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.
1. Introduction
First we recall some basic definitions and notations that will be used in this paper. Let be an real matrix and be the eigenvalues of . Since is not symmetric in general, the eigenvalues may be complex numbers. Without loss of generality, we assume that , and then the spectral radius of is defined as ; 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) is simple if is a nonnegative irreducible matrix.
Let be a graph (digraph) with vertex set and edge set (arc set ). A graph (digraph ) is simple if it has no loops and multiple edges (arcs). For any pairs of vertices , if there is a (directed) path from to , the graph (digraph ) is called (strongly) connected. In this paper, we consider finite, simple graphs and digraphs.
Let be a graph and be the diagonal matrix of vertex degrees of , where is the degree of vertex .
Let be a digraph; and denote the in-neighbors and out-neighbors of , respectively. Let and denote the indegree and outdegree of the vertex in , respectively, and be the diagonal matrix of the vertex outdegrees of .
Let be the adjacency matrix of , where Let denote the adjacency matrix of , where is equal to the number of arcs .
Then the signless Laplacian matrix of () is defined as The spectral radii of and ( and ), denoted by and ( and ), are called the (adjacency) spectral radius of () and the signless Laplacian spectral radius of (), respectively.
Let be a connected graph and be a strong connected digraph. For , the distance from to , denoted by , is the length of the shortest (directed) path from to in (). For , the transmission of vertex in () is the sum of distances from to all other vertices of (), denoted by . The distance matrix of () is the matrix , where , where ). In fact, for , the transmission of vertex , , is just the th row sum of . For convenience, we also call the distance degree (outdegree) of vertex in (), denoted by (); that is, . Similarly, we define .
Let be the diagonal matrix of vertex transmissions of , and let be the diagonal matrix of vertex transmissions of . The distance signless Laplacian matrix of is the matrix defined by Aouchiche and Hansen as [1] The spectral radii of and and ), denoted by and ( and ), are called the distance spectral radius of () and the distance signless Laplacian spectral radius of (), respectively.
Let be a connected graph. The reciprocal distance matrix (also called the Harary matrix) of is the matrix, where if and for . Clearly, the reciprocal distance matrix is nonnegative and symmetric.
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 [2–4].
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 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.
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]). If is an nonnegative matrix with the spectral radius and row sums , then . Moreover, if is irreducible, then one of the equalities holds if and only if the row sums of are all equal.
Theorem 2. Let be an nonnegative matrix with row sums , where , and let be the smallest diagonal element, be the smallest nondiagonal element, and be the spectral radius of . Take and for , Let for some . Then . Moreover, if is irreducible, then (1) if and only if . (2) with if and only if satisfies the following conditions: (i) for ; (ii) for ; (iii).
Proof. If , then for any by . Thus by Lemma 1 and , we have , and if is irreducible, if and only if .
Now we consider the case .
Firstly, we show for all .
Since is a nonnegative matrix, then for . ThusLetIt is easy to show that . Take and let be a diagonal matrix of order . Let , and then and have the same eigenvalues, and .
Now we consider the row sums of , say, .
Case 1 . Considerwith equality if and only if (a) and (b) hold: (a) and if with and (b) .
Case 2 . Considerwith equality if and only if (c) and (d) hold: (c) if and (d) .
Noting thatthen, by Lemma 1, we have .
Noting that by , thus , where for some .
Let be irreducible; for some .
Case 1 . For , by and , we have . ThenOn the other hand, by Lemma 1 and , we haveBy (11), (12), and , (1) holds.
Case 2 for some . Then and by .
If , then by the above arguments and Lemma 1; thus (a) and (b) hold for and (c) and (d) hold for . Thus for and for . Now (i), (ii), and (iii) follow.
Conversely, if (i), (ii), and (iii) hold, it is easy to show that equality holds.
Corollary 3. Let be an nonnegative matrix with row sums , where , and let be the smallest diagonal element, be the smallest nondiagonal element, and be the spectral radius of . Take and, for , Let for some . Then Moreover, if is irreducible with or is irreducible and symmetric, then
Proof. We complete the proof by the following two cases.
Case 1 . It is obvious by the proof of Theorem 2.
Case 2 . By (i) and (ii), is symmetric and is the smallest nondiagonal element. We have . It is a contradiction by the fact .
Similar to the proof of Theorem 2 (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. Let be an nonnegative matrix with row sums , where , and let be the largest diagonal element, be the largest nondiagonal element, and be the spectral radius of . Take and, for ,Let for some . Then . Moreover, if is irreducible, then (1) if and only if . (2) with if and only if satisfies the following conditions: (i) for ; (ii) for ; (iii).
3. Various Spectral Radii of a Graph
Let be a graph. In Section 1, the (adjacency) matrix , the signless Laplacian matrix , the distance matrix (if is connected), the distance signless Laplacian matrix (if is connected), the reciprocal distance matrix (if is connected), the (adjacency) 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 are defined. Now, in this section, we will apply Theorem 2, Corollary 3, and Theorem 4 to , , , , and and obtain some new results or known results.
3.1. Adjacency Spectral Radius of a Graph
Let be a graph. By applying Corollary 3 and Theorem 4 to the (adjacency) matrix with , , , , and for any , we have the following.
Corollary 5. Let be a graph on vertices with degree sequence , where . 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 such that is a bidegreed graph with .
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 , , , , and for any , we have the following.
Corollary 7. Let be a graph on vertices with degree sequence , where . 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 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].
3.3. Distance Spectral Radius of a Graph
Let be a connected graph and be the diameter of . Then the distance matrix is nonnegative and symmetric. By applying Corollary 3 and Theorem 4 to the distance matrix with , , , , and for any , we note that 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 such that . LetThen 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].
By applying Theorem 2 and Corollary 3 to the distance matrix with , , and for , we have the following.
Corollary 11 (see [16, Theorem 2]). Let be a connected graph on vertices with distance degree sequence such that for some . Then
3.4. Distance Signless Laplacian Spectral Radius of a Graph
Let be a connected graph and be the diameter of . Then the distance matrix is nonnegative and symmetric. By applying Corollary 3 and Theorem 4 to the distance matrix with , , , , and for , we note that implies a contradiction. Then we have the following.
Corollary 12. Let be a connected graph on vertices with distance degree sequence such that and be the diameter of . Let Then one has Moreover, one of the equalities holds if and only if .
Remark 13. The right inequality in Corollary 12 is the result of Theorem 3.8 in [9].
By applying Theorem 2 and Corollary 3 to the distance matrix with , , and for , we have the following.
Corollary 14. Let be a connected graph on vertices with distance degree sequence such that for some . Then
3.5. Spectral Radius of the Reciprocal Distance Matrix
By applying Corollary 3 and Theorem 4 to the reciprocal distance matrix with , , , , and for , we have the following.
Corollary 15. Let be a connected graph on vertices, be the diameter of , , and the row sum sequence be of satisfying . LetThen Moreover, the left equality holds if and only if , and the right equality holds if and only if either or there exists some with such that is a graph with vertices of degree and the remaining vertices have equal degree less than .
Remark 16. The right inequality in Corollary 15 is the result (i) of Theorem 4 in [16].
4. Various Spectral Radii of a Digraph
Let be a strong connected digraph. In Section 1, the adjacency matrix , the signless Laplacian matrix , the distance matrix (if is connected), the distance signless Laplacian matrix (if is connected), the adjacency spectral radius , the signless Laplacian spectral radius , the distance spectral radius , and the distance signless Laplacian spectral radius are defined. Now, in this section, we will apply Theorem 2, Corollary 3, and Theorem 4 to , , , and , 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
Let be a digraph. By applying Corollary 3 and Theorem 4 to the (adjacency) matrix with , , , and for , we have the following.
Corollary 17. Let be a digraph on vertices with outdegree sequence such that . Then one has Moreover, if is a strong connected digraph, then the left equality holds if and only if is a regular digraph, the right equality holds if and only if is a regular digraph, or there exists some with such that is a bidegreed digraph with and the indegrees .
4.2. Signless Laplacian Spectral Radius of a Digraph
Let be a digraph. By applying Corollary 3 and Theorem 4 to the signless Laplacian matrix with , , and for , we have the following.
Corollary 18. Let be a digraph on vertices with outdegree sequence such that . Then one has Moreover, if is a strong connected digraph, then the left equality holds if and only if is a regular digraph, the right equality holds if and only if is a regular digraph, or there exists some with such that is a bidegreed digraph with and the indegrees .
Remark 19. The left inequality in Corollary 18 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]). Let be a strong connected digraph on vertices with outdegree sequence such that . Then one has Moreover, if , the equality holds if and only if is a regular digraph. If , the equality holds if and only if is a regular digraph or a bidegreed digraph in which and .
The following example shows that the result of Proposition 20 is incorrect.
Example 21. Let and is shown in Figure 1. For , the outdegree sequence is and the indegree . We have by direct computation. It is clear that
4.3. Distance Spectral Radius of a Digraph
Let be a strong connected digraph and be the diameter of . By applying Theorems 2 and 4 to the distance matrix with , , and for , we note that implies a contradiction. Then we have the following.
Corollary 22. Let be a strong connected digraph on vertices with distance outdegree sequence such that , and let be the diameter of . LetThen one has Moreover, the left equality holds if and only if or there exists some with such that and and the right equality holds if and only if .
4.4. Distance Signless Laplacian Spectral Radius of a Digraph
Let be a strong connected digraph and be the diameter of . By applying Theorems 2 and 4 to the distance signless Laplacian matrix with , , and for , we note two facts: the first fact is that (i) and (iii) of (2) in Theorem 2 cannot hold at the same time by and , and the second fact is that implies a contradiction. Then we have the following.
Corollary 23. Let be a strong connected digraph on vertices with distance outdegree sequence such that , and let be the diameter of . LetThen one hasMoreover, one of the equalities holds if and only if .
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Lihua You’s research is supported by the National Natural Science Foundation of China (Grant no. 11571123) and the Guangdong Provincial Natural Science Foundation (Grant no. 2015A030313377); Danping Huang’s research is supported by the Scientific Research Foundation of Graduate School of South China Normal University (Grant no. 2015lkxm19). The authors would like to thank Yafei chen for the valuable comments, corrections, and suggestions, which lead to an improvement of the original paper.