof. Since m ≤ ∆(n − ∆), it follows that The equality holds if and only if G is a K ∆,∆ .