Abstract
Let be a simple and undirected graph. The eigenvalues of the adjacency matrix of are called the eigenvalues of . In this paper, we characterize all the -vertex graphs with some eigenvalue of multiplicity and , respectively. Moreover, as an application of the main result, we present a family of nonregular graphs with four distinct eigenvalues.
1. Introduction
All graphs here considered are simple, undirected, and connected. Let be a graph with vertex set . The set of all the neighbors of a vertex is denoted by . Two adjacent vertices and are denoted by . The adjacency matrix of is a real symmetric matrix, and if there is an edge joining the vertices and ; otherwise, . The eigenvalues of are called the eigenvalues of , denoted by . The rank of the adjacency matrix of is called the rank of , written as . The rank of a matrix is also written as . An independent set of is a subset of such that there is no edge between any two vertices. The number of vertices in a maximum independent set of is called the independent number of , denoted by . The distance between two vertices and , denoted by , is the length of a shortest path between and . Denote by the diameter of , then . Let be the multiplicity of an eigenvalue of a graph .
The multiplicity of an eigenvalue of a graph has attracted much attention. Rowlinson gives an extensive investigation in this topic [1–8]. Let be a graph of order with an eigenvalue . In [1], the author proved that if and , then with . This upper bound was extended to with (or equivalently, ) in [2]. The graphs satisfying were discussed in [5]. In [3, 4, 6–8], the authors studied the multiplicity of an eigenvalue of a graph in some special graph classes. Moreover, Fonseca [9] proved many relations between the multiplicities of an eigenvalue whenever a path is removed from the graph. Bu et al. [10] gave two upper bounds on eigenvalue multiplicity of unicyclic graphs and trees. Wong et al. [11] improved an upper bound on the multiplicity of a positive eigenvalue of a tree in [3].
Notice that the upper bounds in [1, 2] are established for the multiplicity of the eigenvalue not equal to 0 or -1. In other words, the multiplicities of the eigenvalues 0 and -1 cannot be bounded easily. Then, it is interesting to study the multiplicities of the eigenvalues 0 and -1 of graphs. Here, we are interested in searching the graphs with the eigenvalue -1 or 0 of large multiplicity. It is well known that the multiplicity of the eigenvalue 0 is called the nullity of a graph, which has been studied intensively. Hence, attention may be paid to the graphs with the eigenvalue -1 of large multiplicity. More generally, in this paper, we investigate the graphs with some eigenvalue of large multiplicity because they are related to the graphs with few distinct eigenvalues, which have been investigated intensively (see [12–18], for example).
Denote the set of all -vertex connected graphs with some eigenvalue of multiplicity by . The following are the main conclusions of this paper.
Theorem 1. Let be a graph of order , then if and only if is the complete bipartite graph with .
Theorem 2. Let be a graph of order , then if and only if is the complete tripartite graph with or the graph (see Figure 1) with and .
2. Proofs
Before showing the proofs of Theorems 1 and 2, we first present some known results as lemmas.
Lemma 3 (interlacing theorem, [19]). For a real symmetric matrix of order , let be a principal submatrix of with order . Then, where is the th largest eigenvalue.
Let be a symmetric real matrix, whose block form is where the transpose of is . Let be the average row sum of , then is the quotient matrix of . If the row sum of is constant, then we say has an equitable partition.
Lemma 4 (see [19]). Let be a symmetric real matrix having an equitable partition and be the quotient matrix of . Then, each eigenvalue of is an eigenvalue of .
Lemma 5 (see [20, 21]). Let be a graph, then if and only if is a complete bipartite graph, and if and only if is a complete tripartite graph.
Lemma 6 (see [2]). Let be a graph of order and be an eigenvalue of multiplicity . If , then or equivalently, with .
Lemma 7. Let be a graph with vertices and induces a clique in such that , then −1 is an eigenvalue of with multiplicity at least .
Proof. Since induces a clique of and , then the first rows of the matrix are identical, where is the identity matrix. Thus contains as an eigenvalue of multiplicity at least , which indicates that is an eigenvalue of with multiplicity at least .
In the following, we present the proofs of Theorems 1 and 2.
2.1. Proof of Theorem 1
Let be a graph of order . If is the complete bipartite graph with , then it is easy to know that all the eigenvalues of are with multiplicities , respectively. Thus, .
Now suppose that . We will show that must be a complete bipartite graph. Let be the eigenvalue of with multiplicity . First, assume that , then the rank of is 2, and thus, is a complete bipartite graph from Lemma 5. Next, assume that (this case cannot happen from the following proof). Then, with as the identity matrix, which indicates that the independent number (otherwise, clearly, a contradiction). Moreover, we claim that is a cograph, i.e., contains no path as an induced subgraph. Otherwise, assume that contains as an induced subgraph, and then, (resp., ) is a principal submatrix of (resp., ). Thus, one can obtain that a contradiction. As a result, . If , is the complete graph whose eigenvalues are and with multiplicity 1 and , respectively. Obviously, . Suppose that and is an arbitrary connected subgraph with order 4 of in the following. For the eigenvalues of and of , it follows from Lemma 3 that
Since , we obtain that also contains as an eigenvalue of multiplicity at least 2. Recalling that , , and is a cograph, then must be isomorphic to one of the graphs (see Figure 2). However, by direct calculation, contains no nonzero eigenvalue of multiplicity at least 2 from Table 1, a contradiction.
Consequently, the proof is completed.
2.2. Proof of Theorem 2
Let be a graph of order . We first show the sufficiency part. If is the complete tripartite graph with , then from Lemma 5, it is clear that with eigenvalue 0 of multiplicity . Suppose that is the graph with and in Figure 1. From Lemma 7, contains −1 as an eigenvalue of multiplicity at least . According to the partition , the quotient matrix of is
By calculation, the determinant of the matrix is which implies that −1 is not an eigenvalue of the quotient matrix . Applying Lemma 4, we obtain that −1 is an eigenvalue of with multiplicity ; that is, .
We now prove the necessity part. Suppose that and is the eigenvalue of with multiplicity . First, if , then and is a complete tripartite graph with from Lemma 5. Next, suppose that , then . We claim that the independent number . Assume on the contrary that . If , is the complete graph and from the proof of Theorem 1. Suppose that with an independent set of , and let be the principal submatrix of indexed by . Then, contradicting with .
Now suppose that with a maximum independent set of , which yields that each vertex out of must be adjacent to at least one of . To complete the proof, the following claims are necessary.
Claim 1. The eigenvalue .
Recalling that , further, if , then from Lemma 6 and ,
contradicting with . Hence, .
Claim 2. There exists no vertex adjacent to exactly two of .
Without loss of generality, suppose for a contradiction that there exists a vertex such that , and . Let be the principal submatrix of indexed by , then is a principal submatrix of and
Denote by the row of indexed by the vertex . Since , it is clear that are linearly independent, which yields that any other rows of can be written as a linear combination of . Let
Applying (11) to the first, second, and fourth columns of , we get which yields that , contradicting with Claim 1.
Claim 3. There exists no vertex adjacent to each of .
Suppose for a contradiction that there exists a vertex such that . Analogous with the proof of Claim 2, let be the principal submatrix of indexed by , then
As , then clearly are linearly independent, which span the row space of . Let
Applying (14) to the columns of , we get which implies that , contradicting with Claim 1. Combining the above claims, we see that if , then is not connected, a contradiction. As a result, . Recalling the discussions before, it can be proved that .
In what follows, we prove that contains no induced path , i.e., is a cograph. If contains as an induced subgraph, then by considering an induced subgraph of order 5 of , we see that must contain some (see Figure 3) as an induced subgraph (noting that ). Applying Lemma 3 and Claim 1, we obtain that contains as an eigenvalue of multiplicity at least 2. However, by direct calculation, it follows that the multiplicity of −1 as an eigenvalue of is not more than one (see Table 2), a contradiction. Therefore, is a cograph and the diameter .
Now we are in a position to complete the proof. Note that and from the above process. Let be a diameter of , then is a maximum independent set of and each vertex out of is adjacent to at least one of . Let then any vertex out of belongs precisely to one of . The following claims are needed for us.
Claim 4. Each vertex of (resp., ) is adjacent to each one of .
Suppose and such that . Then, the vertices induce a path , a contradiction. The proof for the case of is parallel, omitted.
Claim 5. All the vertices of (resp., ) induce a clique of .
We only prove the case of . If and , then induce an independent set of , contradicting with .
Claim 6. All the vertices of induce a clique of .
Assume that and . Considering an induced subgraph of order 5 of , we obtain that is isomorphic to one of (see Figure 4). It follows from Lemma 3 that contains as an eigenvalue of multiplicity at least 2. But contains no eigenvalue of multiplicity 2 from Table 3, a contradiction.
From Claims 4–6 and the facts and , we derive that is isomorphic to the graph in Figure 1, as required. The proof is completed.
van Dam [14] and Huang and Huang [18] investigated the regular graphs with four distinct eigenvalues. Here, as an application of Theorem 2, we obtain a family of nonregular graphs with four distinct eigenvalues.
Corollary 8. The graph with and (see Figure 1) contains four distinct eigenvalues, which is not a regular graph.
Proof. From the proof of Theorem 2, we see that −1 is an eigenvalue of with multiplicity and the remaining three eigenvalues of are those of the quotient matrix of . Since , then contains two positive eigenvalues and one negative eigenvalue. By the Perron-Frobenius theorem, the largest eigenvalue of is simple; then, the largest eigenvalue of (resp., ) is simple. Thus, contains three distinct eigenvalues, that is, contains three distinct eigenvalues. Recalling that −1 is not an eigenvalue of , then contains four distinct eigenvalues. Moreover, it is clear that is not a regular graph.
Data Availability
In this study, we use the theoretical model method to carry out our research. Our conclusions are obtained primarily by using theoretical deduction and numerical study. Of these, numerical study data are derived from the author’s assumptions, also illustrated in Tables 1–3. We thereby declare that no further external data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant numbers 11771443 and 71902105) and the Visiting Scholar and Teacher Development Project from the Education Department of Zhejiang Province (Grant number FX2018113).