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) [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].