Bounds on the Spectral Radius of a Nonnegative Matrix and Its Applications

First we recall some basic definitions and notations that will be used in this paper. Let A be an n × n real matrix and λ1, λ2, . . . , λn be the eigenvalues of A. Since A is not symmetric in general, the eigenvalues may be complex numbers. Without loss of generality, we assume that |λ1| ≥ |λ2| ≥ ⋅ ⋅ ⋅ ≥ |λn|, and then the spectral radius of A is defined as ρ(A) = |λ1|; that is, it is the largest modulus of the eigenvalues of A. By the Perron-Frobenius theorem, we have the following: (1) ρ(A) is an eigenvalue of A if A is a nonnegative matrix; (2) ρ(A) = λ1 is simple if A is a nonnegative irreducible matrix. Let G = (V, E) (?⃗? = (V, E)) be a graph (digraph) with vertex set V = V(G) (= V(?⃗?)) = {V1, V2, . . . , Vn} and edge set E = E(G) (arc set E = E(?⃗?)). A graphG (digraph ?⃗?) is simple if it has no loops and multiple edges (arcs). For any pairs of vertices Vi, Vj ∈ V, if there is a (directed) path from Vi to Vj, the graphG (digraph ?⃗?) is called (strongly) connected. In this paper, we consider finite, simple graphs and digraphs. Let G be a graph and diag (G) = diag (d1, d2, . . . , dn) be the diagonal matrix of vertex degrees of G, where di is the degree of vertex Vi. Let ?⃗? be a digraph; N− ?⃗? (Vi) = {Vj ∈ V(?⃗?) | (Vj, Vi) ∈ E(?⃗?)} and N+ ?⃗? (Vi) = {Vj ∈ V(?⃗?) | (Vi, Vj) ∈ E(?⃗?)} denote the in-neighbors and out-neighbors of Vi, respectively. Let d− i = |N− ?⃗?(Vi)| and d+ i = |N+ ?⃗?(Vi)| denote the indegree and outdegree of the vertex Vi in ?⃗?, respectively, and diag (?⃗?) = diag (d+ 1 , d+ 2 , . . . , d+ n ) be the diagonal matrix of the vertex outdegrees of ?⃗?. LetA(G) = (aij) be the (0, 1) adjacencymatrix ofG, where


Introduction
First we recall some basic definitions and notations that will be used in this paper.Let  be an  ×  real matrix and  1 ,  2 , . . .,   be the eigenvalues of .Since  is not symmetric in general, the eigenvalues may be complex numbers.Without loss of generality, we assume that | 1 | ≥ | 2 | ≥ ⋅ ⋅ ⋅ ≥ |  |, and then the spectral radius of  is defined as () = | 1 |; that is, it is the largest modulus of the eigenvalues of .By the Perron-Frobenius theorem, we have the following: (1) () is an eigenvalue of  if  is a nonnegative matrix; (2) () =  1 is simple if  is a nonnegative irreducible matrix.
Let  be a graph and diag () = diag ( 1 ,  2 , . . .,   ) be the diagonal matrix of vertex degrees of , where   is the degree of vertex V  .
Let  be a graph and ⃗  be a digraph; we call  ( ⃗ ) regular if each vertex of  ( ⃗ ) has the same degree (outdegree).Other definitions, terminology, and notations not in the article can be found in [2][3][4].
In recent decades, there are many results on the bounds of the spectral radius of a nonnegative matrix and the various spectral radii of a graph or a digraph, including the spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius, and the spectral radius of the reciprocal distance matrix; see [5][6][7][8][9][10][11][12][13][14][15][16] and so on.
In this paper, we obtain the sharp bounds for the spectral radius of a nonnegative (irreducible) matrix in Section 2 and then obtain some known results or new results by applying these bounds to a graph in Section 3 or a digraph in Section 4; we revise and improve two known results.

Main Results
In this section, we will obtain the sharp bounds for the spectral radius of a nonnegative (irreducible) matrix and revise and improve the result of Theorem 2.9 in [9].The techniques used in this section are motivated by [7,9,14] and so on.
Case 2 ( is symmetric and  > 0).By (i) and (ii),  is symmetric and  is the smallest nondiagonal element.We have It is a contradiction by the fact  −1 ≥   .

Various Spectral Radii of a Graph
Let  be a graph.In Section 1, the (adjacency) matrix (), the signless Laplacian matrix (), the distance matrix D() (if  is connected), the distance signless Laplacian matrix Q() (if  is connected), the reciprocal distance matrix () (if  is connected), the (adjacency) spectral radius (), the signless Laplacian spectral radius (), the distance spectral radius  D (), the distance signless Laplacian spectral radius  D (), and the spectral radius of the reciprocal distance matrix (()) are defined.Now, in this section, we will apply Theorem 2, Corollary 3, and Theorem 4 to (), (), D(), Q(), and () and obtain some new results or known results.

Corollary 5.
Let  be a graph on  vertices with degree sequence  1 ,  2 , . . .,   , where  1 ≥  2 ≥ ⋅ ⋅ ⋅ ≥   .Then one has Moreover, if  is connected, then the left equality holds if and only if  is a regular graph, the right equality holds if and only if  is a regular graph, or there exists some  with 2 ≤  ≤  such that  is a bidegreed graph in which Remark 8.The left inequality in Corollary 7 can be obtained by Lemma 1 immediately, and the right inequality in Corollary 7 is the result of Theorem 3.2 in [15].

Distance Spectral Radius of a Graph.
Let  be a connected graph and  be the diameter of .Then the distance matrix D() = (  ) is nonnegative and symmetric.By applying Corollary 3 and Theorem 4 to the distance matrix D() with  = 0,  = 1,  = 0,  = , and   =   for any 1 ≤  ≤ , we note that  21 = ⋅ ⋅ ⋅ =  1 =  implies a contradiction.Then we have the following.

Corollary 9.
Let  be a connected graph on  vertices and  be the diameter of , with distance degree sequence Then one has

Moreover, one of the equalities holds if and only if
Remark 10.The right inequality in Corollary 9 is the result of Corollary 1.8 in [6].
1 ,  2 , . . .,   ) be the diagonal matrix of vertex transmissions of , and let Tr( ⃗ Let  be a graph on  vertices with degree sequence  1 ,  2 , . . .,   , where  1 ≥  2 ≥ ⋅ ⋅ ⋅ ≥   .Then one has [13]eover, if  is connected, then the left equality holds if and only if  is a regular graph, the right equality holds if and only if  is a regular graph, or there exists some  with 2 ≤  ≤  such that  is a bidegreed graph with 1 = ⋅ ⋅ ⋅ =  −1 =  − 1 >   = ⋅ ⋅ ⋅ =   .Remark 6.The left inequality in Corollary 5 can be obtained by Lemma 1 immediately, and the right inequality in Corollary 5 is the result of Theorem 2.2 in[13].3.2.Signless Laplacian Spectral Radius of a Graph.Let  be a graph.By applying Corollary 3 and Theorem 4 to the signless Laplacian matrix () with  =   ,  = 0,  =  1 ,  = 1, and   = 2  for any 1 ≤  ≤ , we have the following.Corollary 7.