Extremal trees for the geometric-arithmetic index with the maximum degree

Abstract For a graph G, the geometric-arithmetic index of G, denoted by GA(G), is defined as the sum of the quantities 2 √ dx × dy/(dx + dy) over all edges xy ∈ E(G). Here, dx indicates the vertex degree of x. For every tree T of order n ≥ 3, Vukičević and Furtula [J. Math. Chem. 46 (2009) 1369–1376] demonstrated that GA(T ) ≤ 4 √ 2 3 + (n − 3). This result is extended in the present paper. Particularly, for any tree T of order n ≥ 5 and maximum degree ∆, it is proved that


Introduction
Let G be a simple connected graph with the vertex set V (G) and the edges set E(G). The order n(G) of G is the cardinality of the vertex set V (G) and the size m(G) of G is the cardinality of the edge set E(G). The set N (y) = {x ∈ V (G) | xy ∈ E(G)} is the open neighbourhood of a vertex y ∈ V (G). Also, for a vertex y ∈ V (G), its degree d y is equal to the cardinality of N (y). The maximum and minimum degrees of a graph G are denoted by ∆(G) and δ(G), respectively. A vertex of degree one in a tree is known as a leaf. A vertex adjacent to a leaf is referred to as a stem. A strong stem is a vertex adjacent to two or more leaves. An end stem is a vertex whose only neighbours are leaves, except one. A rooted tree is a directed rooted tree with a distinct vertex ν (see [9]).
Many degree-based topological indices have been widely studied; for example, see [5,6,11]. In this paper, we study one of the degree-based topological indices, namely the geometric-arithmetic index proposed in [12]. The definition of the geometric-arithmetic (GA) index for a simple connected graph G is In [2,14], the first three minimum and maximum values of the GA index were determined, and some lower and upper bounds on the GA index for molecular graphs were obtained. Additional details about the mathematical study of the GA index can be found in [1,3,4,7,10].
In [12], Vukičević and Furtula demonstrated the upper bound, given in the next theorem, on the geometric-arithmetic index for trees. Theorem 1.1. If T is a tree with n vertices, then where the upper bound is achieved if and only if T is the path graph.
In this work, we extend Theorem 1.1 by providing an upper bound on the geometric-arithmetic index for trees T in terms of order and maximum degree of T . We also characterize all the extremal trees attaining the bound.

Lemma 2.2.
Let n be the order of the tree T and ∆ be its maximum degree. If T contains at least one vertex of degree three, a stem, and it differs from ν; then there exists a tree T of maximum degree ∆ and order n such that GA(T ) < GA(T ).
The vertex x is not an end-stem by Lemma 2.1. Let T i be the component of T \ ν that contains v i . Let w be the parent of the leaf q i ∈ V (T i ) and q i be the maximum distance from v i (see Figure 2). Then, by Lemma 2.1, d(w) = 2.
Let T be the tree formed by adding a pendent edge q i y to T \ xy (see Figure 2) and S = {q i w, xy, xx 1 , xx 2 , . . . , xx δ−1 }. Obviously, T is a tree of order n with ∆(T ) = ∆(T ). We have Figure 2: The trees T and T used in the proof of Lemma 2.2. and We obtain GA(T ) < GA(T ) by combining (3) and (4). This completes the proof. Proof. Let y = ν and d(y) = δ ≥ 3, i.e., d(y, ν) should probably be large. Let is a leaf and w be its parent, such that it has the maximum distance from v i (see Figure 3) We assume that d(w) = 2, by Lemma 2.1 and 2.2 and by using the option of y and that, apart from leaves, every descendant of y has degree two. We examine two cases.

Case 1. δ = 3.
Let T be the tree formed by adding an edge q i x 0 1 to T \ yx 0 1 (see Figure 3) and S = {yx 0 1 , yx 0 2 , yx 3 , q i w}. Obviously, T is a tree of order n with ∆(T ) = ∆(T ). We have and We obtain GA(T ) < GA(T ) by combining (5) and (6). and From (7) and (8), we obtain GA(T ) < GA(T ). Next, let q i = x u δ−1 δ−1 . Let T be the tree formed by adding edges x 0 By definition, we have and Note that Thus, from (9) and (10) we conclude that GA(T ) < GA(T ). A spider tree (also known as a star-like tree) [8,13] has at most one vertex (known as the center) of degree greater than two; see [9] for more information. A leg of a spider tree is a path that connects its center to a vertex of degree one. Proof. Let the center of T be ν and N (ν) = {v 1 , v 2 , . . . , v k }. Let ν be the root of T . Let d(v 1 ) = 1, without loss of generality, and let the leg v k z 1 z 2 . . . z u be a longest one in T . Let the tree T be formed by adding a pendent edge v 1 z u to T \ z u z u−1 . Assume that S = {v 1 ν, z u z u−1 , z u−2 z u−1 }. By definition, we have and From (11) and (12), we see that GA(T ) < GA(T ).

Main result
In this section, we present the main result of this paper. Figure 4 represents trees with the maximum GA index among trees of maximum degree 4 and order n for n = 7, 9, 10.
with equality if and only if T is a spider tree and all of its legs have lengths at most two or all of its legs have lengths at least two.
Proof. Let GA(T 1 ) = max{GA(T ) | T be a tree of order n and maximum degree ∆}. Let ν be the root of T 1 and its degree be ∆. If ∆ = 2, then T is the path of order n, and the result follows from Theorem 1.1. Next, we assume that ∆ ≥ 3. By using Lemmas 2.1, 2.2 and 2.3, we conclude that T 1 is a spider tree with center ν. By Lemma 2.4, all the legs of T 1 have lengths either at most two or at least two. Suppose that all the legs of T 1 have lengths at least two. Then, it is obvious that ∆ ≤ n−1 2 and Next, assume that all the legs of T 1 have lengths at most two. Suppose T 1 has at least one leg of length one. The result is immediate if T 1 is a star graph. If T 1 is not a star, then (2∆ + 1 − n) √ ∆ is the number of leaves adjacent to ν and thence GA(T ) = 2 (2∆ + 1 − n) The proof is completed.