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

## Abstract

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.

## 1 Introduction

Let $$G=(V,E)$$ be a graph with vertex set $$V(G)=\{v_{1}, \ldots, v_{n}\}$$ and edge set $$E(G)$$. Let $$N=\{1, \ldots, n\}$$, for $$i \in N$$. We assume that $$d_{i}$$ is the degree of vertex $$v_{i}$$. Let $$D(G) = \operatorname{diag}(d_{1}, d_{2}, \ldots, d_{n})$$ be the degree diagonal matrix of the graph G and $$A(G) = (a_{ij})$$ be the adjacency matrix of the graph G. Then the matrix $$Q(G) = D(G)+ A(G)$$ is called the signless Laplacian matrix of the graph G. The largest modulus of eigenvalues of $$Q(G)$$ is denoted by $$\rho(G)$$, which is also called the signless Laplacian spectral radius of G.

Let $$\overrightarrow{G}=(V,E)$$ be a digraph with vertex set $$V(\overrightarrow{G})=\{v_{1}, \ldots, v_{n}\}$$ and arc set $$E(\overrightarrow{G})$$. Let $$d_{i}^{+}$$ be the out-degree of vertex $$v_{i}$$, $$D(\overrightarrow{G}) = \operatorname{diag}(d_{1}^{+}, d_{2}^{+}, \ldots, d_{n}^{+})$$ be the out-degree diagonal matrix of the digraph $$\overrightarrow{G}$$, and $$A(\overrightarrow{G}) = (a_{ij})$$ be the adjacency matrix of the digraph $$\overrightarrow{G}$$. Then the matrix $$Q(\overrightarrow{G}) = D(\overrightarrow{G})+ A(\overrightarrow{G})$$ is called the signless Laplacian matrix of the digraph $$\overrightarrow{G}$$. The largest modulus of eigenvalues of $$Q(\overrightarrow{G})$$ is denoted by $$\rho (\overrightarrow{G})$$, which is also called the signless Laplacian spectral radius of $$\overrightarrow{G}$$.

In recent decades, there are many bounds on the signless Laplacian spectral radius of a graph (digraph) [1â€“3]. Let $$m_{i} = \frac{{\sum_{i\sim j} {d_{j} } }}{{d_{i} }}$$ be the average degree of the neighbours of $$v_{i}$$ in G and $$m_{i}^{+} = \frac{{\sum_{i\sim j} {d_{j}^{+} } }}{{d_{i}^{+} }}$$ be the average out-degree of the out-neighbours of $$v_{i}$$ in $$\overrightarrow{G}$$. In this paper, we assume that the graph (digraph) is simple and connected (strong connected).

In 2013, Maden, Das, and Cevik [4] obtained the following bounds for the signless Laplacian spectral radius of a graph:

$$\rho(G) \leq\max_{i\sim j} \biggl\{ \frac{d_{i}+ 2d_{j} -1+ \sqrt {(d_{i}-2d_{j}+1)^{2}+ 4d_{i}}}{2} \biggr\} .$$
(1)

In 2016, Xi and Wang [5] obtained the following bounds for the signless Laplacian spectral radius of a digraph:

$$\rho(\overrightarrow{G}) \leq\max_{i\sim j} \biggl\{ \frac{d_{i}^{+}+ 2d_{j}^{+} -1+ \sqrt{(d_{i}^{+}-2d_{j}^{+}+1)^{2}+ 4d_{i}^{+}}}{2} \biggr\} .$$
(2)

In this paper, we improve the bounds for the signless Laplacian spectral radius of a graph (digraph) that are given in (1) and (2).

## 2 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.

### Lemma 2.1

([6])

Let A be a nonnegative matrix with the spectral radius $$\rho(A)$$ and the row sum $$r_{1}, r_{2}, \ldots, r_{n}$$. Then $$\mathop{\min} _{1 \le i \le n} r_{i} \le\rho(A) \le\mathop{\max} _{1 \le i \le n} r_{i}$$. Moreover, if the matrix A is irreducible, then the equalities hold if and only if

$$r_{1}=r_{2}= \cdots=r_{n}.$$

### Theorem 2.1

Let $$A=(a_{ij})$$ be an irreducible and nonnegative matrix with $$a_{ii} = 0$$ for all $$i \in N$$ and the row sum $$r_{1}, r_{2}, \ldots, r_{n}$$. Let $$B = A + M$$, where $$M = \operatorname{diag}(t_{1}, t_{2}, \ldots, t_{n})$$ with $$t_{i} \geq0$$ for any $$i \in N$$, $$s_{i} = \sum_{j = 1}^{n} {a_{ij} r_{j} }$$, $$s_{ij} = s_{i}-a_{ij}r_{j}$$. Let $$\rho(B)$$ be the spectral radius of B and let

$$f(i,j) = \frac{t_{i}+t_{j}+\frac{s_{ij}}{r_{i}}+\sqrt{ (t_{i}-t_{j}+\frac {s_{ij}}{r_{i}} )^{2}+\frac{4s_{j}a_{ij}}{r_{i}}}}{2},$$

for any $$i,j \in N$$. Then

$$\mathop{\min} _{1 \le i \le n} \mathop{\max_{1 \le j \le n }}_{ j \ne i } \bigl\{ f(i,j),a_{ij}\neq0\bigr\} \le\rho(B) \le\mathop{\max} _{1 \le i \le n} \mathop{\min _{1 \le j \le n }}_{ j \ne i } \bigl\{ f(i,j),a_{ij} \neq0\bigr\} .$$
(3)

Moreover, either of the equalities in (3) holds if and only if $$t_{i}+\frac{s_{i}}{r_{i}}= t_{j}+\frac{s_{j}}{r_{j}}$$ for any distinct $$i,j \in N$$.

### Proof

Let $$R = \operatorname{diag}(r_{1}, r_{2}, \ldots, r_{n})$$. Since the matrix A is nonnegative irreducible, the matrix $$R^{-1}BR$$ is also nonnegative and irreducible. By the famous Perron-Frobenius theorem [6], there is a positive eigenvector $$x =(x_{1}, x_{2}, \ldots, x_{n})^{T}$$ corresponding to the spectral radius of $$R^{-1}BR$$.

Upper bounds: Let $$x_{p}>0$$ be an arbitrary component of x, $$x_{q}=\max\{ x_{k}, 1\leq k \leq n\}$$. Obviously, $$p\neq q$$, $$a_{pq}\neq0$$. By $$R^{-1}BRx = \rho(B)x$$, we have

$$\rho(B)x_{p}=t_{p}x_{p}+ \sum _{k = 1,k \ne p}^{n} {\frac{{a_{pk} r_{k} x_{k} }}{{r_{p} }}}\leq t_{p}x_{p} + \frac{x_{q}}{r_{p}}\sum _{k = 1}^{n} a_{pk} r_{k}\leq t_{p}x_{p} + \frac{x_{q}s_{p}}{r_{p}}.$$
(4)

Similarly, we have

$$\rho(B)x_{q}=t_{q}x_{q}+ \sum _{k = 1,k \ne q}^{n} {\frac{{a_{qk} r_{k} x_{k} }}{{r_{q} }}}\leq \biggl(t_{q} + \frac{s_{q}-a_{qp}r_{p}}{r_{q}} \biggr)x_{q} + \frac{a_{qp}r_{p}}{r_{q}}x_{p}.$$
(5)

By (4), (5), and $$\rho(B) - t_{p} > 0$$, $$\rho(B) - t_{q} > 0$$, we have

$$\bigl(\rho(B)-t_{p}\bigr) \biggl(\rho(B)- t_{q} - \frac{s_{q}-a_{qp}r_{p}}{r_{q}} \biggr)\leq\frac{s_{p}a_{qp}}{r_{q}}.$$

Therefore,

$$\rho(B)\leq\frac{t_{p}+t_{q} + \frac{s_{qp}}{r_{q}}+\sqrt{ (t_{p}-t_{q} - \frac{s_{qp}}{r_{q}} )^{2}+\frac{4s_{p}a_{qp}}{r_{q}}}}{2}.$$
(6)

This must be true for every $$p\neq q$$. Then

$$\rho(B)\leq\mathop{\min} _{j \ne q} \frac{t_{j}+t_{q} + \frac {s_{qj}}{r_{q}}+\sqrt{ (t_{j}-t_{q} - \frac{s_{qj}}{r_{q}} )^{2}+\frac {4s_{j}a_{qj}}{r_{q}}}}{2}.$$
(7)

This must be true for any $$q\in N$$. Then

$$\rho(B)\leq\mathop{\max} _{1 \leq i \leq n} \mathop{\min} _{j \ne i} \biggl\{ \frac{t_{i}+t_{j}+\frac{s_{ij}}{r_{i}}+\sqrt{ (t_{i}-t_{j}+\frac{s_{ij}}{r_{i}} )^{2}+\frac{4s_{j}a_{ij}}{r_{i}}}}{2}, a_{ij}\neq 0 \biggr\} .$$
(8)

Lower bounds: Let $$x_{p}>0$$ be an arbitrary component of x, $$x_{q}=\min\{ x_{k}, 1\leq k \leq n\}$$. Obviously, $$p\neq q$$, $$a_{pq}\neq0$$. By $$R^{-1}BRx = \rho(B)x$$, we have

$$\rho(B)x_{p}=t_{p}x_{p}+ \sum _{k = 1,k \ne p}^{n} {\frac{{a_{pk} r_{k} x_{k} }}{{r_{p} }}}\geq t_{p}x_{p} + \frac{x_{q}}{r_{p}}\sum _{k = 1}^{n} a_{pk} r_{k}\geq t_{p}x_{p} + \frac{x_{q}s_{p}}{r_{p}}.$$
(9)

Similarly, we have

$$\rho(B)x_{q}=t_{q}x_{q}+ \sum _{k = 1,k \ne q}^{n} {\frac{{a_{qk} r_{k} x_{k} }}{{r_{q} }}}\geq \biggl(t_{q} + \frac{s_{q}-a_{qp}r_{p}}{r_{q}} \biggr)x_{q} + \frac{a_{qp}r_{p}}{r_{q}}x_{p}.$$
(10)

By (9), (10), and $$\rho(B) - t_{p} > 0$$, $$\rho(B) - t_{q} > 0$$, we have

$$\bigl(\rho(B)-t_{p}\bigr) \biggl(\rho(B)- t_{q} - \frac{s_{q}-a_{qp}r_{p}}{r_{q}} \biggr)\geq\frac{s_{p}a_{qp}}{r_{q}}.$$
(11)

Therefore,

$$\rho(B)\geq\frac{t_{p}+t_{q} + \frac{s_{qp}}{r_{q}}+\sqrt{ (t_{p}-t_{q}-\frac{s_{qp}}{r_{q}} )^{2}+\frac{4s_{p}a_{qp}}{r_{q}}}}{2}.$$
(12)

This must be true for every $$p\neq q$$. Then

$$\rho(B)\geq\mathop{\max} _{j \ne q} \frac{t_{j}+t_{q} + \frac {s_{qj}}{r_{q}}+\sqrt{ (t_{j}-t_{q}-\frac{s_{qj}}{r_{q}} )^{2}+\frac {4s_{j}a_{qj}}{r_{q}}}}{2}.$$
(13)

This must be true for all $$q\in N$$. Then

$$\rho(B)\geq\mathop{\min} _{1 \leq i \leq n} \mathop{\max} _{j \ne i} \biggl\{ \frac{t_{i}+t_{j}+\frac{s_{ij}}{r_{i}}+\sqrt{ (t_{i}-t_{j}+\frac{s_{ij}}{r_{i}} )^{2}+\frac{4s_{j}a_{ij}}{r_{i}}}}{2}, a_{ij}\neq 0 \biggr\} .$$
(14)

From (4), (5), and $$x_{p}>0$$ 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 (8) for $$t_{i}+\frac{s_{i}}{r_{i}}= t_{j}+\frac{s_{j}}{r_{j}}$$ for any distinct $$i,j \in N$$. The proof of the left equality in (3) is similar to the proof of the right equality, and we omit it here.

Thus, we complete the proof.â€ƒâ–¡

## 3 Signless Laplacian spectral radius of a graph

In this section, we will apply TheoremÂ 2.1 to obtain some new results on the signless Laplacian spectral radius $$\rho(G)$$ of a graph.

### Theorem 3.1

Let $$G = (V, E)$$ be a simple connected graph on n vertices. Then

\begin{aligned} & \mathop{\min} _{1 \le i \le n} \mathop{\max} _{ i \sim j } \biggl\{ \frac{d_{i}+ 2d_{j} -1+ \sqrt {(d_{i}-2d_{j}+1)^{2}+ 4d_{i}}}{2} \biggr\} \\ &\quad\leq\rho(G) \leq\mathop{\max} _{1 \le i \le n} \mathop{\min} _{ i \sim j } \biggl\{ \frac{d_{i}+ 2d_{j} -1+ \sqrt{(d_{i}-2d_{j}+1)^{2}+ 4d_{i}}}{2} \biggr\} . \end{aligned}
(15)

Moreover, one of the equalities in (15) holds if and only if G is a regular graph.

### Proof

We apply TheoremÂ 2.1 to $$Q(G)$$. Let $$t_{i}=0$$ for any $$i \in N$$. Then $$f(i,j)= \frac{d_{i}+ 2d_{j} -1+ \sqrt {(d_{i}-2d_{j}+1)^{2}+ 4d_{i}}}{2}$$. Thus (15) holds.

And the equality holds in (15) for regular graphs if and only if G is a regular graph.â€ƒâ–¡

### Remark 3.1

Obviously, we have

\begin{aligned} &\mathop{\max} _{1 \le i \le n} \mathop{\min} _{ i \sim j } \biggl\{ \frac{d_{i}+ 2d_{j} -1+ \sqrt{(d_{i}-2d_{j}+1)^{2}+ 4d_{i}}}{2} \biggr\} \\ &\quad \leq\mathop{\max} _{ i \sim j} \biggl\{ \frac{d_{i}+ 2d_{j} -1+ \sqrt{(d_{i}-2d_{j}+1)^{2}+ 4d_{i}}}{2} \biggr\} . \end{aligned}

That is to say, our upper bound in TheoremÂ 3.1 is always better than the upper bound (1) in [4].

### Theorem 3.2

Let $$G = (V, E)$$ be a simple connected graph on n vertices. Then

$$\rho(G) \geq\mathop{\min} _{1 \le i \le n} \mathop{\max} _{ i \sim j } \biggl\{ {\frac{{d_{i} + d_{j} + m_{j} - {{d_{i} }/ {d_{j} + \sqrt{ ( {d_{i} - d_{j} - m_{j} + {{d_{i} } / {d_{j} }}} ) + 4d_{i} } }}}}{2}} \biggr\}$$
(16)

and

$$\rho(G) \leq\mathop{\max} _{1 \le i \le n} \mathop{\min} _{ i \sim j } \biggl\{ {\frac{{d_{i} + d_{j} + m_{j} - {{d_{i} }/ {d_{j} + \sqrt{ ( {d_{i} - d_{j} - m_{j} + {{d_{i} }/ {d_{j} }}} ) + 4d_{i} } }}}}{2}} \biggr\} .$$
(17)

Moreover, one of the equalities in (16), (17) holds if and only if G is a regular graph or a bipartite semi-regular graph.

### Proof

We apply TheoremÂ 2.1 to $$Q(G)$$. Let $$t_{i}=d_{i}$$, $$s_{i} =\sum_{j = 1}^{n} {a_{ij} r_{j} } = d_{i}m_{i}$$ for any $$1 \leq i \leq n$$. Then $$f(i,j)= {\frac{{d_{i} + d_{j} + m_{j} - {{d_{i} }/ {d_{j} + \sqrt{ ( {d_{i} - d_{j} - m_{j} + {{d_{i} }/ {d_{j} }}} ) + 4d_{i} } }}}}{2}}$$. Thus (16), (17) hold.

And the equality holds if and only if G is a regular graph or a bipartite semi-regular graph.â€ƒâ–¡

## 4 Signless Laplacian spectral radius of a digraph

In this section, we will apply TheoremÂ 2.1 to obtain some new results on the signless Laplacian spectral radius $$\rho(\overrightarrow{G})$$ of a digraph.

### Theorem 4.1

Let $$\overrightarrow{G} = (V, E)$$ be a strong connected digraph on n vertices. Then

\begin{aligned} &\mathop{\min} _{1 \le i \le n} \mathop{\max} _{ i \sim j} \biggl\{ \frac{d_{i}^{+}+ 2d_{j}^{+} -1+ \sqrt {(d_{i}^{+}-2d_{j}^{+}+1)^{2}+ 4d_{i}^{+}}}{2} \biggr\} \\ &\quad \leq\rho(\overrightarrow{G}) \leq\mathop{\max} _{1 \le i \le n} \mathop{\min} _{ i \sim j} \biggl\{ \frac{d_{i}^{+}+ 2d_{j}^{+} -1+ \sqrt{(d_{i}^{+}-2d_{j}^{+}+1)^{2}+ 4d_{i}^{+}}}{2} \biggr\} . \end{aligned}
(18)

Moreover, one of the equalities in (18) holds if and only if $$\overrightarrow{G}$$ is a regular digraph.

### Proof

We apply TheoremÂ 2.1 to $$Q(\overrightarrow{G})$$. Let $$t_{i}=0$$ for any $$1 \leq i \leq n$$. Then $$f(i,j)=\frac{d_{i}^{+}+ 2d_{j}^{+} -1+ \sqrt{(d_{i}^{+}-2d_{j}^{+}+1)^{2}+ 4d_{i}^{+}}}{2}$$. Then the inequality (18) holds.

And the equality holds in (18) if and only if $$\overrightarrow{G}$$ is a regular digraph.â€ƒâ–¡

### Remark 4.1

Obviously, we have

\begin{aligned} &\mathop{\max} _{1 \le i \le n} \mathop{\min} _{ i \sim j} \biggl\{ \frac{d_{i}^{+}+ 2d_{j}^{+} -1+ \sqrt{(d_{i}^{+}-2d_{j}^{+}+1)^{2}+ 4d_{i}^{+}}}{2} \biggr\} \\ &\quad \leq\mathop{\max} _{ i \sim j} \biggl\{ \frac{d_{i}^{+}+ 2d_{j}^{+} -1+ \sqrt{(d_{i}^{+}-2d_{j}^{+}+1)^{2}+ 4d_{i}^{+}}}{2} \biggr\} . \end{aligned}

That is to say, our upper bound in TheoremÂ 4.1 is always better than the upper bound (2) in [5].

### Theorem 4.2

Let $$\overrightarrow{G} = (V, E)$$ be a strong connected digraph on n vertices. Then

$$\rho(\overrightarrow{G}) \geq\mathop{\min} _{1 \le i \le n} \mathop{\max} _{ i \sim j} \biggl\{ {\frac{{d_{i}^{+} + d_{j}^{+} + m_{j}^{+} - {{d_{i}^{+} }/ {d_{j}^{+} + \sqrt{ ( {d_{i}^{+} - d_{j}^{+} - m_{j}^{+} + {{d_{i}^{+} }/ {d_{j}^{+} }}} ) + 4d_{i}^{+} } }}}}{2}} \biggr\}$$
(19)

and

$$\rho(\overrightarrow{G}) \leq\mathop{\max} _{1 \le i \le n} \mathop {\min} _{ i \sim j} \biggl\{ {\frac{{d_{i}^{+} + d_{j}^{+} + m_{j}^{+} - {{d_{i}^{+} }/ {d_{j}^{+} + \sqrt{ ( {d_{i}^{+} - d_{j}^{+} - m_{j}^{+} + {{d_{i}^{+} } / {d_{j}^{+} }}} ) + 4d_{i}^{+} } }}}}{2}} \biggr\} .$$
(20)

Moreover, one of the equalities in (19), (20) holds if and only if $$\overrightarrow{G}$$ is a regular digraph or a bipartite semi-regular digraph.

### Proof

We apply TheoremÂ 2.1 to $$Q(\overrightarrow{G})$$. Let $$t_{i}=d_{i}^{+}$$, $$s_{i} =\sum_{j = 1}^{n} {a_{ij} r_{j} } = d_{i}^{+}m_{i}^{+}$$ for any $$1 \leq i \leq n$$. Then $$f(i,j)={\frac{{d_{i}^{+} + d_{j}^{+} + m_{j}^{+} - {{d_{i}^{+} }/ {d_{j}^{+} + \sqrt{ ( {d_{i}^{+} - d_{j}^{+} - m_{j}^{+} + {{d_{i}^{+} }/ {d_{j}^{+} }}} ) + 4d_{i}^{+} } }}}}{2}}$$. Thus (19), (20) hold.

One sees easily that the equality holds if and only if $$\overrightarrow {G}$$ is a regular digraph or a bipartite semi-regular digraph.â€ƒâ–¡

## 5 Conclusion

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 which are better than the bounds in [4, 5].

## References

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3. Cui, SY, Tian, GX, Guo, JJ: A sharp upper bound on the signless Laplacian spectral radius of graphs. Linear Algebra Appl. 439, 2442-2447 (2013)

4. Maden, AD, Das, KC, Cevik, AS: Sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph. Appl. Math. Comput. 219, 5025-5032 (2013)

5. Xi, W, Wang, L: Sharp upper bounds on the signless Laplacian spectral radius of strongly connected digraphs. Discuss. Math., Graph Theory 36, 977-988 (2016)

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## Acknowledgements

Jun He is supported by the Science and Technology Foundation of Guizhou Province (Qian ke he Ji Chu [2016]1161); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2016]255); the Doctoral Scientific Research Foundation of Zunyi Normal College (BS[2015]09); High-level Innovative Talents of Guizhou Province (Zun Ke He Ren Cai[2017]8). Yan-Min Liu is supported by National Science Foundations of China (71461027); Science and Technology Talent Training Object of Guizhou Province outstanding youth (Qian ke he ren zi [2015]06); Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2014]295); 2013, 2014 and 2015 Zunyi 15851 Talents Elite Project funding; Zhunyi Innovative Talent Team (Zunyi KH(2015)38). Tian is supported by Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2015]451); Science and Technology Foundation of Guizhou Province (Qian ke he J zi [2015]2147). Xiang-Hu Liu is supported by the Guizhou Province Department of Education fund (KY[2015]391, [2016]046); Guizhou Province Department of Education Teaching Reform Project [2015]337; Guizhou Province Science and Technology fund (Qian Ke He Ji Chu[2016]1160).

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Correspondence to Jun He.

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He, J., Liu, YM., Tian, JK. et al. Some new sharp bounds for the spectral radius of a nonnegative matrix and its application. J Inequal Appl 2017, 260 (2017). https://doi.org/10.1186/s13660-017-1536-3