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

Authors: Danping Huang; Lihua You
Journal: Journal of Applied Mathematics
 (2016)

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## Abstract

We obtain the sharp bounds for the spectral radius of a nonnegative matrix and then obtain some known results or new results by applying these bounds to a graph or a digraph and revise and improve two known results.


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## Body

## 1. Introduction

First we recall some basic definitions and notations that will be used in this paper. LetA 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.LetG=V,EG→=V,E be a graph (digraph) with vertex set V=VG=VG→=v1,v2,…,vn and edge set E=E(G) (arc set E=E(G→)). A graph G (digraph G→) 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 graph G (digraph G→) is called (strongly) connected. In this paper, we consider finite, simple graphs and digraphs.LetG be a graph and diagG=diagd1,d2,…,dn be the diagonal matrix of vertex degrees of G, where di is the degree of vertex vi.LetG→ be a digraph; NG→-(vi)=vj∈VG→∣(vj,vi)∈EG→ and NG→+(vi)=vj∈VG→∣vi,vj∈EG→ denote the in-neighbors and out-neighbors of vi, respectively. Let di-=NG→-vi and di+=NG→+vi denote the indegree and outdegree of the vertex vi in G→, respectively, and diagG→=diagd1+,d2+,…,dn+ be the diagonal matrix of the vertex outdegrees of G→.LetA(G)=(aij) be the (0,1) adjacency matrix of G, where (1)aij=1,ifviandvjare adjacent;0,otherwise.Let A(G→)=(aij) denote the adjacency matrix of G→, where aij is equal to the number of arcs (vi,vj).Then the signless Laplacian matrix ofG (G→) is defined as (2)QG=diagG+AGQG→=diagG→+AG→.The spectral radii of A(G) and Q(G) (A(G→) and Q(G→)), denoted by ρ(G) and q(G) (ρ(G→) and q(G→)), are called the (adjacency) spectral radius of G (G→) and the signless Laplacian spectral radius of G (G→), respectively.LetG=(V,E) be a connected graph and G→=(V,E) be a strong connected digraph. For u,v∈V, the distance from u to v, denoted by dG(u,v)dG→u,v, is the length of the shortest (directed) path from u to v in G (G→). For u∈V, the transmission of vertex u in G (G→) is the sum of distances from u to all other vertices of G (G→), denoted by TrG(u)TrG→u. The distance matrix of G (G→) is the n×n matrix D(G)=(dij), where dij=dG(vi,vj)(D(G→)=(dij), where dij=dG→vi,vj). In fact, for 1≤i≤n, the transmission of vertex vi, TrG(vi)TrG→vi, is just the ith row sum of D(G)DG→. For convenience, we also call TrG(vi)TrG→vi the distance degree (outdegree) of vertex vi in G (G→), denoted by Di (Di+); that is, Di=∑j=1ndij=TrG(vi)Di+=∑j=1ndij=TrG→vi. Similarly, we define Di-=∑j=1ndji.LetTrG=diagD1,D2,…,Dn be the diagonal matrix of vertex transmissions of G, and let Tr(G→)=diagD1+,D2+,…,Dn+ be the diagonal matrix of vertex transmissions of G→. The distance signless Laplacian matrix of GG→ is the n×n matrix defined by Aouchiche and Hansen as [1] (3)QG=TrG+DGQG→=TrG→+DG→.The spectral radii of D(G) and QG(DG→ and QG→), denoted by ρD(G) and qD(G)  (ρD(G→) and qDG→), are called the distance spectral radius of G  (G→) and the distance signless Laplacian spectral radius of G  (G→), respectively.LetG be a connected graph. The reciprocal distance matrix (also called the Harary matrix) R(G)=(rij) of G is the n×n matrix, where (rij)=1/dij if i≠j and rii=0 for i=1,…,n. Clearly, the reciprocal distance matrix R(G) is nonnegative and symmetric.LetG be a graph and G→ be a digraph; we call G (G→) regular if each vertex of G (G→) has the same degree (outdegree). Other definitions, terminology, and notations not in the article can be found in [2–4].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 [5–16] and so on.In this paper, we obtain the sharp bounds for the spectral radius of a nonnegative (irreducible) matrix in Section2 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.


## 2. 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.Lemma 1 (see [2]).
IfA is an n×n nonnegative matrix with the spectral radius λ(A) and row sums r1,r2,…,rn, then min1≤i≤n⁡ri≤λ(A)≤max1≤i≤n⁡ri. Moreover, if A is irreducible, then one of the equalities holds if and only if the row sums of A are all equal.Theorem 2.
LetA=(aij) be an n×n nonnegative matrix with row sums r1,r2,…,rn, where r1≥r2≥⋯≥rn, and let S be the smallest diagonal element, T be the smallest nondiagonal element, and λ(A) be the spectral radius of A. Take ϕ1=rn and for 2≤l≤n, (4)ϕl=rn+S-T+rn+T-S2+4l-1rl-1-rnT2. Let ϕt=max1≤l≤n⁡ϕl for some 1≤t≤n. Then λ(A)≥ϕt. Moreover, if A is irreducible, then(1)
λ(A)=ϕ1=rn if and only if r1=r2=⋯=rn.(2)
λ(A)=ϕt>rn with 2≤t≤n if and only if A satisfies the following conditions:
(i)
aii=S for 1≤i≤t-1;(ii)
aij=T>0 for 1≤i≤n,1≤j≠i≤t-1;(iii)
r1=r2=⋯=rt-1>rt=rt+1=⋯=rn.Proof.
IfT=0, then ϕl=ϕ1=rn for any 2≤l≤n by rn≥S. Thus by Lemma 1 and r1≥r2≥⋯≥rn, we have λ(A)≥rn=max1≤l≤n⁡ϕl=ϕ1, and if A is irreducible, λ(A)=ϕ1=rn if and only if r1=r2=⋯=rn.
Now we consider the caseT>0.
Firstly, we showλ(A)≥ϕl for all 2≤l≤n.
SinceA is a nonnegative matrix, then ap,q≥T>0 for 1≤p≠q≤n. Thus(5)∑j=1l-1aij≥S+l-2T,if1≤i≤l-1;l-1T,ifl≤i≤n.
Let(6)x=S-rn+2l-3T+rn+T-S2+4l-1rl-1-rnT2l-1T.It is easy to show that x>1. Take (7)xj=x,if1≤j≤l-1,1,ifl≤j≤n,and let U=diagx1,x2,…,xn be a diagonal matrix of order n. Let B=U-1AU, and then B and A have the same eigenvalues, and λB=λA.
Now we consider the row sums ofB, say, s1,s2,…,sn.
Case  1 1 ≤ i ≤ l - 1. Consider(8)si=∑j=1nxjxiaij=∑j=1l-1aij+1x∑j=lnaij=1x∑j=1naij+1-1x∑j=1l-1aij=1xri+1-1x∑j=1l-1aij≥1xri+1-1xS+l-2T=1xri-S+S+1-1xl-2T≥1xrl-1-S+S+1-1xl-2T,with equality if and only if (a) and (b) hold: (a) aii=S and aij=T if 1≤j≤l-1 with j≠i and (b) ri=rl-1.
Case  2 l ≤ i ≤ n. Consider(9)si=∑j=1nxjxiaij=x∑j=1l-1aij+∑j=lnaij=∑j=1naij+x-1∑j=1l-1aij=ri+x-1∑j=1l-1aij≥ri+x-1l-1T≥rn+x-1l-1T,with equality if and only if (c) and (d) hold: (c) aij=T if 1≤j≤l-1 and (d) ri=rn.
Noting that(10)rn+x-1l-1T=1xrl-1-S+S+1-1xl-2T=S+rn-T+rn+T-S2+4l-1rl-1-rnT2=ϕl,then, by Lemma 1, we have λA=λB≥min⁡s1,s2,…,sn≥ϕl.
Noting thatϕl≥ϕ1=rn by rn+T≥S, thus λ(A)≥ϕt, where ϕt=max1≤l≤n⁡ϕl for some 1≤t≤n.
LetA be irreducible; ϕt=max1≤l≤nϕl for some 1≤t≤n.
Case  1 λ ( A ) = ϕ 1. For 2≤l≤n, by ϕl≥ϕ1 and T>0, we have ϕl=ϕ1⇔rl-1=rn. Then(11)ϕt=ϕ1⟺ϕl=ϕ1∀2≤l≤n⟺r1=r2=⋯=rn.
On the other hand, by Lemma1 and r1≥r2≥⋯≥rn, we have(12)λA=rn⟺r1=r2=⋯=rn.By (11), (12), and ϕ1=rn, (1) holds.
Case  2 λ A = ϕ t > ϕ 1 for some 2 ≤ t ≤ n. Then rt-1>rn and T>0 by ϕt>ϕ1=rn.
Ifλ(A)=ϕt, then s1=s2=⋯=sn=ϕt by the above arguments and Lemma 1; thus (a) and (b) hold for 1≤i≤t-1 and (c) and (d) hold for t≤i≤n. Thus aii=S for 1≤i≤t-1,r1=r2=⋯=rt-1>rt=rt+1=⋯=rn and aij=T>0 for 1≤i≤n,1≤j≠i≤t-1. Now (i), (ii), and (iii) follow.
Conversely, if (i), (ii), and (iii) hold, it is easy to show that equality holds.Corollary 3.
LetA=(aij) be an n×n nonnegative matrix with row sums r1,r2,…,rn, where r1≥r2≥⋯≥rn, and let S be the smallest diagonal element, T be the smallest nondiagonal element, and λ(A) be the spectral radius of A. Take ϕ1=rn and, for 2≤l≤n, (13)ϕl=rn+S-T+rn+T-S2+4l-1rl-1-rnT2. Let ϕt=max1≤l≤n⁡ϕl for some 1≤t≤n. Then λ(A)≥ϕt. Moreover, if A is irreducible with T=0 or A is irreducible and symmetric, then(14)λA=ϕtifft=1,r1=r2=⋯=rn.Proof.
We complete the proof by the following two cases.
Case  1 T = 0. It is obvious by the proof of Theorem 2.
Case  2 A  i s  s y m m e t r i c  a n d  T > 0. By (i) and (ii), A is symmetric and T is the smallest nondiagonal element. We have r1=r2=⋯=rt-1=S+n-1T<rt=⋯=rn. It is a contradiction by the fact rt-1≥rt.Similar to the proof of Theorem2 (so we omit the proof of Theorem 4), we can show Theorem 4 which revises and improves the result of Theorem  2.9 in [9].Theorem 4.
LetA=aij be an n×n nonnegative matrix with row sums r1,r2,…,rn, where r1≥r2≥⋯≥rn, and let M be the largest diagonal element, N be the largest nondiagonal element, and λ(A) be the spectral radius of A. Take ϕ1=r1 and, for 2≤l≤n,(15)ϕl=rl+M-N+rl+N-M2+4l-1r1-rlN2.Let ϕt=min1≤l≤n⁡ϕl for some 1≤t≤n. Then λ(A)≤ϕt. Moreover, if A is irreducible, then(1)
λ(A)=ϕ1=r1 if and only if r1=r2=⋯=rn.(2)
λ(A)=ϕt<r1 with 2≤t≤n if and only if A satisfies the following conditions:
(i)
aii=M for 1≤i≤t-1;(ii)
aij=N>0 for 1≤i≤n,1≤j≠i≤t-1;(iii)
r1=r2=⋯=rt-1>rt=rt+1=⋯=rn.


## 3. Various Spectral Radii of a Graph

LetG be a graph. In Section 1, the (adjacency) matrix A(G), the signless Laplacian matrix Q(G), the distance matrix D(G) (if G is connected), the distance signless Laplacian matrix Q(G) (if G is connected), the reciprocal distance matrix R(G) (if G is connected), the (adjacency) spectral radius ρ(G), the signless Laplacian spectral radius q(G), the distance spectral radius ρD(G), the distance signless Laplacian spectral radius qD(G), and the spectral radius of the reciprocal distance matrix λ(R(G)) are defined. Now, in this section, we will apply Theorem 2, Corollary 3, and Theorem 4 to A(G), QG, DG, QG, and RG and obtain some new results or known results.


### 3.1. Adjacency Spectral Radius of a Graph

LetG be a graph. By applying Corollary 3 and Theorem 4 to the (adjacency) matrix A(G) with S=0, T=0, M=0, N=1, and ri=di for any 1≤i≤n, we have the following.Corollary 5.
LetG be a graph on n vertices with degree sequence d1,d2,…,dn, where d1≥d2≥⋯≥dn. Then one has (16)dn≤ρG≤min1≤i≤ndi-1+di+12+4i-1d1-di2.Moreover, if G is connected, then the left equality holds if and only if G is a regular graph, the right equality holds if and only if G is a regular graph, or there exists some t with 2≤t≤n such that G is a bidegreed graph with d1=⋯=dt-1=n-1>dt=⋯=dn.Remark 6.
The left inequality in Corollary5 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

LetG be a graph. By applying Corollary 3 and Theorem 4 to the signless Laplacian matrix Q(G) with S=dn, T=0, M=d1, N=1, and ri=2di for any 1≤i≤n, we have the following.Corollary 7.
LetG be a graph on n vertices with degree sequence d1,d2,…,dn, where d1≥d2≥⋯≥dn. Then one has (17)2dn≤qG≤min1≤i≤nd1+2di-1+2di-d1+12+8i-1d1-di2.Moreover, if G is connected, then the left equality holds if and only if G is a regular graph, the right equality holds if and only if G is a regular graph, or there exists some t with 2≤t≤n such that G is a bidegreed graph in which d1=⋯=dt-1=n-1>dt=⋯=dn.Remark 8.
The left inequality in Corollary7 can be obtained by Lemma 1 immediately, and the right inequality in Corollary 7 is the result of Theorem  3.2 in [15].


### 3.3. Distance Spectral Radius of a Graph

LetG be a connected graph and d be the diameter of G. Then the distance matrix D(G)=(dij) is nonnegative and symmetric. By applying Corollary 3 and Theorem 4 to the distance matrix D(G) with S=0, T=1, M=0, N=d, and ri=Di for any 1≤i≤n, we note that d21=⋯=dn1=d implies a contradiction. Then we have the following.Corollary 9.
LetG be a connected graph on n vertices and d be the diameter of G, with distance degree sequence D1,D2,…,Dn such that D1≥D2≥⋯≥Dn. Let(18)fi=Dn-1+Dn+12+4i-1Di-1-Dn2.Then one has (19)max2≤i≤nDn,fi≤ρDG≤min1≤i≤nDi-d+Di+d2+4di-1D1-Di2.Moreover, one of the equalities holds if and only if D1=D2=⋯=Dn.Remark 10.
The right inequality in Corollary9 is the result of Corollary  1.8 in [6].By applying Theorem2 and Corollary 3 to the distance matrix D(G) with S=0, T=1, and ri=Di for i=1,2,…,n, we have the following.Corollary 11 (see [16, Theorem  2]).
LetG be a connected graph on n vertices with distance degree sequence D1,D2,…,Dn such that D1≥D2≥Di-1>Di≥⋯≥Dn  for some 2≤i≤n. Then (20)ρDG>Dn-1+Dn+12+4i-1Di-1-Dn2.


### 3.4. Distance Signless Laplacian Spectral Radius of a Graph

LetG be a connected graph and d be the diameter of G. Then the distance matrix Q(G) is nonnegative and symmetric. By applying Corollary 3 and Theorem 4 to the distance matrix Q(G) with S=Dn, T=1, M=D1, N=d, and ri=2Di for i=1,2,…,n, we note that d21=⋯=dn1=d implies a contradiction. Then we have the following.Corollary 12.
LetG be a connected graph on n vertices with distance degree sequence D1,D2,…,Dn such that D1≥D2≥⋯≥Dn and d be the diameter of G. Let (21)fi=3Dn-1+Dn+12+8i-1Di-1-Dn2,gi=D1+2Di-d+2Di-D1+d2+8di-1D1-Di2.Then one has (22)max2≤i≤n2Dn,fi≤qDG≤min1≤i≤ngi.Moreover, one of the equalities holds if and only if D1=D2=⋯=Dn.Remark 13.
The right inequality in Corollary12 is the result of Theorem  3.8 in [9].By applying Theorem2 and Corollary 3 to the distance matrix Q(G) with S=Dn, T=1, and ri=2Di for i=1,2,…,n, we have the following.Corollary 14.
LetG be a connected graph on n vertices with distance degree sequence D1,D2,…,Dn such that D1≥D2≥Di-1>Di≥⋯≥Dn for some 2≤i≤n. Then qD(G)>f(i).


### 3.5. Spectral Radius of the Reciprocal Distance Matrix

By applying Corollary3 and Theorem 4 to the reciprocal distance matrix R(G) with S=0, T=1/d, M=0, N=1, and ri=Ri for i=1,…,n, we have the following.Corollary 15.
LetG be a connected graph on n vertices, d be the diameter of G, Ri=∑j=1n‍rij, and the row sum sequence be R1,R2,…,Rn of R(G) satisfying R1≥R2≥⋯≥Rn. Let(23)fi=Rn-1/d+Rn+1/d2+4/di-1Ri-1-Rn2,gi=Ri-1+Ri+12+4i-1R1-Ri2.Then (24)max2≤i≤nRn,fi≤λRG≤min1≤i≤ngi.Moreover, the left equality holds if and only if R1=R2=⋯=Rn, and the right equality holds if and only if either R1=R2=⋯=Rn or there exists some t with 2≤t≤n such that G is a graph with t-1 vertices of degree n-1 and the remaining n-t+1 vertices have equal degree less than n-1.Remark 16.
The right inequality in Corollary15 is the result (i) of Theorem  4 in [16].


## 3.1. Adjacency Spectral Radius of a Graph

LetG be a graph. By applying Corollary 3 and Theorem 4 to the (adjacency) matrix A(G) with S=0, T=0, M=0, N=1, and ri=di for any 1≤i≤n, we have the following.Corollary 5.
LetG be a graph on n vertices with degree sequence d1,d2,…,dn, where d1≥d2≥⋯≥dn. Then one has (16)dn≤ρG≤min1≤i≤ndi-1+di+12+4i-1d1-di2.Moreover, if G is connected, then the left equality holds if and only if G is a regular graph, the right equality holds if and only if G is a regular graph, or there exists some t with 2≤t≤n such that G is a bidegreed graph with d1=⋯=dt-1=n-1>dt=⋯=dn.Remark 6.
The left inequality in Corollary5 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

LetG be a graph. By applying Corollary 3 and Theorem 4 to the signless Laplacian matrix Q(G) with S=dn, T=0, M=d1, N=1, and ri=2di for any 1≤i≤n, we have the following.Corollary 7.
LetG be a graph on n vertices with degree sequence d1,d2,…,dn, where d1≥d2≥⋯≥dn. Then one has (17)2dn≤qG≤min1≤i≤nd1+2di-1+2di-d1+12+8i-1d1-di2.Moreover, if G is connected, then the left equality holds if and only if G is a regular graph, the right equality holds if and only if G is a regular graph, or there exists some t with 2≤t≤n such that G is a bidegreed graph in which d1=⋯=dt-1=n-1>dt=⋯=dn.Remark 8.
The left inequality in Corollary7 can be obtained by Lemma 1 immediately, and the right inequality in Corollary 7 is the result of Theorem  3.2 in [15].


## 3.3. Distance Spectral Radius of a Graph

LetG be a connected graph and d be the diameter of G. Then the distance matrix D(G)=(dij) is nonnegative and symmetric. By applying Corollary 3 and Theorem 4 to the distance matrix D(G) with S=0, T=1, M=0, N=d, and ri=Di for any 1≤i≤n, we note that d21=⋯=dn1=d implies a contradiction. Then we have the following.Corollary 9.
LetG be a connected graph on n vertices and d be the diameter of G, with distance degree sequence D1,D2,…,Dn such that D1≥D2≥⋯≥Dn. Let(18)fi=Dn-1+Dn+12+4i-1Di-1-Dn2.Then one has (19)max2≤i≤nDn,fi≤ρDG≤min1≤i≤nDi-d+Di+d2+4di-1D1-Di2.Moreover, one of the equalities holds if and only if D1=D2=⋯=Dn.Remark 10.
The right inequality in Corollary9 is the result of Corollary  1.8 in [6].By applying Theorem2 and Corollary 3 to the distance matrix D(G) with S=0, T=1, and ri=Di for i=1,2,…,n, we have the following.Corollary 11 (see [16, Theorem  2]).
LetG be a connected graph on n vertices with distance degree sequence D1,D2,…,Dn such that D1≥D2≥Di-1>Di≥⋯≥Dn  for some 2≤i≤n. Then (20)ρDG>Dn-1+Dn+12+4i-1Di-1-Dn2.


## 3.4. Distance Signless Laplacian Spectral Radius of a Graph

LetG be a connected graph and d be the diameter of G. Then the distance matrix Q(G) is nonnegative and symmetric. By applying Corollary 3 and Theorem 4 to the distance matrix Q(G) with S=Dn, T=1, M=D1, N=d, and ri=2Di for i=1,2,…,n, we note that d21=⋯=dn1=d implies a contradiction. Then we have the following.Corollary 12.
LetG be a connected graph on n vertices with distance degree sequence D1,D2,…,Dn such that D1≥D2≥⋯≥Dn and d be the diameter of G. Let (21)fi=3Dn-1+Dn+12+8i-1Di-1-Dn2,gi=D1+2Di-d+2Di-D1+d2+8di-1D1-Di2.Then one has (22)max2≤i≤n2Dn,fi≤qDG≤min1≤i≤ngi.Moreover, one of the equalities holds if and only if D1=D2=⋯=Dn.Remark 13.
The right inequality in Corollary12 is the result of Theorem  3.8 in [9].By applying Theorem2 and Corollary 3 to the distance matrix Q(G) with S=Dn, T=1, and ri=2Di for i=1,2,…,n, we have the following.Corollary 14.
LetG be a connected graph on n vertices with distance degree sequence D1,D2,…,Dn such that D1≥D2≥Di-1>Di≥⋯≥Dn for some 2≤i≤n. Then qD(G)>f(i).


## 3.5. Spectral Radius of the Reciprocal Distance Matrix

By applying Corollary3 and Theorem 4 to the reciprocal distance matrix R(G) with S=0, T=1/d, M=0, N=1, and ri=Ri for i=1,…,n, we have the following.Corollary 15.
LetG be a connected graph on n vertices, d be the diameter of G, Ri=∑j=1n‍rij, and the row sum sequence be R1,R2,…,Rn of R(G) satisfying R1≥R2≥⋯≥Rn. Let(23)fi=Rn-1/d+Rn+1/d2+4/di-1Ri-1-Rn2,gi=Ri-1+Ri+12+4i-1R1-Ri2.Then (24)max2≤i≤nRn,fi≤λRG≤min1≤i≤ngi.Moreover, the left equality holds if and only if R1=R2=⋯=Rn, and the right equality holds if and only if either R1=R2=⋯=Rn or there exists some t with 2≤t≤n such that G is a graph with t-1 vertices of degree n-1 and the remaining n-t+1 vertices have equal degree less than n-1.Remark 16.
The right inequality in Corollary15 is the result (i) of Theorem  4 in [16].


## 4. Various Spectral Radii of a Digraph

LetG→ be a strong connected digraph. In Section 1, the adjacency matrix A(G→), the signless Laplacian matrix Q(G→), the distance matrix D(G→) (if G→ is connected), the distance signless Laplacian matrix Q(G→) (if G→ is connected), the adjacency spectral radius ρ(G→), the signless Laplacian spectral radius q(G→), the distance spectral radius ρD(G→), and the distance signless Laplacian spectral radius qD(G→) are defined. Now, in this section, we will apply Theorem 2, Corollary 3, and Theorem 4 to A(G→), Q(G→), D(G→), and Q(G→), obtain some new results or known results, and revise and improve the result of Theorem  2.5 in [11].


### 4.1. Adjacency Spectral Radius of a Digraph

LetG→ be a digraph. By applying Corollary 3 and Theorem 4 to the (adjacency) matrix A(G→) with S=0,T=0, M=0, N=1, and ri=di+ for i=1,…,n, we have the following.Corollary 17.
LetG→ be a digraph on n vertices with outdegree sequence d1+,d2+,…,dn+ such that d1+≥d2+≥⋯≥dn+. Then one has (25)dn+≤ρG→≤min1≤i≤ndi+-1+di++12+4i-1d1+-di+2.Moreover, if G→ is a strong connected digraph, then the left equality holds if and only if G→ is a regular digraph, the right equality holds if and only if G→ is a regular digraph, or there exists some t with 2≤t≤n such that G→ is a bidegreed digraph with d1+=⋯=dt-1+>dt+=⋯=dn+ and the indegrees d1-=⋯=dt-1-=n-1.


### 4.2. Signless Laplacian Spectral Radius of a Digraph

LetG→ be a digraph. By applying Corollary 3 and Theorem 4 to the signless Laplacian matrix Q(G→) with S=dn+,T=0, M=d1+,N=1, and ri=2di+ for i=1,…,n, we have the following.Corollary 18.
LetG→ be a digraph on n vertices with outdegree sequence d1+,d2+,…,dn+ such that d1+≥d2+≥⋯≥dn+. Then one has (26)2dn+≤qG→≤min1≤i≤nd1++2di+-1+2di+-d1++12+8i-1d1+-di+2.Moreover, if G→ is a strong connected digraph, then the left equality holds if and only if G→ is a regular digraph, the right equality holds if and only if G→ is a regular digraph, or there exists some t with 2≤t≤n such that G→ is a bidegreed digraph with d1+=⋯=dt-1+>dt+=⋯=dn+ and the indegrees d1-=⋯=dt-1-=n-1.Remark 19.
The left inequality in Corollary18 can be obtained by Lemma 1 immediately, and the right inequality in Corollary 18 revises and improves Proposition 20.Proposition 20 (see [11, Theorem  2.5]).
LetG→ be a strong connected digraph on n vertices with outdegree sequence d1+,d2+,…,dn+ such that d1+≥d2+≥⋯≥dn+. Then one has (27)qG→≤min1≤i≤nd1++2di+-1+2di+-d1++12+8i-1d1+-di+2.Moreover, if i=1, the equality holds if and only if G→ is a regular digraph. If 2≤i≤n, the equality holds if and only if G→ is a regular digraph or a bidegreed digraph in which d1+=d2+=⋯=di-1+=n-1 and di+=⋯=dn+=δ+.The following example shows that the result of Proposition20 is incorrect.Example 21.
Letn≥5 and D1 is shown in Figure 1. For D1, the outdegree sequence is 3=d1+>d2+=d3+=⋯=dn+=2 and the indegree d1-=n-1. We have q(D1)=3+3 by direct computation. It is clear that(28)qD1=3+3=min1≤i≤n⁡d1++2di+-1+2di+-d1++12+8i-1d1+-di+2.Figure 1
The digraphsD1.


### 4.3. Distance Spectral Radius of a Digraph

LetG→ be a strong connected digraph and d be the diameter of G→. By applying Theorems 2 and 4 to the distance matrix D(G→) with S=0,T=1, M=0,N=d, and ri=Di+ for i=1,…,n, we note that d21=⋯=dn1=d implies a contradiction. Then we have the following.Corollary 22.
LetG→ be a strong connected digraph on n vertices with distance outdegree sequence D1+,D2+,…,Dn+ such that D1+≥D2+≥⋯≥Dn+, and let d be the diameter of G→. Let(29)fi=Dn+-1+Dn++12+4i-1Di-1+-Dn+2,gi=Di+-d+Di++d2+4di-1D1+-Di+2.Then one has (30)max2≤i≤nDn+,fi≤ρDG→≤min1≤i≤ngi.Moreover, the left equality holds if and only if D1+=⋯=Dn+ or there exists some t with 2≤t≤n such that D1+=⋯=Dt-1+>Dt+=⋯=Dn+ and D1-=⋯=Dt-1-=n-1 and the right equality holds if and only if D1+=⋯=Dn+.


### 4.4. Distance Signless Laplacian Spectral Radius of a Digraph

LetG→ be a strong connected digraph and d be the diameter of G→. By applying Theorems 2 and 4 to the distance signless Laplacian matrix Q(G→) with S=Dn+,T=1, M=D1+,N=d, and ri=2Di+ for i=1,…,n, we note two facts: the first fact is that (i) and (iii) of (2) in Theorem 2 cannot hold at the same time by aii=Di+=∑1≤j≤n‍dij and ri=2Di+, and the second fact is that d21=⋯=dn1=d implies a contradiction. Then we have the following.Corollary 23.
LetG→ be a strong connected digraph on n vertices with distance outdegree sequence D1+,D2+,…,Dn+ such that D1+≥D2+≥⋯≥Dn+, and let d be the diameter of G→. Let(31)fi=3Dn+-1+Dn++12+8i-1Di-1+-Dn+2,gi=D1++2Di+-d+2Di+-D1++d2+8di-1D1+-Di+2.Then one has(32)max2≤i≤nDn+,fi≤qDG→≤min1≤i≤ngi.Moreover, one of the equalities holds if and only if D1+=⋯=Dn+.


## 4.1. Adjacency Spectral Radius of a Digraph

LetG→ be a digraph. By applying Corollary 3 and Theorem 4 to the (adjacency) matrix A(G→) with S=0,T=0, M=0, N=1, and ri=di+ for i=1,…,n, we have the following.Corollary 17.
LetG→ be a digraph on n vertices with outdegree sequence d1+,d2+,…,dn+ such that d1+≥d2+≥⋯≥dn+. Then one has (25)dn+≤ρG→≤min1≤i≤ndi+-1+di++12+4i-1d1+-di+2.Moreover, if G→ is a strong connected digraph, then the left equality holds if and only if G→ is a regular digraph, the right equality holds if and only if G→ is a regular digraph, or there exists some t with 2≤t≤n such that G→ is a bidegreed digraph with d1+=⋯=dt-1+>dt+=⋯=dn+ and the indegrees d1-=⋯=dt-1-=n-1.


## 4.2. Signless Laplacian Spectral Radius of a Digraph

LetG→ be a digraph. By applying Corollary 3 and Theorem 4 to the signless Laplacian matrix Q(G→) with S=dn+,T=0, M=d1+,N=1, and ri=2di+ for i=1,…,n, we have the following.Corollary 18.
LetG→ be a digraph on n vertices with outdegree sequence d1+,d2+,…,dn+ such that d1+≥d2+≥⋯≥dn+. Then one has (26)2dn+≤qG→≤min1≤i≤nd1++2di+-1+2di+-d1++12+8i-1d1+-di+2.Moreover, if G→ is a strong connected digraph, then the left equality holds if and only if G→ is a regular digraph, the right equality holds if and only if G→ is a regular digraph, or there exists some t with 2≤t≤n such that G→ is a bidegreed digraph with d1+=⋯=dt-1+>dt+=⋯=dn+ and the indegrees d1-=⋯=dt-1-=n-1.Remark 19.
The left inequality in Corollary18 can be obtained by Lemma 1 immediately, and the right inequality in Corollary 18 revises and improves Proposition 20.Proposition 20 (see [11, Theorem  2.5]).
LetG→ be a strong connected digraph on n vertices with outdegree sequence d1+,d2+,…,dn+ such that d1+≥d2+≥⋯≥dn+. Then one has (27)qG→≤min1≤i≤nd1++2di+-1+2di+-d1++12+8i-1d1+-di+2.Moreover, if i=1, the equality holds if and only if G→ is a regular digraph. If 2≤i≤n, the equality holds if and only if G→ is a regular digraph or a bidegreed digraph in which d1+=d2+=⋯=di-1+=n-1 and di+=⋯=dn+=δ+.The following example shows that the result of Proposition20 is incorrect.Example 21.
Letn≥5 and D1 is shown in Figure 1. For D1, the outdegree sequence is 3=d1+>d2+=d3+=⋯=dn+=2 and the indegree d1-=n-1. We have q(D1)=3+3 by direct computation. It is clear that(28)qD1=3+3=min1≤i≤n⁡d1++2di+-1+2di+-d1++12+8i-1d1+-di+2.Figure 1
The digraphsD1.


## 4.3. Distance Spectral Radius of a Digraph

LetG→ be a strong connected digraph and d be the diameter of G→. By applying Theorems 2 and 4 to the distance matrix D(G→) with S=0,T=1, M=0,N=d, and ri=Di+ for i=1,…,n, we note that d21=⋯=dn1=d implies a contradiction. Then we have the following.Corollary 22.
LetG→ be a strong connected digraph on n vertices with distance outdegree sequence D1+,D2+,…,Dn+ such that D1+≥D2+≥⋯≥Dn+, and let d be the diameter of G→. Let(29)fi=Dn+-1+Dn++12+4i-1Di-1+-Dn+2,gi=Di+-d+Di++d2+4di-1D1+-Di+2.Then one has (30)max2≤i≤nDn+,fi≤ρDG→≤min1≤i≤ngi.Moreover, the left equality holds if and only if D1+=⋯=Dn+ or there exists some t with 2≤t≤n such that D1+=⋯=Dt-1+>Dt+=⋯=Dn+ and D1-=⋯=Dt-1-=n-1 and the right equality holds if and only if D1+=⋯=Dn+.


## 4.4. Distance Signless Laplacian Spectral Radius of a Digraph

LetG→ be a strong connected digraph and d be the diameter of G→. By applying Theorems 2 and 4 to the distance signless Laplacian matrix Q(G→) with S=Dn+,T=1, M=D1+,N=d, and ri=2Di+ for i=1,…,n, we note two facts: the first fact is that (i) and (iii) of (2) in Theorem 2 cannot hold at the same time by aii=Di+=∑1≤j≤n‍dij and ri=2Di+, and the second fact is that d21=⋯=dn1=d implies a contradiction. Then we have the following.Corollary 23.
LetG→ be a strong connected digraph on n vertices with distance outdegree sequence D1+,D2+,…,Dn+ such that D1+≥D2+≥⋯≥Dn+, and let d be the diameter of G→. Let(31)fi=3Dn+-1+Dn++12+8i-1Di-1+-Dn+2,gi=D1++2Di+-d+2Di+-D1++d2+8di-1D1+-Di+2.Then one has(32)max2≤i≤nDn+,fi≤qDG→≤min1≤i≤ngi.Moreover, one of the equalities holds if and only if D1+=⋯=Dn+.



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