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) [1–3]. 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. □

3 Signless Laplacian spectral radius of a graph

In this section, we will apply Theorem 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 Theorem 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. □

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 Theorem 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 Theorem 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. □

4 Signless Laplacian spectral radius of a digraph

In this section, we will apply Theorem 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 Theorem 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. □

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 Theorem 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 Theorem 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. □

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].