Some new sharp bounds for the spectral radius of a nonnegative matrix and its application

In this paper, we give some new sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. Using these bounds, we obtain some new and improved bounds for the signless Laplacian spectral radius of a graph or a digraph.


Introduction
The largest modulus of eigenvalues of Q( be the average out-degree of the out-neighbours of v i in − → G . In this paper, we assume that the graph (digraph) is simple and connected (strong connected).
In , Maden, Das, and Cevik [] obtained the following bounds for the signless Laplacian spectral radius of a graph: In , Xi and Wang [] obtained the following bounds for the signless Laplacian spectral radius of a digraph: In this paper, we improve the bounds for the signless Laplacian spectral radius of a graph (digraph) that are given in () and ().

Main result
In this section, some upper and lower bounds for the spectral radius of a nonnegative irreducible matrix are given. We need the following lemma.
Theorem . Let A = (a ij ) be an irreducible and nonnegative matrix with a ii =  for all i ∈ N and the row sum r  , r  , . . . , r n .
for any i, j ∈ N . Then Moreover, either of the equalities in () holds if and only if t i + s i r i = t j + s j r j for any distinct i, j ∈ N .
Proof Let R = diag(r  , r  , . . . , r n ). Since the matrix A is nonnegative irreducible, the matrix R - BR is also nonnegative and irreducible. By the famous Perron-Frobenius theorem [], there is a positive eigenvector x = (x  , x  , . . . , x n ) T corresponding to the spectral radius of R - BR.
Upper bounds: Let x p >  be an arbitrary component of x, Similarly, we have ρ(B)x q = t q x q + n k=,k =q a qk r k x k r q ≤ t q + s qa qp r p r q x q + a qp r p r q x p .
By (), (), and ρ(B)t p > , ρ(B)t q > , we have Therefore, This must be true for every p = q. Then This must be true for any q ∈ N . Then Lower bounds: Let x p >  be an arbitrary component of x, Similarly, we have ρ(B)x q = t q x q + n k=,k =q a qk r k x k r q ≥ t q + s qa qp r p r q x q + a qp r p r q x p . (   ) By (), (), and ρ(B)t p > , ρ(B)t q > , we have This must be true for every p = q. Then This must be true for all q ∈ N . Then From (), (), and x p >  as an arbitrary component of x, we get x k = x q = x p for all k. Then we see easily that the right equality holds in () for t i + s i r i = t j + s j r j for any distinct i, j ∈ N . The proof of the left equality in () is similar to the proof of the right equality, and we omit it here.
Thus, we complete the proof.

Signless Laplacian spectral radius of a graph
In this section, we will apply Theorem . to obtain some new results on the signless Laplacian spectral radius ρ(G) of a graph.

Moreover, one of the equalities in () holds if and only if G is a regular graph.
Proof We apply Theorem . to Q(G).
. Thus () holds. And the equality holds in () for regular graphs if and only if G is a regular graph.

Remark . Obviously, we have
That is to say, our upper bound in Theorem . is always better than the upper bound () in [].

Theorem . Let G = (V , E) be a simple connected graph on n vertices. Then
Moreover, one of the equalities in (), () holds if and only if G is a regular graph or a bipartite semi-regular graph.
Proof We apply Theorem . to Q(G). Let t i = d i , s i = n j= a ij r j = d i m i for any  ≤ i ≤ n.
And the equality holds if and only if G is a regular graph or a bipartite semi-regular graph.

Signless Laplacian spectral radius of a digraph
In this section, we will apply Theorem . to obtain some new results on the signless Laplacian spectral radius ρ( − → G ) of a digraph.
Theorem . Let − → G = (V , E) be a strong connected digraph on n vertices. Then