Sharp bounds on the zeroth-order general Randi\'c index of trees in terms of domination number

The zeroth-order general Randi\'c index of graph $G=(V_G,E_G)$, denoted by $^0R_{\alpha}(G)$, is the sum of items $(d_{v})^{\alpha}$ over all vertices $v\in V_G$, where $\alpha$ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $^0R_{\alpha}$ of trees with a domination number $\gamma$, in intervals $\alpha\in(-\infty,0)\cup(1,\infty)$ and $\alpha\in(0,1)$, respectively. The corresponding extremal graphs of these bounds are also characterized.


Introduction
Let G be a graph with vertex vertex V G and edge set E G .The general Randić index is defined as where d v denotes the degree of a vertex v ∈ V (G), and α is an arbitrary real number.It's widely known that R − 1 , i.e., the Randić index in original sense, was introduced by the chemist Milan Randić [18] under the name connectivity index or branching index in 1975, which has a good correlation with a variety of physico-chemical properties of alkanes, such as enthalpy of formation, boiling point, parameters in the Antoine equation, surface area and solubility in water, etc.In the past 30 to 40 years, the Randić index has been widely utilized in physics, chemistry, biology, and complex networks [5,19], and many interesting mathematical properties have been obtained [4,11,13].In 1998, Bollobás and Erdös [1] generalized this index by replacing − 1  2 with a real number α, and called it the general Randić index, denoted by R α = R α (G).
Moreover, there are also many variants of Randić index [6,9,20].In [8], Kier and Hall In some bibliographies, 0 R is also called the modified first Zagreb index ( m M 1 ).Pavlović [16] determined the extremal (n, m)-graphs of 0 R with maximum value.Almost at the same time, Lang et al. considered similar problems in [12] for the first Zagreb index (M 1 ), which is defined as In 2005, Li and Zheng [14] constructed the zeroth-order general Randić index, written 0 R α , is the sum of items (d v ) α over all vertices v ∈ V G , where α is an pertinently chosen real number.Note that 0 R − 1 2 = 0 R = m M 1 , and 0 R 2 = M 1 in the mathematical sense.For the zeroth-order general Randić index of trees, Li and Zhao [15] determined the first three maximum and minimum values with exponent α 0 , −α 0 , 1 α 0 , 1 α 0 , where α 0 ≥ 2 is an integer.In 2007, Hu et al. [7] investigated connected (n, m)-graphs with extremal values of 0 R α .Two years later, in [17], Pavlović et al. corrected some errors in the work of Hu et al.
Recently, the relationships between Randić-type indices and domination number has attracted much attention of many researchers.In 2016, Borovićanin and Furtula [2] gave the precise upper and lower bounds on the first Zagreb index (M 1 ) of trees in terms of domination number and characterized the corresponding extremal trees.Later, Bermudo et al. [3] and Liu et al. [10] answered the same question regarding the Randić index (R − 1

2
) and the modified first Zagreb index ( m M 1 ), respectively.Motivate by [2,3,10], in this paper, we intend to establish connections between the zeroth-order general Randić index of trees and domination number.
For convenience, we first introduce some graph-theoretic terminology and notions.The number of vertices and edges of graph G are called the order and size of G, respectively.For each v ∈ V G , the set of neighbours of this vertex is denoted by called a pendent vertex or a leaf vertex.The maximum vertex degree in G is denote by ∆(G).The diameter of a tree is the longest path between two pendent vertices.The dominating set of graph G is a vertex subset in V G such that every vertex in V G \ D is adjacent at least one vertex in D. A subset D is called minimum dominating set of G if D contains least vertices among all dominating sets.Domination number γ is defined as γ = |D|.
Based on the above consideration, the structure of this paper is arranged as below.In Section 2, we prove a fundamental lemma and simplify the mathematical formula of several bounds on 0 R α .Then in Section 3 and 4, sharp upper and lower bounds on 0 R α of trees with a given domination number for α ∈ (−∞, 0) ∪ (1, ∞) and α ∈ (0, 1) are obtained, respectively.Furthermore, the corresponding extremal trees are characterized.

Preliminaries
Now, we present a basic lemma, and then show the simplified mathematical formula of bounds on 0 R α .
By repeating Lemma 2.1, finally, we can obtain the following corollary.
Note that D is a minimum dominating set in a tree T with order n and domination number γ, and (2.1) Now the zeroth-order general Randić index 0 R α (T ) can be given by (2.3) If α ∈ (0, 1), then by Corollary 2.2, we can see the sum (2.2) necessarily attains maximum when degrees Bearing in mind previous discussion, one can check that the formula shown in (2.2) will attain its maximum if D contains t vertices with degree q + 1 and γ − t vertices with degree q, and D contains t vertices with degree q + 1 and n − γ − t vertices with degree q .Thus, we have (2.4) For fixed n and γ, the right-hand side of the inequality (2.4) can be viewed as the function ).So far, we have obtained a simplified formula of bounds on 0 R α .
3 Bounds for the 0 R α∈(0,1) of trees in terms of domination number In this section, several upper and lower bounds for the zeroth-order general Randić index 0 R α∈(0,1) of trees are determined.To characterize extremal n-vertex trees of 0 R 2 with a given domination number γ, Borovićanin and Furtula [2] defined three trees family, denoted by F 1 (n, γ), F 2 (n, γ) and F 3 (n, γ) in here, which will be shown below.(iii) F 3 (n, γ) is a set of trees with order n and domination number γ, which are obtained from the star S n−γ+1 by attaching a pendant edge to its γ − 1 pendent vertices.
with equality holding if and only if T ∈ F 2 (n, γ).
Proof.(i).For path P 3 , the theorem holds.Suppose n ≥ 3, then by 1 There are two possible cases to be consider.
Note that 0 R α∈(0,1) (T ) attains its maximum if and only if T is a path (See [13] Theorem 4.2), we conclude that an extremal T , whose 0 R α∈(0,1) is maximum, only consist of vertices with degree 1, 2 and 3. To determine a sharp upper bound on 0 R α , we must investigate further to find a feasible value of l 2 − l 3 .For a minimum dominating set D of tree T , the number of vertices with degree 2 and 3 are denoted by s 2 and s 3 , respectively.Also, for the set D, the number of vertices with degree 1 and 2are denoted by s 1 and s 2 , respectively.It's holds (3.14) Based on (2.4) and system (3.14), the function h(l 2 − l 3 ) becomes One can easily check that the following relations hold.
By the analogous derivation, the function h(l 2 − l 3 ) can be given by In [2], Borovićanin and Furtula have proved that s 1 ≥ 3γ − n for trees, and s 1 > 3γ − n always holds if there exists a vertex in V T has two pendent neighbours.It's obvious that 0 R α of trees attains its maximum value if s 1 = 3γ − n, i.e., l 2 − l 3 = 5γ − 2n + 1.In such a case, one can check that corresponding extremal trees all belong to F 2 (n, γ).Now, the function h(l 2 − l 3 ) becomes , implying all vertices in D has degrees 2, where D is an arbitrary minimum dominating set.Consequently, all vertices in D have degree 1 and 2, i.e., T ∼ = P n , a contradiction, since γ ≥ n+3 3 .
Next, we assume l 3 − l 2 ≥ n − 2γ + 2, then by (2.4), we get Analogously, we have to find the minimum realizable value of l 2 −l 3 .For an arbitrary minimum dominating set D of T , the number of vertices with degree 1, and 2 are denoted by s 1 and s 2 , respectively, and for the set D, the number of vertices with degree 2, and 3 are denoted by s 2 and s 3 , respectively.Apparently, it holds s 2 − s 2 = 2γ − n − 2 and l 2 − l 3 = 2γ − n − s 1 + 1.Hence, the function h(l 2 − l 3 ) can be given by Based on previous discussions, we can determined the only possible value of s 1 , that is 3γ − n, implying l 2 − l 3 = 1 − γ, such that there exists a corresponding extremal trees with order n and domination number γ, where n+3 3 ≤ γ ≤ n 2 , satisfying its all vertices in an arbitrary minimum dominating set D have degrees 1 and 2, while all vertices in D have degrees 2 and 3.Then, the function h(l 2 − l 3 ) can be written as At this time, we have l 1 = n − γ, l 2 = 0, and l 3 = γ − 1.Based on previous considerations, one can check that the corresponding extremal trees all belong to F 2 (n, γ).

Definition 3 . 1 .
(i) F 1 (n, γ) is a graph family contains some n-vertex trees with domination number γ which consists of the stars of orders n−γ γ and n−γ γ with exactly γ − 1 pairs of adjacent pendent vertices in neighbouring stars.(ii) F 2 (n, γ) is a graph family contains some n-vertex trees T with domination number γ such that each vertex in V T has at most one pendent neighbour and T satisfies: (1) there exists a minimum dominating set D of T has 3γ − n − 2 vertices with degree 3 and 2(n − 2γ) vertices with degree 2, while D has n − 2γ + 2 vertices with degree 2 and 3γ − n pendent vertices, or (2) there exists a minimum dominating set D of T has n − 2γ vertices with degree 2 and 3γ − n pendent vertices, while D has 2(n − 2γ + 1) vertices with degree 2, 3γ − n − 2 with degree 3, and each vertex in D has only one neighbour in domination set D.