The effect of removing a 2-downer edge or a cut 2-downer edge triangle for an eigenvalue

Kenji Toyonaga

doi:10.1515/spma-2022-0186

Artikel Open Access

The effect of removing a 2-downer edge or a cut 2-downer edge triangle for an eigenvalue

Kenji Toyonaga

Veröffentlicht/Copyright: 15. März 2023

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Special Matrices Band 11 Heft 1

Abstract

Edges in the graph associated with a square matrix over a field may be classified as to how their removal affects the multiplicity of an identified eigenvalue. There are five possibilities: + 2 (2-Parter); + 1 (Parter); no change (neutral); − 1 (downer); and − 2 (2-downer). Especially, it is known that 2-downer edges for an eigenvalue comprise cycles in the graph. We investigate the effect for the statuses of other edges or vertices by removing a 2-downer edge. Then, we investigate the change in the multiplicity of an eigenvalue by removing a cut 2-downer edge triangle.

Keywords: eigenvalue; graph of a matrix; 2-downer edge cycle

MSC 2010: 15A18; 05C50

1 Introduction

Throughout, G denotes a simple, undirected graph on n vertices, and we denote by S ( G ) the set of all n -by- n real symmetric matrices, ℋ ( G ) the set of all n -by- n Hermitian matrices, the graph whose off-diagonal entries has an edge { i , j } iff a i j a j i ≠ 0 for A = [ a i j ] ; no restriction is placed on the diagonal entries, other than that they be real. For a given matrix A , we denote the multiplicity of an eigenvalue λ of A by m A ( λ ) .

When a graph G is a tree T , there are many papers relating the structure of T to the multiplicities of the eigenvalues of the matrices in S ( T ) .

In this article, we consider a general simple graph G . There are papers that relate the structure of G to the multiplicities of the eigenvalues of the matrices in S ( G ) , [1,2,4,6,8–12, etc.]. When a vertex v is removed from G , the remaining graph is denoted by G ( v ) , and we denote by A ( v ) the ( n − 1 ) -by- ( n − 1 ) principal submatrix of A ∈ S ( G ) , resulting from deletion of the row and column corresponding to v . When an edge e i j is removed from G , we denote the remaining graph by G ( e i j ) ; then A ( e i j ) ∈ S ( G ( e i j ) ) denotes the matrix obtained from A by changing the entries corresponding to e i j to zero. Further, when a vertex v and an edge e i j are removed from G , we denote the remaining graph by G ( v , e i j ) and the corresponding submatrix by A ( v , e i j ) . For an identified A ∈ S ( G ) , we often speak interchangeably about the graph and the matrix, and we identify vertices in a graph with indices of a matrix, for convenience.

From the interlacing inequalities for a symmetric matrix, the multiplicity of an eigenvalue may change by at most 1, when a particular vertex is deleted. A vertex v of a graph G is called Parter (respectively neutral, downer) in G for an eigenvalue λ of A ∈ S ( G ) , if

m A ( u ) ( λ ) = m A ( λ ) + 1 ( resp . m A ( λ ) , m A ( λ ) − 1 ) .

We call these the status (or classifications) of the vertex v for the eigenvalue λ relative to A . We denote Parter, neutral, and downer by P , N , and D , respectively.

The change in the multiplicity of an eigenvalue when an edge is removed was investigated in [3].

Lemma 1

[3] Let G be a graph, and A ∈ ℋ ( G ) . For an edge e i j in G , and λ ∈ σ ( A ) ,

m A ( λ ) − 2 ≤ m A ( e i j ) ( λ ) ≤ m A ( λ ) + 2 .

We can define the status of an edge. An edge e i j is called 2-Parter (resp. Parter, neutral, downer, 2-downer) for λ relative to A , if for A ∈ ℋ ( G ) and A ( e i j ) ∈ ℋ ( G ( e i j ) ) ,

m A ( e i j ) ( λ ) − m A ( λ ) = 2 ( resp . 1 , 0 , − 1 , − 2 ) .

If the status of a vertex or an edge is neutral, Parter, or 2-Parter, then the status of it is called at least neutral.

The classification number of a vertex or an edge is described in [6]. Given A ∈ S ( G ) , and λ ∈ σ ( A ) , the classification number of a vertex is defined in a natural numerical way, Parter is 1, neutral 0, and downer − 1 . In particular, we may numerically classify a vertex i in A relative to the identified eigenvalue λ , S A ( i ) , as follows:

S A ( i ) = m A ( i ) ( λ ) − m A ( λ ) .

An edge’s classification number is defined as 2 , 1 , 0 , − 1 , and − 2 , depending on whether the edge is 2-Parter, Parter, neutral, downer, or 2-downer, respectively. We may numerically classify an edge e i j , S A ( e i j ) , as follows:

S A ( e i j ) = m A ( e i j ) ( λ ) − m A ( λ ) .

There is a relationship between the classification number of an edge and the classification number of the incident vertices,

(1) S A ( e i j ) = S A ( i ) − S A ( e i j ) ( i ) = S A ( j ) − S A ( e i j ) ( j ) .

If vertex j is a neighbor of vertex i in G and j is a downer vertex for λ in A ( i ) , then we call j a downer neighbor of i for A ∈ S ( G ) and λ [5].

Definition 2

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . If there is a cycle Γ in G whose edges on Γ are all 2-downer edges for λ relative to A , then we call Γ a 2-downer edge cycle for λ relative to A.

As a simple example of a 2-downer edge cycle, there is a cycle C n with n vertices whose eigenvalues are 2 cos 2 π n j ( j = 1 , … , n ) , and every edge of C n can be a 2-downer edge of a certain double eigenvalue.

Definition 3

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . If Γ is a 2-downer edge cycle for λ relative to A and it is not connected to other 2-downer edge cycles with 2-downer edges for λ , then we call Γ a primitive 2-downer edge cycle for λ relative to A.

As an example of a primitive 2-downer edge cycle, we refer to Example 1 in Section 4, in which the triangle 1, 2, 3, and the triangle 8, 9, 10 are primitive 2-downer edge cycles for the eigenvalue 1.

We note that if Γ is a primitive 2-downer edge cycle for λ , it is separated from other 2-downer edge cycles, then edges incident to vertices on Γ that are not on Γ are edges other than 2-downer edges.

We are interested in a 2-downer edge cycle in this article, and in Section 2, we investigate the effect of removing a 2-downer edge from the cycle, the change of the statuses of other edges or vertices. Then we investigate the change in the multiplicity of an eigenvalue by removing a cut 2-downer edge triangle in Section 3.

2 Removing a 2-downer edge from a cycle

In this article, we are particular about in a 2-downer edge cycle and focus on the effect of removing a 2-downer edge in a cycle or a 2-downer edge triangle. In [11], the possible classification for an edge e i j for λ relative to A ∈ ℋ ( G ) is given as in Table 1, when the classifications of adjacent vertices i and j are known. Here, we refer to the two theorems we require later.

Table 1

Possible classification for edges in ℋ ( G )

i	j	Possible classifications for edge e i j
P	P	2-Parter or neutral
P	N	Parter or neutral
P	D	Neutral
N	N	Parter or neutral
N	D	Downer
D	D	2-downer, downer, or neutral

P : Parter vertex, N : neutral vertex, D : downer vertex.

Theorem 4

[11, Theorem 6] Let G be a connected graph, A ∈ ℋ ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . An edge e i j is 2-downer for λ in G if and only if the status of i is downer for λ in G , and j is a downer neighbor of i in G. Here, i and j are interchangeable.

Theorem 5

[11, Theorem 7] Let G be a graph, A ∈ ℋ ( G ) , and λ ∈ σ ( A ) . If an edge e i j is 2-Parter for λ in G , then each of i and j are Parter for λ in G, and each is a downer neighbor for the other in G.

We need a necessary and sufficient condition for a Parter vertex.

If a graph is a tree T , a Parter vertex v for λ relative to A ∈ S ( T ) is characterized by the existence of a downer branch at v [5]. However, when G is a general graph, a necessary and sufficient condition for a vertex to be a Parter vertex is given in [12]. We give the proof of it to be self-contained here.

Theorem 6

[12, Theorem 3] A vertex i is Parter for λ in G relative to A ∈ S ( G ) if and only if there is a downer neighbor j at i and the edge e i j is at least neutral for λ in A (i.e., m A ( e i j ) ( λ ) ≥ m A ( λ ) ).

Proof

Wlog, we may assume that the index of i is 1, λ = 0 and that A has the following form:

(2) A = α x T x B ,

in which the x is a nonzero column vector, α is a scalar, and the B is a square matrix.

If the index 1 is Parter for λ in A , there has to be at least one downer neighbor j adjacent to 1 in B . Because if there is no downer neighbor adjacent to 1 in B , then all adjacent vertices to 1 are neutral or Parter for λ in B . Then, every column (resp. row) relative to the adjacent vertex in B is not a linear combination of the remaining columns (resp. rows). Let RS ( B ) (resp. CS ( B ) ) denote the row space (resp. column space)of B . Then, e k T ∈ RS ( B ) , (resp. e k ∈ CS ( B ) ), in which e k is a normal unit vector and k corresponds to some indices to which index 1 is adjacent in B . Since x T is a linear combination of some e k T s ( k ≥ 1 ), x T ∈ RS ( B ) , a contradiction because the index 1 is Parter in A . Furthermore, e 1 j cannot be a downer or a 2-downer edge in A , since 1 is Parter. Therefore, e 1 j is at least neutral for λ in A .

Next, we give a proof for sufficiency. Suppose that there is a downer neighbor j of 1 in B and e 1 j is at least neutral.

To reach a contradiction, suppose that the index 1 is not Parter for λ in A satisfying the aforementioned conditions. If the index 1 is neutral for λ in A , then x T ∈ RS ( B ) . When e 1 j is removed from G , x T − a 1 j e j T ∉ RS ( B ) , since e j T ∉ RS ( B ) , because j is a downer neighbor of 1 in B . Then index 1 becomes Parter in A ( e 1 j ) and S A ( e 1 j ) = S A ( 1 ) − S A ( e 1 j ) ( 1 ) = − 1 . This means the edge e 1 j is a downer edge in A , a contradiction to the assumption.

Next, if index 1 is downer for λ in A satisfying the aforementioned conditions, then x T ∈ RS ( B ) . When e 1 j is removed from G , x T − a 1 j e j T ∉ RS ( B ) , since e j T ∉ RS ( B ) because j is a downer neighbor at 1 in B . Then index 1 becomes Parter in A ( e 1 j ) . Therefore, S A ( e 1 j ) = S A ( 1 ) − S A ( e 1 j ) ( 1 ) = − 2 , which means the edge e 1 j is a 2-downer edge in A , a contradiction to the assumption.

Thus, when the conditions are satisfied, index 1 must be Parter in A .□

In [12], it was observed that if there is a 2-downer edge in a graph G for λ relative to A ∈ S ( G ) , then there is a 2-downer edge cycle for λ in the graph. To be self-contained in this article, we give the proof here.

Theorem 7

[12] Suppose G is a graph, A ∈ ℋ ( G ) and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Then each 2-downer edge for λ is contained in a 2-downer edge cycle of G or it is on a path connecting 2-downer edge cycles.

Proof

Let e i j be a 2-downer edge for λ in A ∈ ℋ ( G ) . Then, i and j are downer for λ in A and downer neighbors for each other. When the edge e i j is removed from G , the status of i changes to Parter . Then, there has to be a downer neighbor k distinct from j in A ( e i j ) by Theorem 6. Then, we note that k is also a downer neighbor of i in A , and i is originally downer in A . Thus, e i k is a 2-downer edge in A by Theorem 4.

From a similar argument, there must be another 2-downer edge incident to e i j at j . Thus, 2-downer edges are connected sequentially and compose a cycle in G in the end.□

2.1 Change in status by removing a 2-downer edge

First, we see that when a vertex on a 2-downer edge cycle Γ is removed, other 2-downer edges on Γ in G are not 2-downer in the remaining graph G ( v ) .

Lemma 8

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let Γ be a primitive 2-downer edge cycle in G for λ relative to A. If a vertex v on Γ is removed from G, then the rest of the edges on Γ in G ( v ) are not 2-downer for λ relative to A ( v ) .

Proof

If there is a 2-downer edge on Γ ( v ) in G ( v ) , then there must be a 2-downer cycle in G ( v ) . Then there is a 2-downer edge e i k in which i is on Γ ( v ) and k is outside Γ ( v ) .

Since e i k is 2-downer in G ( v ) , i is downer in G ( v ) and k is downer in G ( v , i ) by Theorem 4. Then,

(3) m A ( v , i , k ) ( λ ) = m A ( λ ) − 3 .

However, k is not downer in G ( i ) from the assumption that Γ is a primitive 2-downer edge cycle in G and k is not on Γ . Then k is neutral or Parter in G ( i ) . Thus, m A ( i , k , v ) ( λ ) > m A ( λ ) − 3 has to hold. That is a contradiction to (3). Therefore, the rest of the edges on Γ in G ( v ) are not 2-downer edges in G ( v ) .□

Let Γ be a primitive 2-downer edge cycle for λ in a general simple graph G , and Γ ′ be a subgraph of Γ obtained by removing an edge e i j on Γ . Then let G ′ = G ( e i j ) .

Lemma 9

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let Γ be a primitive 2-downer edge cycle in G for λ relative to A. If a 2-downer edge e i j on Γ is removed from G, then the statuses of vertices and edges on Γ ′ are at least neutral in G ′ for λ relative to A ( e i j ) .

Proof

Let v be a vertex on Γ , then v is downer for λ . If v is removed from G , all the edges on Γ ′ are not 2-downer for λ by Lemma 8. Then

m A ( v , e i j ) ( λ ) ≠ m A ( λ ) − 3 .

Since m A ( e i j ) ( λ ) = m A ( λ ) − 2 , v cannot be downer in A ( e i j ) . Then the status of vertices on Γ ′ is at least neutral in G ′ .

If an edge on Γ ′ is downer or 2-downer for λ , the status of incident vertices has to be ( N , D ) or ( D , D ) (cf. Table 1). From the preceding argument, the statuses of vertices on Γ ′ are not downer. Therefore, the edges on Γ ′ are at least neutral for λ in G ′ .□

It is known that when a neutral vertex is removed from G , a downer vertex in G stays in the remaining graph, and vice versa [4].

Lemma 10

[4] Let A be an n -by- n Hermitian matrix. If i is neutral, then j ≠ i is downer for A if and only if j is downer for A ( i ) .

Theorem 11

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let Γ be a primitive 2-downer edge cycle in G for λ relative to A, and Γ ′ be a subgraph of Γ obtained by removing an edge e i j on Γ . The statuses of all edges in G ′ incident to the vertices on Γ ′ are at least neutral for λ relative to A ( e i j ) .

Proof

From Lemma 9, the edges on Γ ′ are at least neutral for λ . We examine the status of an edge outside Γ ′ that is incident to a vertex on Γ ′ .

We know that the statuses of vertices on Γ ′ are Parter or neutral from Lemma 9. If a vertex on Γ ′ is Parter, it is obvious that edges incident to it are at least neutral from Table 1. Therefore, we suppose that there is a neutral vertex k on Γ ′ that is not i or j because i and j are Parter in G ( e i j ) by (1).

To attain a contradiction, we suppose that there is a downer edge incident to k in G ( e i j ) . Let the edge be e k l . Then the status of l has to be downer in G ( e i j ) (cf. Table 1). Then

(4) m A ( e i j , k , l ) ( λ ) = m A ( λ ) − 3 ,

because a downer vertex is still downer after removing a neutral vertex by Lemma 10.

On the other hand, k is downer in G since k is on Γ , and the status of l is not downer in G ( k ) since Γ is primitive. So l is Parter or neutral for λ in G ( k ) . If l is Parter in G ( k ) , m A ( k , l , e i j ) ( λ ) > m A ( λ ) − 3 , then (4) does not hold. Thus, in the case for (4) to hold, l must be neutral in G ( k ) and e i j has to be 2-downer in G ( k , l ) , then we note that

(5) m A ( k , l , i , j ) ( λ ) = m A ( λ ) − 3

holds by Theorem 4.

Next, we focus on the status of the vertex i in G ( k ) .

If i is Parter in G ( k ) , m A ( k , i ) ( λ ) = m A ( λ ) , then m A ( k , i , j , l ) ( λ ) > m A ( λ ) − 3 , so i has to be neutral or downer in G ( k ) for (5) to hold. We note that l is neutral in G ( k ) from the prior argument. If i is neutral in G ( k ) , then in G ( k , i ) , l is neutral or Parter, because l cannot be downer in G ( k , i ) from Lemma 10. Then we have m A ( k , i , l , j ) ( λ ) > m A ( λ ) − 3 . So (5) does not hold. Therefore, for (5) to hold, i has to be downer in G ( k ) . By Lemma 8, e i j is not a 2-downer edge in G ( k ) , so j is not downer in G ( k , i ) , then j is neutral or Parter in G ( k , i ) .

If j is neutral in G ( k , i ) , then l can be Parter or neutral in G ( k , i , j ) , because l cannot be downer from Lemma 10. Then m A ( k , i , j , l ) ( λ ) ≠ m A ( λ ) − 3 .

If j is Parter in G ( k , i ) , then m A ( k , i , j ) ( λ ) = m A ( λ ) − 1 , so m A ( k , i , j , l ) ( λ ) ≠ m A ( λ ) − 3 . Therefore, (5) cannot hold. It is a contradiction.

So, we have a conclusion that e k l cannot be a downer edge in G ( e i j ) .□

If edges in G ′ are incident to the removed 2-downer edge in G , we can see their statuses more accurately.

Corollary 12

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let Γ be a primitive 2-downer edge cycle in G for λ relative to A and Γ ′ be a subgraph of Γ obtained by removing an edge e i j on Γ . The edge incident to i or j on Γ ′ is Parter or 2-Parter, and other edges incident to i or j are neutral in G ′ for λ relative to A ( e i j ) .

Proof

The status of i and j is Parter in G ( e i j ) by (1). Let k and m be vertices on Γ ′ adjacent to i and j , respectively. We refer to the following figure.

If e i k is neutral in G ( e i j ) , then i is Parter in G ( e i j , e i k ) by (1). However, since Γ is primitive, there is no downer neighbor at i in G ( e i j , e i k ) . So i cannot be Parter in G ( e i j , e i k ) by Theorem 6. It is a contradiction. So, e i k is Parter or 2-Parter for λ in G ( e i j ) . By a similar argument, e j m is Parter or 2-Parter for λ in G ( e i j ) .

Let e i l be an edge incident to i that is not on Γ ′ . Then l is not downer neighbor at i in G ( e i j ) , since Γ is primitive. So e i l cannot be a 2-Parter edge from Theorem 5. Next, we show that e i l cannot be a Parter edge for λ in G ( e i j ) . If e i l is a Parter edge with ( P , N ) in G ( e i j ) , i is neutral and l is downer in G ( e i j , e i l ) . Further, when i is removed from G ( e i j , e i l ) , l is downer in G ( e i j , e i l , i ) by Lemma 10. It means that l is a downer neighbor at i in G ( e i j ) , a contradiction. So, e i l cannot be a Parter edge in G ( e i j ) . Then e i l is neutral by Theorem 11.□

Next, we investigate the status of vertices adjacent to Γ ′ .

Theorem 13

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let Γ be a primitive 2-downer edge cycle for λ relative to A , and Γ ′ be a graph obtained by removing an edge e i j on Γ . The statuses of all vertices adjacent to Γ ′ in G ′ are at least neutral for λ relative to A ( e i j ) . Furthermore, the edges between the vertices adjacent to Γ ′ are also at least neutral for λ relative to A ( e i j ) .

Proof

From Lemma 9, vertices on Γ ′ are at least neutral in G ′ . Let k be a vertex on Γ ′ and l be a vertex that is not on Γ ′ and adjacent to k .

To reach a contradiction, if we suppose that l is downer in G ′ for λ , then m A ( e i j , l ) ( λ ) = m A ( λ ) − 3 . Then, m A ( l , e i j ) ( λ ) = m A ( λ ) − 3 , so l has to be downer for λ in G and e i j be 2-downer in G ( l ) .

We note that k is not downer in G ( l ) , because l is not downer in G ( k ) . Since k is not downer in G ( l ) , Γ is not a 2-downer edge cycle in G ( l ) .

If e i j is a 2-downer edge for λ in G ( l ) , then there is a 2-downer edge e m r in which the vertex m (possibly i or j ) is on Γ and the vertex r is adjacent to m that is not on Γ . Then we have m A ( l , m , r ) ( λ ) = m A ( λ ) − 3 since l is downer in G and e m r is 2-downer in G ( l ) from the assumption. Then m A ( m , r ) ( λ ) = m A ( λ ) − 2 must hold to be m A ( l , m , r ) ( λ ) = m A ( λ ) − 3 , then r is a downer neighbor at m in G . But r cannot be a downer neighbor at m in G , since Γ is primitive. It is a contradiction, so e i j is not a 2-downer edge in G ( l ) . Therefore, l is not downer in G ( e i j ) , then l is at least neutral in G ( e i j ) .

When the vertices adjacent to Γ ′ are at least neutral in G ′ , the edges between them are at least neutral from Table 1. So, the edge e l r in the figure is also at least neutral in G ′ .□

2.2 The same status by removing an edge

Next, we observed that when an edge of a certain status for λ relative to A is removed from G , there are edges or vertices whose statuses stay same in the resulting graph.

Theorem 14

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let e i j be a 2-downer edge for λ relative to A. Let G ˜ be a graph obtained by removing the edge e i j from G . The statuses of neutral or Parter vertices in G for λ relative to A stay in G ˜ for λ relative to A ( e i j ) .

Proof

Since e i j is a 2-downer edge in G , i and j are downer for λ in G , and m A ( i , j ) ( λ ) = m A ( λ ) − 2 . Let k be a Parter (resp. neutral) vertex in G . We note that k is Parter (resp. neutral) in A ( i , j ) , because Parter (resp. neutral) vertices stay after removing a downer vertex. Then,

(6) m A ( i , j , k ) ( λ ) = m A ( λ ) − 1 , ( resp . m A ( λ ) − 2 ) .

On the other hand, i is downer for λ in A ( k ) , because downer vertices stay after removing a Parter (resp. neutral) vertex (cf. Table 1 [8]). Then j has to be downer in A ( k , i ) for (6) to hold. Thus, j is a downer neighbor at i in A ( k ) . Then e i j is a 2-downer edge in A ( k ) . Therefore, 2-downer edges stay after removing a Parter (resp. neutral) vertex from G .

Through converse consideration, a Parter (resp. neutral) vertex stays after removing a 2-downer edge from G , because the change in multiplicity is consistent after removing a 2-downer edge first and a Parter (resp. neutral) vertex.□

In Theorem 14, it was observed that the status of a Parter (resp. neutral) vertex does not change after removing a 2-downer edge for λ from G .

Furthermore, we noticed that there is another case in which the status of the vertex does not change after removing an identified status of edge.

Theorem 15

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) . If a 2-Parter edge or a Parter edge for λ relative to A is removed from G, then the statuses of downer vertices for λ in G stay in the resulting graph for λ relative to the corresponding matrix.

Proof

Let e i j be a 2-Parter edge in G . Let k be a downer vertex in G . If we assume that the status of k changes to neutral or Parter after removing e i j from G , then

(7) m A ( e i j , k ) ( λ ) ≥ m A ( λ ) + 2 .

However, since k is downer for λ in G , m A ( k ) ( λ ) = m A ( λ ) − 1 . Then, m A ( k , e i j ) ( λ ) ≤ m A ( λ ) + 1 . That is a contradiction to (7). Therefore, k stays downer in G ( e i j ) .

Next, let e i j be a Parter edge in G and k be a downer vertex in G . A pair of statuses of i and j can be ( P , N ) or ( N , N ) from Table 1.

If we suppose the status of k changes to Parter in G ( e i j ) , then

(8) m A ( e i j , k ) ( λ ) = m A ( λ ) + 2 .

However, since k is downer for λ in G , m A ( k ) ( λ ) = m A ( λ ) − 1 . Then, m A ( k , e i j ) ( λ ) ≤ m A ( λ ) + 1 . That is a contradiction to (8). So, k cannot be Parter for λ in G ( e i j ) .

Next, we suppose that k changes to neutral in G ( e i j ) . Wlog, let j be neutral for λ in G since i or j is neutral when e i j is a Parter edge. In G ( e i j ) , j is downer by (1), then

(9) m A ( e i j , j , k ) ( λ ) = m A ( λ ) ,

because the status of k stays neutral in G ( e i j , j ) , it can be said by using Lemma 10. However, we note that

(10) m A ( e i j , j , k ) ( λ ) = m A ( j , k ) ( λ ) = m A ( λ ) − 1 ,

because k is downer and j is neutral in G , and k stays downer in G ( j ) . This is a contradiction to (9). So, k cannot become neutral in G ( e i j ) . Therefore, k stays downer in G ( e i j ) .□

If we conversely see Theorem 15, then we have the next result. The change in the multiplicity of an eigenvalue is independent of the order of removing a vertex and an edge.

Corollary 16

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) . If a downer vertex for λ relative to A is removed from G, then 2-Parter edges or Parter edges for λ in G stay in the resulting graph for λ relative to the corresponding matrix.

Next, we observe that when a 2-downer edge is removed from G , some edges do not change in their statuses.

Theorem 17

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let e i j be a 2-downer edge in G for λ relative to A. Let G ˜ be a graph obtained by removing the edge e i j from G . The statuses of 2-Parter edges or Parter edges in G for λ relative to A stay in G ˜ for λ relative to A ( e i j ) .

Proof

Let e k l be a 2-Parter edge in G . Then the statuses of k and l are Parter. When a 2-downer edge e i j is removed from G , the status of a Parter vertex stays in G ˜ by Theorem 14. So, e k l is 2-Parter or neutral in G ( e i j ) from Table 1. However, we show that e k l cannot be neutral in G ( e i j ) . If we suppose that e k l is neutral in G ( e i j ) , then

(11) m A ( e i j , e k l ) ( λ ) = m A ( λ ) − 2 .

However, since e k l is 2-Parter for λ ,

(12) m A ( e k l , e i j ) ( λ ) ≥ m A ( λ ) .

considering Lemma 1. So, (11) is contradictory to (12). Therefore, e k l is a 2-Parter edge for λ in G ( e i j ) .

Next, we consider Parter edges in G . Let e k l be a Parter edge for λ in G . There are two types of Parter edges with ( P , N ) and ( N , N ) as a pair of the statuses of the adjacent vertices from Table 1. When the 2-downer edge e i j is removed from G , Parter vertices and neutral vertices in G stay in G ( e i j ) by Theorem 14. So the status of e k l is a Parter edge or a neutral edge in G ( e i j ) with ( P , N ) or ( N , N ) by Table 1. However, we show that it cannot be a neutral edge in G ( e i j ) .

If we suppose that e k l is neutral in G ( e i j ) , then

(13) m A ( e i j , e k l ) ( λ ) = m A ( λ ) − 2 .

But, since e k l is a Parter edge in G ,

(14) m A ( e k l , e i j ) ( λ ) ≥ m A ( λ ) − 1 ,

then (13) is contradictory to (14). Therefore, e k l is a Parter edge in G ( e i j ) .□

Considering Theorem 17 conversely, when a 2-Parter edge or a Parter edge is removed from G , 2-downer edges stay in the resulting graph. Thus, we have the next result.

Corollary 18

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . If a 2-Parter edge or a Parter edge for λ relative to A is removed from G, original 2-downer edge cycles for λ in G stay in the resulting graph for λ relative to the corresponding matrix.

3 Removing a cut 2-downer edge triangle

Next, we focus on a 2-downer edge cycle with three vertices, called a 2-downer edge triangle here. When three edges on a triangle are removed from G , if the number of components of G increases, then we call it a cut triangle in G . Now, we give a simple observation for a cut triangle that is used later.

Lemma 19

Let Γ be a cut triangle in G, then there is an edge on Γ , which is a cut edge after removing the rest of the edges on Γ from G .

Proof

Let the vertices of Γ be i , j , and k . Wlog, we suppose G is a connected graph. Let G ′ = G ( e i j ) , then G ′ is connected. We refer to the following figure as a part of G that includes Γ . If e i k is not a cut edge in G ′ , then e j k is a cut edge in G ( e i j , e i k ) since Γ is a cut triangle.

If e i k is a cut edge in G ′ , then e k j is an edge that is included in one component of G ( e i j , e i k ) . Then we note that e i k is also a cut edge in G ( e i j , e j k ) .□

When a cut edge in G is removed, the change in multiplicity of an identified eigenvalue is observed in [7, Lemma 19].

Lemma 20

[7, Lemma 19] Let G be a graph and A ∈ ℋ ( G ) . If e i j is a cut-edge in G and λ ∈ σ ( A ) , then

m A ( λ ) − 1 ≤ m A ( e i j ) ( λ ) ≤ m A ( λ ) + 2 .

This lemma indicates that if there is a cut edge in G , it cannot be a 2-downer edge for an eigenvalue of A ∈ ℋ ( G ) .

The condition for a cut edge to be a downer edge is shown in [7, Theorem 20].

Lemma 21

[7, Theorem 20] Let G be a graph, A ∈ ℋ ( G ) , and λ ∈ σ ( A ) . A cut-edge e i j in G is downer for λ relative to A if and only if the statuses of i and j in G are ( D , D ) for λ relative to A .

Lemma 22

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let Γ be a cut primitive 2-downer edge triangle in G for λ relative to A. If one edge on Γ is removed from G, then the rest of the edges on Γ are Parter edges in the remaining graph for λ relative to the corresponding matrix.

Proof

Let the vertices of Γ be i , j , and k . From Corollary 12, if an edge e i j is removed, the rest of the edges e i k and e j k are Parter or 2-Parter.

We note that i and j are Parter vertices in G ( e i j ) . If e i k is a Parter edge, then k has to be neutral for λ , and if e i k is 2-Parter, then k has to be Parter for λ (cf. Table 1). Thus, the statuses of two edges e i k and e j k have to be the same when e i j is removed. We note that if e i k and e j k are Parter in G ( e i j ) , then e i j and e j k are Parter in G ( e i k ) .

We can observe that if one edge on Γ is removed, the other two edges on Γ cannot be 2-Parter edges in G ( e i j ) .

Wlog, we suppose that e j k is a cut edge in G ( e i j , e i k ) by Lemma 19. To reach a contradiction, we suppose that e i k and e j k are 2-Parter for λ in G ( e i j ) . Then j is a downer neighbor at k in G ( e i j ) by Theorem 5 since e j k is 2-Parter in G ( e i j ) . Then, k is downer for λ in G ( e i j , e i k ) . We note that j is a downer neighbor at k also in G ( e i j , e i k ) . Then e j k has to be a 2-downer edge by Theorem 4, and it is a cut edge in G ( e i j , e i k ) . So it is a contradiction to Lemma 20, because a cut edge cannot be a 2-downer edge. Therefore, e i k and e j k cannot be 2-Parter in G ( e i j ) , then they are Parter edges for λ in G ( e i j ) .□

We investigate the change in multiplicity of an identified eigenvalue λ , when all edges on a cut primitive 2-downer edge triangle are removed.

Theorem 23

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let Γ be a cut primitive 2-downer edge triangle in G for λ relative to A. Let G ˜ be a subgraph of G obtained by removing all edges on Γ from G and A ˜ a corresponding matrix. Then

m A ˜ ( λ ) = m A ( λ ) − 2 ,

and the statuses of three vertices on Γ are neutral in G ˜ for λ relative to A ˜ .

Proof

Let the vertices of Γ be i , j , and k . Since Γ is a cut 2-downer edge triangle, we may suppose that wlog e j k is a cut edge after removing e i j and e i k by Lemma 19.

From Lemma 22, when the edge e i j is removed from G , e i k and e j k are Parter edges for λ , then i and j are Parter by (1) and k is neutral in G ( e i j ) since e i k is a Parter edge with vertices ( P , N ) in G ( e i j ) . Then if e i k is removed from G ( e i j ) , k is downer by (1), then we next show that j becomes downer in G ( e i j , e i k ) . When e i k is removed from G , e i j and e j k are Parter by Lemma 22, and j is neutral because when an edge is a Parter edge, if one vertex is Parter, then the other vertex has to be neutral. So, m A ( e i k , j ) ( λ ) = m A ( λ ) − 2 . On the other hand, m A ( e i k , j ) ( λ ) = m A ( e i j , e i k , j ) ( λ ) = m A ( λ ) − 2 . Since m A ( e i j , e i k ) ( λ ) = m A ( λ ) − 1 , j has to be downer in G ( e i j , e i k ) .

Since the statuses of j and k are downer for λ in G ( e i j , e i k ) , and e j k is a cut edge in G ( e i j , e i k ) , then e k j has to be a downer edge by Lemma 21. We conclude

m A ˜ ( λ ) = m A ( e i j , e i k , e j k ) ( λ ) = m A ( λ ) − 2 .

We note that e j k is a downer edge with ( D , D ) in G ( e i j , e i k ) , and i is neutral in G ( e i j , e i k ) for λ . If the edge e j k is removed from G ( e i j , e i k ) , then the statuses of j and k change to neutral.

Next, we observe i is still neutral in G ( e i j , e i k , e j k ) . When i is removed from G ( e i j , e i k ) , k and j are downer by Lemma 10 and e j k is a cut edge, so e j k is downer in G ( e i j , e i k , i ) = G ( i ) . Then m A ( e i j , e i k , i , e j k ) ( λ ) = m A ( i , e j k ) ( λ ) = m A ( λ ) − 2 . Since e j k was downer in G ( e i j , e i k ) , m A ( e i j , e i k , e j k ) ( λ ) = m A ( λ ) − 2 .

Therefore, i is also neutral in G ( e i j , e i k , e j k ) .□

Next, we investigate the change in the multiplicity of an identified eigenvalue λ when all vertices on a cut 2-downer edge triangle are removed.

Theorem 24

Let G be a graph, A ∈ S ( G ) , and λ ∈ σ ( A ) with m A ( λ ) ≥ 2 . Let Γ be a cut 2-downer edge triangle in G for λ relative to A . Let G ˜ be a subgraph of G obtained by removing all vertices on Γ from G and A ˜ a corresponding matrix. Then

m A ˜ ( λ ) = m A ( λ ) − 2 .

Proof

Wlog, we may suppose that e j k is a cut edge in G ( e i j , e i k ) by Lemma 19. Then we note that e j k is also a cut edge in G ( i ) . Since j and k are downer in G ( i ) and e j k is a cut edge in G ( i ) , e j k is a downer edge in G ( i ) .

If e j k is removed from G ( i ) , then m A ( i , e j k ) ( λ ) = m A ( λ ) − 2 . Then k and j is neutral in G ( i , e j k ) , and k and j belong to different components. So, m A ( i , e j k , j , k ) ( λ ) = m A ( i , j , k ) ( λ ) = m A ( λ ) − 2 .□

We note that in Theorem 24, the cut 2-downer edge triangle does not have to be primitive in G .

4 Example

Example 1

We give an example to sketch Theorems 11 and 13.

Let

A = 1 1 1 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 2 1 1 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 1 1 2 ,

whose graph G is as follows.

A has an eigenvalue λ = 1 with multiplicity 3. The triangle Γ whose vertices are 1 , 2 , 3 is a 2-downer edge triangle for λ = 1 . When one edge e 12 on Γ is removed from G , the status of edges and vertices on G ′ are shown in the following figure. The statuses of edges and vertices are indicated in small letters and capital letters, respectively.

Example 2

We can find a simple example to sketch Theorems 23 and 24. Let

A = 1 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 ,

whose graph G is as follows.

A has an eigenvalue λ = 1 + 5 2 with multiplicity 2. The center triangle is a cut primitive 2-downer edge triangle for λ . When edges e 12 , e 23 , and e 13 are removed from G , let

A ˜ = 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 .

A ˜ does not have λ = 1 + 5 2 as an eigenvalue of A ˜ . So, the multiplicity of λ decreases by 2 in A ˜ .

If vertices 1 , 2 , and 3 are removed from G , isolated vertices 4 , 5 , and 6 do not have λ as an eigenvalue. So, the multiplicity of λ decreases by 2.

Acknowledgement

The author would like to thank to referees for their helpful comments that have improved the article.

Funding information: This work was supported by JSPS KAKENHI (Grant Number JP21K03361).
Conflict of interest: The author states no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during this study.

References

[1] C. M. da Fonseca, A note on the multiplicities of the eigenvalues of a graph, Linear Multilinear Algebra 53 (2005), no. 4, 303–307. 10.1080/03081080500092307Suche in Google Scholar

[2] C. M. da Fonseca, A lower bound for the number of distinct eigenvalues of some real symmetric matrices, Electronic J. Linear Algebra 21 (2010), 3–11. 10.13001/1081-3810.1409Suche in Google Scholar

[3] C. R. Johnson and P. R. McMichael, The change in multiplicity of an eigenvalue of a Hermitian matrix associated with the removal of an edge from its graph, Discrete Math. 311 (2011), 166–170. 10.1016/j.disc.2010.10.010Suche in Google Scholar

[4] C. R. Johnson and B. D. Sutton, Hermitian matrices, eigenvalue multiplicities, and eigenvector components, SIAM J. Matrix Anal. Appl. 26 (2004), no. 2, 390–399. 10.1137/S0895479802413649Suche in Google Scholar

[5] C. R. Johnson and C. M. Saiago, Eigenvalues, multiplicities and graphs, Cambridge Tracts in Mathematics, Cambridge University Press, United Kingdom, 2018. 10.1017/9781316155158Suche in Google Scholar

[6] C. R. Johnson, C. M. Saiago, and K. Toyonaga, Classification of vertices and edges with respect to the geometric multiplicity of an eigenvalue in a matrix, with a given graph, over a field, Linear Multilinear Algebra 66 (2018), no. 11, 2168–2182. 10.1080/03081087.2017.1389848Suche in Google Scholar

[7] C. R. Johnson, C. M. Saiago, and K. Toyonaga, The change in multiplicity of an eigenvalue due to adding or removing edges, Linear Algebra Appl. 560 (2019), 86–99. 10.1016/j.laa.2018.09.017Suche in Google Scholar

[8] C. R. Johnson, C. M. Saiago, and K. Toyonaga, Change in vertex status after removal of another vertex in the general setting, Linear Algebra Appl. 612 (2021), 128–145. 10.1016/j.laa.2020.11.023Suche in Google Scholar

[9] K. Toyonaga and C. R. Johnson, The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value, Spec. Matrices 5 (2017), 51–60. 10.1515/spma-2017-0004Suche in Google Scholar

[10] K. Toyonaga, The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree, Spec. Matrices 7 (2019), 257–262. 10.1515/spma-2019-0019Suche in Google Scholar

[11] K. Toyonaga and C. R. Johnson, Classification of edges in a general graph associated with the change inmultiplicity of an eigenvalue, Linear Multilinear Algebra 69 (2021), no. 10, 1803–1812. 10.1080/03081087.2020.1825606Suche in Google Scholar

[12] K. Toyonaga and C. R. Johnson, Parter vertices and generalization of the Downer branch mechanism in the general setting, Linear and Multilinear Algebra, 2023. 10.1080/03081087.2023.2176414. Suche in Google Scholar

Received: 2021-11-28

Revised: 2023-01-21

Accepted: 2023-02-13

Published Online: 2023-03-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/spma-2022-0186

Schlagwörter für diesen Artikel

eigenvalue; graph of a matrix; 2-downer edge cycle

Creative Commons

BY 4.0