ON THE HARMONIC INDEX AND THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

. The harmonic index of a conected graph G is deﬁned as H ( G ) = P uv ∈ E ( G ) 2 d ( u )+ d ( v ) , where E ( G ) is the edge set of G , d ( u ) and d ( v ) are the degrees of vertices u and v , respectively. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M . Let q ( G ) be the spectral radius of the signless Laplacian matrix Q ( G ) = D ( G ) + A ( G ), where D ( G ) is the diagonal matrix having degrees of the vertices on the main diagonal and A ( G ) is the (0 , 1) adjacency matrix of G . The harmonic index of a graph G and the spectral radius of the matrix Q ( G ) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q ( G ). We prove that for a connected graph G with n vertices, we have and the bounds are best possible.


Introduction
A lot of research has been done on topological indices due to their chemical importance. Chemical-based experiments show that there is a strong relationship between the properties of chemical compounds and their molecular structures. Topological indices are used for modelling properties of chemical compounds and biological activities in chemistry, biochemistry and nanotechnology. We study the harmonic index which is one of the most known topological indices.
Let G be a simple connected graph with vertex set V (G) and edge set E(G). The degree of a vertex v ∈ V (G), d (v), is the number of edges incident with v. A tree is a connected graph containing no cycles and a unicyclic graph is a connected graph containing exactly one cycle. A bicyclic graph is a connected graph G having n + 1 edges where n is the number of vertices of G. Let us denote the complete graph, the star and the path having n vertices by K n , S n and P n , respectively.
The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M . Let q(G) be the spectral radius of the signless Laplacian matrix Q(G) = D(G) + A(G), where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. We denote the spectral radius (of the adjacency matrix A(G)) of a graph G by λ(G).
The Randić index of a graph G is defined as .
This topological index has been successfully related to chemical and physical properties of organic molecules, and become one of the most important molecular descriptors. The Randić index was introduced by Randić [16] and generalized by Bollobás and Erdős [3]. Using the AutoGraphiX2 system, Aouchiche, Hansen and Zheng [1,2] studied lower and upper bounds on R(G) ⊕ i(G) in terms of the number of vertices of G, where i(G) is one of the following invariants: the maximum, minimum and average degree, diameter, girth, algebraic and vertex connectivity, matching number and the spectral radius of G, and ⊕ denotes one of the four operations +, −, ×, /. The harmonic index of a graph G was introduced by Fajtlowicz [8]. Hansen and Vukicević [12] studied the connection between the Randić index and the chromatic number of graphs. Deng et al. [6] considered the relation connecting the harmonic index and the chromatic number and strengthened the result relating the Randić index and the chromatic number conjectured by the system AutoGraphiX and proved in [12]. Favaron, Mahéo and Saclé [9] considered the relationship between the harmonic index and eigenvalues of a graph. Using the AutoGraphiX system, Hansen and Lucas [11] gave a conjecture saying that if G is a connected graph having n ≥ 4 vertices, then with equality if and only if G is K n for 4 ≤ n ≤ 12 and G is S n for n ≥ 13. Recently, Ning and Peng [15] solved this conjecture. Motivated by the work of [15] we study the relationship between the harmonic index H(G) of a graph G and the spectral radius of the signless Laplacian matrix Q(G). In particular, we prove the following theorem.

Theorem 1.1. Let G be a connected graph having n vertices. Then
with equality if and only if G is S n for n ≥ 6 and G is K n for 4 ≤ n ≤ 5.

Preliminaries
In this section, we present known results, which will be used in the proofs of our theorems. Upper bounds on the spectral radius of the signless Laplacian matrix and the adjacency matrix of a graph were given in [10] and [13], respectively.

Lemma 2.1 ([10]
). Let G be a connected graph with n vertices, m edges and let q(G) be the spectral radius of the signless Laplacian matrix of G. Then with equality if and only if G is K n or S n .

Lemma 2.2 ([13]
). Let G be a connected graph G with n vertices, m edges and let λ(G) be the spectral radius of the adjacency matrix of G. Then with equality if and only if G is K n or S n .
Let us present three lower bounds on the harmonic index H(G) of a graph G for general graphs, unicyclic graphs and bicyclic graphs.

Results
First we consider graphs with n ≥ 6 vertices, m edges and the harmonic index at least 2(n−1) Theorem 3.1. Let G be a connected graph with n ≥ 6 vertices and m edges. If .

Lemma 3.1 ([4]). The maximum spectral radius λ(G) of a connected graph G with n ≥ 4 vertices and m edges is the maximum root of
We use Lemma 3.1 in the proof of Theorem 3.5.  Let us present bounds for graphs having 4 and 5 vertices. For m = 7 we have G = K 5 − {e 1 , e 2 , e 3 } where e 1 , e 2 , e 3 ∈ E(K 5 ). There are four non-isomorphic graphs having 7 edges. If G[e 1 , e 2 , e 3 ] is K 3 , then H(K 5 −{e 1 , e 2 , e 3 }) = If m = 4, 5 or 6, it can be proved similarly that q(G) H(G) < 16 5 . From Theorems 3.6 and 3.7, and Corollaries 3.1 and 3.2 we obtain our main result (Theorem 1.1).