New Bounds for the Randi´c Index of Graphs

The Randi´c index of a graph G is deﬁned as the sum of weights 1/ ���� d u d v 􏽰 over all edges uv of G , where d u and d v are the degrees of the vertices u and v in G , respectively. In this paper, we will obtain lower and upper bounds for the Randi´c index in terms of size, maximum degree, and minimum degree. Moreover, we obtain a generally lower and a general upper bound for the


Introduction
Let G be a simple graph with a vertex set V � V(G) and edge set E(G). e integers n � n(G) � |V(G)| and m � m(G) � |E(G)| are the order and the size of the graph G, respectively. e open neighborhood of vertex v is { }, and the degree of v is d G (v) � d v � |N(v)|. One of the most active fields of research in contemporary chemical graph theory is the study of topological indices of graph invariants that can be used for describing and predicting physicochemical and pharmacologic properties of organic compounds. Topological indices have been used and have been shown to give a high degree of predictability of pharmaceutical properties. In 1947, Wiener [1] conceived the first molecular graph-based structure descriptor, eventually named the "Wiener index." e information on the chemical constitution of the molecule is conventionally represented by a molecular graph. Graph theory was successfully provided by the chemist with a variety of very useful tools, namely, topological indices. Among the several hundred presently existing graphbased molecular structure descriptors [2], the Randić index R(G) of a graph was introduced by the chemist Randić under the name of "branching index" in 1975 [3] as the sum of Also, it was designed in 1975 to measure the extent of branching of the carbon-atom skeleton of saturated hydrocarbons. It was demonstrated that the Randić index is well correlated with a variety of physicochemical properties of alkanes, such as boiling point, enthalpy of formation, surface area, and solubility in water.
e Randić index is certainly the most widely applied in chemistry and pharmacology, in particular for designing quantitative structure-property and structure-activity relations. Randić proposed this index to "quantitatively characterize the degree of molecular branching." According to him, "the degree of branching of the molecular skeleton is a critical factor" for some molecular properties such as "boiling points of hydrocarbons and the retention volumes and the retention times obtained from chromatographic studies" (all citations are taken from [3]).
Zhou et al. [4] obtained lower and upper bounds for the general Randić index, and Du et al. [5] obtained new lower and upper bounds for the Randić index in terms of other topology indices; for other bounds, see [6,7]. en, in this paper, we will obtain new lower and upper bounds for the Randić index.

Main Results
In this section, we present lower and upper bounds for the Randić index.
We make use of the following lemmas in this paper to obtain the results.
Lemma 1 (see [8]). Let x i and y i , i � 1, . . . , n, be real numbers such that Xx i ≤ y i ≤ Yx i for each i � 1, . . . , n. en, with equality holding if and only if either y i � Xx i or y i � Ya i for each i � 1, . . . , n.
Lemma 2 (see [9]). Let s 1 , s 2 , . . . , s n be nonnegative real numbers with the property Further, let be real numbers, and assume that there are t, T ∈ R such that Equality in equation (5) is obtained if and only if T � b 1 � · · · � b k ≥ b k+1 � · · · � b n � t for some k, 1 ≤ k ≤ n. Theorem 1. Let G be a connected graph of size m, maximum degree Δ, and minimum degree δ. en, and the equality holds in (6) if and only if G is a regular graph.
Proof. By the definition of the Randić index, we can write For each edge v i v j ∈ E(G), it holds that where the left-hand side equality is attained if and only if d i � d j � δ for v i v j ∈ E(G) and the right-hand side equality is attained if and only if .
and the proof is completed. Suppose that equality holds in (6). en, the equality holds in (8). From the equality in (8) or for each edge v i v j ∈ E(G). By the equality condition in (9), Conversely, one can easily see that equality holds in (6) for regular graph.
For any nontrivial connected graph G, R − 2 (G) ≥ m/Δ 4 and M 2 (G) ≥ mδ 2 with either equality if and only if G is regular. From eorem 1, it then follows immediately the following consequence. □ Corollary 1. Let G be a connected graph of size m, maximum degree Δ, and minimum degree δ. en, Next, we present an upper bound for the Randić index in terms of size m, maximum degree Δ, and minimum degree δ. Theorem 2. Let G be a graph of size m, with maximum degree Δ and minimum degree δ. en, the following equality holds: if and only if G � K 2 .
Proof. Note that for each By Inequality (13), we have erefore, by definition of the Randić index and Inequality (15), we have and this leads to the desired bound. Suppose that equality holds in (13). en, the equality holds in (14). From the equality in (14), we have Note that Δ and δ are natural numbers; hence, we have Conversely, one can easily see that equality holds in (13) for G � K 2 . Now, we present a lower bound for the Randić index in terms of size m, maximum degree Δ, and minimum degree δ. □ Theorem 3. Let G be a graph of size m, with maximum degree Δ and minimum degree δ. en, the following inequality holds: if and only if G is a regular graph.
Proof. Note that for each edge v i v j ∈ E(G), we have By Inequality (20), we have erefore, by definition of the Randić index and Inequality (21), we have and this leads to the desired bound. Suppose that equality holds in (19). en, the equality holds in (20). From the equality in (20), we have By the equalities condition in (23) and (24), we must i.e., G is the regular graph.
Conversely, one can easily see that equality holds in (19) for regular graph.
Here, we present a lower bound for the Randić index in terms of size m, maximum degree Δ, and minimum degree δ. □ Theorem 4. Let G be a graph of size m, with maximum degree Δ and minimum degree δ. en, the following equality holds: if and only if G is a regular graph.
Proof. Since for any x, y ∈ R + , we have By the definition of the Randić index and setting x � d i and y � d j in Inequality (26), we can write that Journal of Mathematics 3 as desired.
Suppose that equality holds in (25). en, the equality holds in (26). From the equality in (26), we have or Note that x, y > 0; hence, we get x � y; therefore, Also, suppose that equality holds in (25). en, the equality holds in (27). From the equality in (27), we have By equality (31), erefore, d i � d j for each edge v i v j ∈ E(G). Conversely, one can easily see that equality holds in (25) for regular graph.
For any real number α, the general Randić index, R α , is defined in [10] as e concept of the first general Zagreb index introduced by Li et al. [11] is defined as where α ∈ R.
In [12], the following upper bounds for ID were also established: We need the following lemma to prove the next theorem.
□ Lemma 3 (see [13]). Let G be an undirected, simple graph of order n ≥ 2 with no isolated vertices. en, Equality holds if and only if G is a k-regular graph, 1 ≤ k ≤ n − 1.
Applying Inequalities (35)-(40), we establish upper bounds for the Randić index in terms of the general first and second Zagreb indices maximum degree Δ and minimum degree δ. Theorem 5. Let G be a graph of order n, size m, maximum degree Δ, minimal degree δ, and with no isolated vertices. en,

Journal of Mathematics
Proof. By the definition of the Randić index, we can write By Lemma 3, we have and from (35)-(40) and (49), we arrive at (42)-(47) directly.
Here, we obtain the upper bound for the inverse degree index that helps us to obtain next result. □ Lemma 4. Let G be a simple graph of order n ≥ 2, with m edges and with no isolated vertices. en, the following equality holds: if and only if G is a k-regular graph, 1 ≤ k ≤ n − 1.
By Inequality (49) and Lemma 3, we get the next result. □ Corollary 2. . Let G be a graph of order n, size m, maximum degree Δ, and minimal degree δ. en, In [4], the following upper bounds for R − 1 were also established: where d � 2m 2 − (n − 1)mΔ where By using Inequality (48) and Inequalities (55)-(57), we get the next results.

Corollary 3.
Let G be an undirected, simple graph of order n, size m, maximum degree Δ, and minimal degree δ. en, Denote by A the adjacency matrix of the graph G and by D the diagonal matrix of its vertex degrees. e eigenvalues adjacency matrix is e energy of graph G is defined as is concept was introduced by Gutman and is intensively studied in chemistry since it can be used to approximate the total π-electron energy of a molecule (see, e.g., [14,15]). e normalized Laplacian matrix of a graph G is denoted by L and L � I − D − 1/2 AD − 1/2 . Its eigenvalues are normalized Laplacian eigenvalues of graph G. e normalized Laplacian energy (or L-energy) of a graph G is Journal of Mathematics 5 In [16], the following upper bounds for R − 1 were also established: Applying Inequalities (48), (64), and (65), we establish upper bounds for the Randić index in terms of the energy and the normalized Laplacian energy.

Corollary 4.
Let G be a graph with maximum degree Δ and minimum degree δ. en, In [17], the following lower bounds for R − 1 were also established: Theorem 6. Let G be graph of order n and minimal degree δ. en, Equality holds if and only if G � K n .
Proof. By the definition of the Randić index and by Inequality (67), we can write and the proof is completed. Suppose that equality holds in (68). en, the equality holds in (69) and (70). From the equality in (69), we have d i � d j � δ for each edge v i v j ∈ E(G). From the equality in (70), we have G � K n .
Conversely, one can easily see that equality holds in (68) for graph K n . Denote where R(G) is the Randić index. Next, we present a general upper bound for the Randić index in terms of other topological indices. □ Theorem 7. Let G be a graph with the Randić index R (G). en, the following equality holds: if and only if G � K n .
Proof. For x, y, z ≥ 0 and x ≤ z, we have and by solving this inequality, we get (76) Suppose that equality holds in (73). en, the equality holds in (74); hence, we have By the equality condition in (77), we obtain (x + y) � 0 or (z − x) � 0.