Characterization of Graphs with an Eigenvalue of Large Multiplicity

Let G be a simple and undirected graph. The eigenvalues of the adjacency matrix of G are called the eigenvalues of G. In this paper, we characterize all the n-vertex graphs with some eigenvalue of multiplicity n−2 and n−3, respectively. Moreover, as an application of the main result, we present a family of nonregular graphs with four distinct eigenvalues.


Introduction
All graphs here considered are simple, undirected, and connected. Let G be a graph with vertex set VðGÞ = fv 1 , v 2 , ⋯, v n g. The set of all the neighbors of a vertex v i is denoted by N G ðv i Þ. Two adjacent vertices v i and v j are denoted by v i~v j . The adjacency matrix AðGÞ = ½a ij of G is a real symmetric matrix, and a ij = 1 if there is an edge joining the vertices v i and v j ; otherwise, a ij = 0. The eigenvalues of AðGÞ are called the eigenvalues of G, denoted by ρ 1 ≥ ρ 2 ≥⋯≥ρ n . The rank of the adjacency matrix AðGÞ of G is called the rank of G, written as rðGÞ. The rank of a matrix M is also written as rðMÞ. An independent set of G is a subset of VðGÞ such that there is no edge between any two vertices. The number of vertices in a maximum independent set of G is called the independent number of G, denoted by μðGÞ. The distance between two vertices v i and v j , denoted by d ðv i , v j Þ, is the length of a shortest path between v i and v j . Denote by diamðGÞ the diameter of G, then diamðGÞ = max fdðv i , v j Þ: v i , v j ∈ VðGÞg. Let mðρ i Þ be the multiplicity of an eigenvalue ρ i of a graph G.
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][13][14][15][16][17][18], for example).
Denote the set of all n-vertex connected graphs with some eigenvalue of multiplicity k by Gðn, kÞ. The following are the main conclusions of this paper. Theorem 1. Let G be a graph of order n > 3, then G ∈ G ðn, n − 2Þ if and only if G is the complete bipartite graph K s,t with s + t = n. Theorem 2. Let G be a graph of order n > 6, then G ∈ G ðn, n − 3Þ if and only if G is the complete tripartite graph K a,b,c with a + b + c = n or the graph Γ (see Figure 1) with s, t, p ≥ 1 and s + t + p = n.

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 A of order n, let M be a principal submatrix of A with order sð≤nÞ. Then, where λ i is the ith largest eigenvalue.
Let H be a symmetric real matrix, whose block form is where the transpose of H ij is H ji . Let q ij be the average row sum of H ij , then Q = ½q ij is the quotient matrix of H. If the row sum of H ij is constant, then we say H has an equitable partition.
Lemma 4 (see [19]). Let H be a symmetric real matrix having an equitable partition and Q be the quotient matrix of H. Then, each eigenvalue of Q is an eigenvalue of H.
Lemma 5 (see [20,21]). Let G be a graph, then rðGÞ = 2 if and only if G is a complete bipartite graph, and rðGÞ = 3 if and only if G is a complete tripartite graph.
Lemma 6 (see [2]). Let G be a graph of order n and ρ be an eigenvalue of multiplicity k. If ρ ∉ f0,−1g, then or Lemma 7. Let G be a graph with n vertices and then the first q rows of the matrix AðGÞ + I are identical, where I is the identity matrix. Thus AðGÞ + I contains 0 as an eigenvalue of multiplicity at least q − 1, which indicates that −1 is an eigenvalue of AðGÞ with multiplicity at least q − 1.
In the following, we present the proofs of Theorems 1 and 2.
2.1. Proof of Theorem 1. Let G be a graph of order n ≥ 4. If G is the complete bipartite graph K s,t with s + t = n, then it is easy to know that all the eigenvalues of K s,t are f ffiffiffi ffi st p ,− ffiffiffi ffi st p , 0g with multiplicities f1, 1, n − 2g, respectively. Thus, G = K s,t ∈ Gðn, n − 2Þ. Now suppose that G ∈ Gðn, n − 2Þ. We will show that G must be a complete bipartite graph. Let θ be the eigenvalue of G with multiplicity n − 2. First, assume that θ = 0, then the rank rðGÞ of G is 2, and thus, G is a complete bipartite graph from Lemma 5. Next, assume that θ ≠ 0 (this case cannot happen from the following proof). Then, rðAðGÞ − θ IÞ = 2 with I as the identity matrix, which indicates that the independent number μðGÞ ≤ 2 (otherwise, rðAðGÞ − θIÞ > 2 clearly, a contradiction). Moreover, we claim that G is a cograph, i.e., G contains no path P 4 as an induced subgraph. Otherwise, assume that G contains P 4 as an induced subgraph, and then, AðP 4 Þ (resp., AðP 4 Þ − θI) is a principal submatrix of AðGÞ (resp., AðGÞ − θI). Thus, one can obtain that a contradiction. As a result, diamðGÞ ≤ 2. If diamðGÞ = 1, G is the complete graph K n whose eigenvalues are n − 1 and −1 with multiplicity 1 and n − 1, respectively. Obviously, K n ∉ Gðn, n − 2Þ. Suppose that diamðGÞ = 2 and H is an arbitrary connected subgraph with order 4 of G in the following. For the eigenvalues ρ i of G and λ i of H, it follows from Lemma 3 that K s K t K p Figure 1: The graph Γ.

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Since G ∈ Gðn, n − 2Þ, we obtain that H also contains θð≠ 0Þ as an eigenvalue of multiplicity at least 2. Recalling that μðGÞ ≤ 2, diamðGÞ = 2, and G is a cograph, then H must be isomorphic to one of the graphs fH 1 , H 2 , H 3 g (see Figure 2). However, by direct calculation, H i ð1 ≤ i ≤ 3Þ contains no nonzero eigenvalue of multiplicity at least 2 from Table 1, a contradiction.
Consequently, the proof is completed.

Proof of Theorem 2.
Let G be a graph of order n > 6. We first show the sufficiency part. If G is the complete tripartite graph K a,b,c with a + b + c = n, then from Lemma 5, it is clear that K a,b,c ∈ Gðn, n − 3Þ with eigenvalue 0 of multiplicity n − 3. Suppose that G is the graph Γ with s, t, p ≥ 1 and s + t + p = n in Figure 1. From Lemma 7, Γ contains −1 as an eigenvalue of multiplicity at least s + t + p − 3 = n − 3. According to the partition VðΓÞ = fVðK s Þ, VðK t Þ, VðK p Þg, the quotient matrix Q of AðΓÞ is By calculation, the determinant of the matrix Q + I is which implies that −1 is not an eigenvalue of the quotient matrix Q. Applying Lemma 4, we obtain that −1 is an eigenvalue of Γ with multiplicity n − 3; that is, Γ ∈ Gðn, n − 3Þ.
We now prove the necessity part. Suppose that G ∈ G ðn, n − 3Þ and θ is the eigenvalue of G with multiplicity n − 3. First, if θ = 0, then rðGÞ = 3 and G is a complete tripartite graph K a,b,c with a + b + c = n from Lemma 5. Next, suppose that θ ≠ 0, then rðAðGÞ − θIÞ = 3. We claim that the independent number μðGÞ = 2. Assume on the contrary that μðGÞ ≠ 2. If μðGÞ = 1, G is the complete graph K n and K n ∉ Gðn, n − 3Þ from the proof of Theorem 1. Suppose that μðGÞ ≥ 4 with fv 1 , v 2 , v 3 , v 4 g an independent set of G, and let M be the principal submatrix of AðGÞ indexed by fv 1 , v 2 , v 3 , v 4 g. Then, contradicting with rðAðGÞ − θIÞ = 3. Now suppose that μðGÞ = 3 with S = fv 1 , v 2 , v 3 g a maximum independent set of G, which yields that each vertex out of S must be adjacent to at least one of S. To complete the proof, the following claims are necessary. Recalling that θ ≠ 0, further, if θ ≠ −1, then from Lemma 6 and n > 6, contradicting with mðθÞ = n − 3. Hence, θ = −1.

Claim 2. There exists no vertex adjacent to exactly two of
Without loss of generality, suppose for a contradiction that there exists a vertex u such that u~v 1 , u~v 2 and u ≁ v 3 . Let M be the principal submatrix of AðGÞ indexed by fv 1 , v 2 , v 3 , ug, then M − θI is a principal submatrix of AðGÞ − θI and Denote by R v i the row of AðGÞ − θI indexed by the vertex v i . Since rðAðGÞ − θIÞ = 3, it is clear that fR v 1 , R v 2 , R v 3 g are linearly independent, which yields that any other rows of AðGÞ − θI can be written as a linear combination Applying (11) to the first, second, and fourth columns of M − θI, we get −aθ = 1, −bθ = 1, which yields that θ 2 = 2, contradicting with Claim 1.
Claim 3. There exists no vertex adjacent to each of fv 1 , v 2 , v 3 g.

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Suppose for a contradiction that there exists a vertex u such that u~v i ði = 1, 2, 3Þ. Analogous with the proof of Claim 2, let M be the principal submatrix of AðGÞ indexed by fv 1 , v 2 , v 3 , ug, then As rðAðGÞ − θIÞ = 3, then clearly fR v 1 , R v 2 , R v 3 g are linearly independent, which span the row space of AðGÞ − θI. Let Applying (14) to the columns of M − θI, we get −aθ = 1, which implies that θ 2 = 3, contradicting with Claim 1.
Combining the above claims, we see that if μðGÞ = 3, then G is not connected, a contradiction. As a result, μðGÞ ≠ 3.
Recalling the discussions before, it can be proved that μ ðGÞ = 2.
In what follows, we prove that G contains no induced path P 4 , i.e., G is a cograph. If G contains P 4 as an induced subgraph, then by considering an induced subgraph of order 5 of G, we see that G must contain some G i ð1 ≤ i ≤ 4Þ (see Figure 3) as an induced subgraph (noting that μðGÞ = 2). Applying Lemma 3 and Claim 1, we obtain that G i ð1 ≤ i ≤ 5Þ contains θ = −1 as an eigenvalue of multiplicity at least 2. However, by direct calculation, it follows that the multiplicity of −1 as an eigenvalue of G i ð1 ≤ i ≤ 5Þ is not more than one (see Table 2), a contradiction. Therefore, G is a cograph and the diameter diamðGÞ = 2.
Now we are in a position to complete the proof. Note that μðGÞ = 2 and diamðGÞ = 2 from the above process. Let P 3 = v 1 v 2 v 3 be a diameter of G, then fv 1 , v 3 g is a maximum independent set of G and each vertex out of fv 1 , v 3 g is adjacent to at least one of fv 1 , v 3 g. Let then any vertex out of VðP 3 Þ belongs precisely to one of S v i ð1 ≤ i ≤ 3Þ. The following claims are needed for us.
Suppose u ∈ S v 1 and w ∈ S v 2 such that u ≁ w. Then, the vertices fu, v 1 , w, v 3 g induce a path P 4 , a contradiction. The proof for the case of S v 3 is parallel, omitted.
Claim 5. All the vertices of S v 1 (resp., S v 3 ) induce a clique of G.
We only prove the case of S v 1 . If x, y ∈ S v 1 and x ≁ y, then fx, y, v 3 g induce an independent set of G, contradicting with μðGÞ = 2. Claim 6. All the vertices of S v 2 induce a clique of G.
Assume that x, y ∈ S v 2 and x ≁ y. Considering an induced subgraph H of order 5 of G, we obtain that H is isomorphic to one of fX 1 , X 2 , X 3 g(see Figure 4). It follows from Lemma 3 that H contains θ as an eigenvalue of multiplicity at least 2. But X i ð1 ≤ i ≤ 3Þ contains no eigenvalue of multiplicity 2 from Table 3, a contradiction.
From Claims 4-6 and the facts μðGÞ = 2 and diamðGÞ = 2, we derive that G is isomorphic to the graph Γ 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. Figure 3: The graphs G i ð1 ≤ i ≤ 5Þ.

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Corollary 8. The graph Γ with s, t, p ≥ 1 and s + t + p = n (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 Γ with multiplicity n − 3 and the remaining three eigenvalues of Γ are those of the quotient matrix Q of AðΓÞ. Since det ðQ + IÞ = −pst < 0, then Q + I contains two positive eigenvalues and one negative eigenvalue. By the Perron-Frobenius theorem, the largest eigenvalue of Γ is simple; then, the largest eigenvalue of Q (resp., Q + I) is simple. Thus, Q + I contains three distinct eigenvalues, that is, Q contains three distinct eigenvalues. Recalling that −1 is not an eigenvalue of Q, then Γ contains four distinct eigenvalues. Moreover, it is clear that Γ 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.