Accepted Manuscript The effect on eigenvalues of connected graphs by adding edges
Ji-Ming Guo, Pan-Pan Tong, Jianxi Li, Wai Chee Shiu, Zhi-Wen Wang
PII: DOI: Reference:
S0024-3795(18)30073-9 https://doi.org/10.1016/j.laa.2018.02.012 LAA 14477
To appear in:
Linear Algebra and its Applications
Received date: Accepted date:
25 September 2017 7 February 2018
Please cite this article in press as: J.-M. Guo et al., The effect on eigenvalues of connected graphs by adding edges, Linear Algebra Appl. (2018), https://doi.org/10.1016/j.laa.2018.02.012
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The effect on eigenvalues of connected graphs by adding edges∗ Ji-Ming Guoa†, Pan-Pan Tonga , Jianxi Lib ,
Wai Chee Shiuc , Zhi-Wen Wanga a Department of Mathematics, East China University of Science and Technology, Shanghai, P.R. China b Department of Mathematics & Information Science, Minnan Normal University, Zhangzhou, Fujian, P.R. China c Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, P.R. China.
Abstract By the well-known Perron-Frobenius Theorem [3], for a connected graph G, its largest eigenvalue strictly increases when an edge is added. We are interested in how the other eigenvalues of a connected graph change when edges are added. Examples show that all cases are possible: increased, decreased, unchanged. In this paper, we consider the effect on the eigenvalues by suitably adding edges in particular families, say the family of connected graphs with clusters. By using the result, we also consider the effect on the energy by suitably adding edges to the graphs of the above families. AMS classification: 05C50 Keywords: graph; eigenvalue; adding an edge; energy ∗
Partially supported by NSF of China(No.11371372); General Research Fund of Hong Kong; Faculty Research Grant of Hong Kong Baptist University; Program for New Century Excellent Talents in Fujian Province University; Project of Fujian Education Department (No.JZ160455); Research Fund of Minnan Normal University (No.MX1603). † Corresponding author; Email addresses:
[email protected](J.-M. Guo),
[email protected](J. Li),
[email protected](W.C. Shiu),
[email protected](Z.-W. Wang)
1
1
Introduction
Let G = (V, E) be a graph with vertex set V = {v1 , v2 , · · · , vn } and edge set E. Let A(G) = (aij ) be the adjacency matrix of G, where aij = 1 if vi is adjacent to vj ; and aij = 0, otherwise. It is easy to see that A(G) is a real symmetric matrix. Denote its eigenvalues by λ1 (G) ≥ λ2 (G) ≥ · · · ≥ λn (G), which are always enumerated in non-increasing order and repeated according to their multiplicity. Let X be a column vector with n entries. It will be convenient to associate with X an assignment of G in which vertex vi is assigned xi (or X(vi )). Such assignment is sometimes called vertex valuations of G. For A = (aij ) an n × n1 matrix and B an m × m1 matrix we denote by A ⊗ B the matrix (aij B) (in block partitioned form) and call it the tensor product of A with B. It is easy to see that A ⊗ B is an nm × n1 m1 matrix and Im ⊗ In = Imn , where Im denotes the identity matrix of order m. By the well-known Perron-Frobenius Theorem, for a connected graph G, its largest eigenvalue strictly increases when an edge is added. We are interested in how the other eigenvalues of a connected graph change when edges are added. Examples show that all cases are possible: increased, decreased, unchanged. In this paper, we consider the effect on the eigenvalues by suitably adding edges in particular families, say the family of connected graphs with clusters. By using the result, we also consider the effect on the energy by suitably adding edges to the graphs of the above families.
2
Lemmas and results The following results can be found in ([5] p.408).
Lemma 2.1. If A and C are m × m matrices, B and D are n × n matrices, then (1). (A + C) ⊗ B = (A ⊗ B) + (C ⊗ B); (2). (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD); (3). There exists a permutation matrix P of order mn such that A ⊗ B = P −1 (B ⊗ A)P ; 2
(4). If A is an m×m matrix and B is an n×n matrix, then (A⊗B)−1 = ⊗ B −1 , provided that A−1 and B −1 exist.
A−1
In this section, we use the tensor product of matrices to study how the eigenvalues of connected graphs change by suitably adding edges to the graphs in certain particular manners as described below. Let N (vi ) = {vj : vi vj ∈ E(G)}. es denotes the s-dimensional column vector with all entries 1. We now consider the property of the eigenvector of some eigenvalue of graphs. Lemma 2.2. Suppose that H1 ∼ = H2 ∼ = ··· ∼ = Hs (s ≥ 2) are s disjoint graphs of order t (t ≥ 1) and they are copies of H, V (Hi ) = {vi1 , vi2 , · · · , vit } and for any 1 ≤ i < j ≤ s and 1 ≤ x, y ≤ t, vix viy ∈ E(Hi ) if and only if vjx vjy ∈ E(Hj ). Let G be a graph with vertices v1 , v2 , · · · , vr , Gs (s ≥ 2) be the graph on n = r + st vertices obtained from G and H1 , H2 , · · · , Hs by adding edges between G and Hi (i = 1, 2, · · · , s) satisfying: N (vih ) ∩ V (G) = N (vjh ) ∩ V (G)
(1 ≤ i < j ≤ s;
Let G∗s be the graph obtained from Gs by adding v1i , v2i , · · · , vsi for each i (1 ≤ i ≤ t). Then
s(s−1) 2
1 ≤ h ≤ t).
edges among vertices
(1). Let X be an eigenvector corresponding to some eigenvalue λ(Gs ) of Gs . If λ(Gs ) ∈ / spec(H), where spec(H) denotes the spectrum of A(H), then X(v1i ) = X(v2i ) = · · · = X(vsi ), (i = 1, 2, · · · , t). G∗s .
(2). Let Y be an eigenvector corresponding to some eigenvalue λ(G∗s ) of If λ(G∗s ) ∈ / spec(G∗s − V (G)), then Y (v1i ) = Y (v2i ) = · · · = Y (vsi ), (i = 1, 2, · · · , t).
Proof. Let G0 be the graph obtained from Gs by deleting all the edges among vertices vi1 , vi2 , · · · , vit (i = 1, 2, · · · , s). Giving a suitable ordering for the vertices of Gs , we can assume that A(G0 ) has the following form: ⎡ ⎤ α1r Os×s · · · Os×s α11 · · · ⎢ .. .. .. .. ⎥ .. .. ⎢ . . . . . . ⎥ ⎢ ⎥ ⎢ Os×s · · · Os×s αt1 ⎥ · · · α tr ⎥, A(G0 ) = ⎢ T ⎢ αT ⎥ ⎢ 11 · · · αt1 ⎥ ⎢ .. ⎥ .. .. ⎣ . ⎦ . . A(G) T α1r
···
T αtr
3
where Om×n denotes the m × n zero matrix and αpq = cpq es (either cpq = 1 or cpq = 0 depending on whether or not v1p (1 ≤ p ≤ t) is adjacent to vq (1 ≤ q ≤ r)) . Let C = (cpq )t×r . Thus we have from (1) of Lemma 2.1 that A(H) ⊗ Is Ots×r A(Gs ) = A(G0 ) + Or×ts Or×r A(H) ⊗ Is C ⊗ es , = (C ⊗ es )T A(G) where Is is the unit matrix of order s. Let X T = (xT1 , xT2 , · · · , xTt ; a1 , a2 , · · · , ar ) be an eigenvector of Gs corresponding to λ(Gs ), where x1 , x2 , · · · , xt are column vectors each of them with s components. Then A(Gs )X = λ(Gs )X. Equating the first ts coordinates of the both sides of this equation, we have ⎡ ⎢ ⎢ (A(H) ⊗ Is ) ⎢ ⎣
x1 x2 .. .
⎤
⎡
⎥ ⎢ ⎥ ⎢ ⎥ = −(C ⊗ es ) ⎢ ⎦ ⎣
xt
a1 a2 .. .
⎤
⎡
x1 x2 .. .
⎥ ⎢ ⎥ ⎢ ⎥ + λ(Gs ) ⎢ ⎦ ⎣
ar
So
⎡ ⎢ ⎢ ((A(H) − λ(Gs )It ) ⊗ Is ) ⎢ ⎣
x1 x2 .. .
⎤
⎡
⎥ ⎢ ⎥ ⎢ = −(C ⊗ e ) ⎥ s ⎢ ⎦ ⎣
a1 a2 .. .
⎤ ⎥ ⎥ ⎥. ⎦
(2.1)
ar
⎤
⎡
⎥ ⎢ ⎥ ⎢ ⎥ ⊗ e1 ) = −(C ⎢ ⎦ ⎣
a1 a2 .. .
⎤ ⎥ ⎥ ⎥) ⊗ es . (2.2) ⎦
ar
Substituting Eq. (2.2) into Eq. (2.1), we have ⎤ ⎡ ⎤ ⎡ ⎡ a1 x1 ⎢ a2 ⎥ ⎢ x2 ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ((A(H) − λ(Gs )It ) ⊗ Is ) ⎢ . ⎥ = −(C ⎢ . ⎥) ⊗ es= ⎢ ⎣ .. ⎦ ⎣ .. ⎦ ⎣
f1 f2 .. .
xt
ft
ar 4
⎥ ⎥ ⎥. ⎦
xt
xt From (2) of Lemma 2.1, we have ⎤ ⎡ ⎡ a1 a1 ⎢ a2 ⎥ ⎢ a2 ⎥ ⎢ ⎢ −(C ⊗ es ) ⎢ . ⎥ = −(C ⊗ es )(⎢ . . ⎣ . ⎦ ⎣ .. ar ar
⎤
⎤ ⎥ ⎥ ⎥ ⊗ es , (2.3) ⎦
where f1 , f2 , · · · , ft are column vectors each of them with s components. / spec(H), then M = A(H) − λ(Gs )It is nonsingular. From Eq. If λ(Gs ) ∈ (2.3) and (2), (4) of Lemma 2.1, we have ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ x1 f1 f1 ⎢ x2 ⎥ ⎢ f2 ⎥ ⎢ f2 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ .. ⎥ = (M −1 ⊗ Is )(⎢ .. ⎥ ⊗ es ) = (M −1 ⎢ .. ⎥) ⊗ es . ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ xt
ft
ft
Then xi = ci es (i = 1, 2, · · · , t), where c1 , · · · , ct are some constants. This completes the proof of (1). Now we prove that (2) holds. By similar reasoning as above, we have It ⊗ A(Ks ) + A(H) ⊗ Is C ⊗ es ∗ , A(Gs ) = (C ⊗ es )T A(G) where Ks denotes the complete graph with s vertices. Let Y T = (y1T , y2T , · · · , ytT ; b1 , b2 , · · · , br ) be an eigenvector of G∗s corresponding to λ(G∗s ), where y1 , y2 , · · · , yt are column vectors each of them with s components. Then A(G∗s )Y = λ(G∗s )Y . Equating the first ts coordinates of the both sides of this equation, we have ⎡ ⎢ ⎢ (It ⊗ A(Ks ) + A(H) ⊗ Is ) ⎢ ⎣
y1 y2 .. .
⎤
⎡
⎥ ⎢ ⎥ ⎢ ⎥ = −(C ⊗ es ) ⎢ ⎦ ⎣
yt
b1 b2 .. .
⎤
⎡
⎥ ⎢ ⎥ ⎢ ⎥ + λ(G∗s ) ⎢ ⎦ ⎣
br
So
⎡
⎢ ⎢ (It ⊗ A(Ks ) + A(H) ⊗ Is − λ(G∗s )Ist ) ⎢ ⎣
y1 y2 .. .
br
br 5
⎤
⎤ ⎥ ⎥ ⎥. ⎦
yt
⎤
⎡
⎥ ⎢ ⎥ ⎢ ⎥ = −(C ⊗ es ) ⎢ ⎦ ⎣
yt From (2) of Lemma 2.1, we have ⎤ ⎡ ⎡ b1 b1 ⎢ b2 ⎥ ⎢ b2 ⎥ ⎢ ⎢ −(C ⊗ es ) ⎢ . ⎥ = −(C ⊗ es )(⎢ . . ⎣ . ⎦ ⎣ ..
y1 y2 .. .
b1 b2 .. .
⎤ ⎥ ⎥ ⎥ . (2.4) ⎦
br ⎡
⎥ ⎢ ⎥ ⎢ ⎥ ⊗ e1 ) = −(C ⎢ ⎦ ⎣
b1 b2 .. . br
⎤ ⎥ ⎥ ⎥ ) ⊗ es . ⎦
(2.5)
Substituting Eq. (2.5) into Eq. (2.4), we have ⎡ ⎢ ⎢ (It ⊗ A(Ks ) + A(H) ⊗ Is − λ(G∗s )Ist ) ⎢ ⎣ ⎡ ⎢ ⎢ = −(C ⎢ ⎣
b1 b2 .. .
⎤
⎡
⎥ ⎢ ⎢ ⎥ ) ⊗ e = ⎥ s ⎢ ⎦ ⎣
br
g1 g2 .. .
y1 y2 .. .
⎤ ⎥ ⎥ ⎥ ⎦
yt
⎤ ⎥ ⎥ ⎥ ⊗ es , ⎦
(2.6)
gt
where g1 , g2 , · · · , gt are column vectors each of them with s components. Note that A(G∗s − V (G)) = It ⊗ A(Ks ) + A(H) ⊗ Is . Thus we have if / spec(G∗s − V (G)), then N = It ⊗ A(Ks ) + A(H) ⊗ Is − λ(G∗s )Ist is λ(G∗s ) ∈ nonsingular. From Eq. (2.6) and (2) of Lemma 2.1, we have ⎤ ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ ⎡ f1 g1 g1 y1 ⎢ f2 ⎥ ⎢ g2 ⎥ ⎢ g2 ⎥ ⎢ y2 ⎥ ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎢ ⎢ .. ⎥ = N −1 (⎢ .. ⎥ ⊗ es ) = (N −1 ⊗ e1 )(⎢ .. ⎥ ⊗ es ) = (N −1 ⎢ .. ⎥) ⊗ es . ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ yt
gt
gt
ft
Then yi = di es (i = 1, 2, · · · , t), where d1 , · · · , dt are some constants. This completes the proof of (2). Now we give an example to illustrate the graphs of Lemma 2.2. Example 1. Let Gs be the graph of Fig. 1, Hi : vi1 vi2 vi3 , (i = 1, 2) be a path with length 2. Then Gs (s = 2) is the graph on n = r + st = 5 + 2 × 3 = 11 vertices obtained from G and H1 , H2 by adding edges v5 v13 , v4 v13 ; v5 v23 , v4 v23 between G and Hi (i = 1, 2), where G is the induced subgraph of Gs induced by vertices v1 , v2 , v3 , v4 and v5 . Furthermore G∗s = Gs + v11 v21 + v12 v22 + v13 v23 .
v2 v5 v13 v12 v11 v1 v3 v4 v23 v22 v21 Fig. 1 Gs : s = 2, t = 3, H ∼ = P3 6
Let C n be the complex vector space of complex n−vectors. By the well known Courant-Fischer theorem, we have the following. Lemma 2.3. ([4]) Let A be a Hermitian matrix of order n, the eigenvalues of A be arranged in decreasing order, that is, λ1 ≥ λ2 ≥ · · · ≥ λn , and let Sk be a given k−dimensional subspace of C n (1 ≤ k ≤ n). If there exists a constant c1 such that x∗ Ax ≥ c1 x∗ x for all x ∈ Sk , then λ1 ≥ λ2 ≥ · · · ≥ λk ≥ c1 . If there exists a constant c2 such that x∗ Ax ≤ c2 x∗ x for all x ∈ Sk , then c2 ≥ λn−k+1 ≥ λn−k+2 ≥ · · · ≥ λn . Now, we give the main result of this paper. Theorem 2.1. Let Gs and G∗s be the two graphs defined in Lemma 2.2. Let G be a graph obtained from Gs by adding ti (0 ≤ ti ≤ s(s−1) 2 ) edges among vertices v1i , v2i , · · · , vsi for each i (1 ≤ i ≤ t). Then (1). If {λ1 (Gs ), λ2 (Gs ), · · · , λk (Gs )} ∩ Spec(H) = ∅, then λi (G ) ≥ λi (Gs ), i = 1, 2, · · · , k. (2). If {λn−k+1 (G∗s ), · · · , λn (G∗s )} ∩ Spec(G∗s − V (G)) = ∅, then λi (G ) ≤ λi (G∗s ), i = n − k + 1, · · · , n. Proof. Let Xi be a unit eigenvector corresponding to λi (Gs ) (i = 1, 2, · · · , k) and X1 , · · · , Xk are linearly independent and orthogonal to each other. Let Sk = {X1 , · · · , Xk } which is a k−dimensional subspace of C n (1 ≤ k ≤ n). If x ∈ Sk , then there exist real numbers a1 , · · · , ak such that x = a1 X1 + · · · + ak Xk . It is easy to see that x∗ x = a21 + · · · + a2k , where x∗ is the transpose of x. From Lemma 2.2, we have x∗ A(G )x = (a1 X1∗ + · · · + ak Xk∗ )A(G )(a1 X1 + · · · + ak Xk )
= (a1 X1∗ + · · · + ak Xk∗ )A(Gs )(a1 X1 + · · · + ak Xk )
+2
t
ti [a1 X1 (v1i ) + a2 X2 (v1i ) + · · · + ak Xk (v1i )]2
i=1
≥ (a1 X1∗ + · · · + ak Xk∗ )(λ1 (Gs )a1 X1 + · · · + λk (Gs )ak Xk ) = λ1 (Gs )a21 X1∗ X1 + · · · + λk (Gs )a2k Xk∗ Xk ≥ λk (Gs )a21 X1∗ X1 + · · · + λk (Gs )a2k Xk∗ Xk = λk (Gs )(a21 + · · · + a2k )
= λk (Gs )x∗ x
7
From Lemma 2.3, we have λk (G ) ≥ λk (Gs ). This completes the proof of (1). We now prove that (2) holds. Let Yi be a unit eigenvector corresponding to λi (G∗s ) (i = n − k + 1, n − k + 2, · · · , n) and Yn−k+1 , · · · , Yn are linearly independent and orthogonal to each other. Let Sk = {Yn−k+1 , · · · , Yn } which is a k−dimensional subspace of C n (1 ≤ k ≤ n). If y ∈ Sk , then there exist real numbers bn−k+1 , · · · , bn such that y = bn−k+1 Yn−k+1 + · · · + bn Yn . It is easy to see that y ∗ y = b2n−k+1 + · · · + b2n . From Lemma 2.2, we have ∗ y ∗ A(G )y = (bn−k+1 Yn−k+1 + · · · + bn Yn∗ )A(G )(bn−k+1 Yn−k+1 + · · · + bn Yn )
∗ + · · · + bn Yn∗ )A(G∗s )(bn−k+1 Yn−k+1 + · · · + bn Yn ) = (bn−k+1 Yn−k+1
t
s(s − 1) ( − ti )[bn−k+1 Yn−k+1 (v1i ) + · · · + bn Yn (v1i )]2 −2 2 i=1
∗ ≤ (bn−k+1 Yn−k+1 + · · · + bn Yn∗ )λn−k+1 (G∗s )(bn−k+1 Yn−k+1 + · · ·
+ λn (G∗s )bn Yn )
∗ Yn−k+1 + · · · + λn (G∗s )b2n Yn∗ Yn = λn−k+1 (G∗s )b2n−k+1 Yn−k+1 ∗ ≤ λn−k+1 (G∗s )b2n−k+1 Yn−k+1 Yn−k+1 + · · · + λn−k+1 (G∗ )b2n Yn∗ Yn
= λn−k+1 (G∗s )(b2n−k+1 + · · · + b2n ) = λn−k+1 (G∗s )y ∗ y
From Lemma 2.3, we have λn−k+1 (G ) ≤ λn−k+1 (G∗s ). This completes the proof of (2). Remark: Let Gs be the graph of Fig. 1. Then the graph G in Theorem 2.1 may be one of the following graphs: Gs , Gs + v11 v21 , Gs + v12 v22 , Gs + v13 v23 , Gs + v11 v21 + v12 v22 , Gs + v11 v21 + v13 v23 , Gs + v12 v22 + v13 v23 , G∗s = Gs + v11 v21 + v12 v22 + v13 v23 . Let G be a simple undirected graph of order n. A cluster [2] in G of order k and degree s, is a pair of vertex subsets (C, S), where C is a set of cardinality |C| = k ≥ 2 of pairwise co-neighbor vertices sharing the same set S of s neighbors. Corollary 2.1. Let G be a connected graph on n vertices with a cluster (C, S) of order k and degree s and V (C) = {v1 , v2 , · · · , vk }, G∗k be the graph obtained from G by adding k(k−1) edges among vertices v1 , v2 , · · · , vk and 2 ) edges among G be the graph obtained from G by adding t (0 ≤ t ≤ k(k−1) 2 vertices v1 , v2 , · · · , vk .
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(i). If λi (G) > 0, then λi (G) ≤ λi (G ). (ii). If {k − 1, −1} ∩ {λn−t+1 (G∗k ), · · · , λn (G∗k )} = ∅, then λi (G ) ≤ λi (G∗k ), i = n − t + 1, · · · , n. Proof. By using (1) of Theorem 2.1 for H is an isolated vertex, (i) holds. From (2) of Theorem 2.1 and note that the eigenvalues of Kk are k − 1 and −1 with multiplicity k − 1, respectively, (ii) follows. The energy of G was first defined by Gutman in 1978 as the sum of the absolute values of its eigenvalues: E(G) =
n
|λi (G)|.
i=1
Just as Brualdi [1] pointed out that since Kn does not have maximum energy, it is not true in general that E(G ) ≤ E(G), where G is a spanning subgraph of G. He proposed the following basic problem: when does this inequality hold? By the well-known Perron-Frobenius Theorem and Corollary 2.1, we have the following. Corollary 2.2. Let G be a connected graph with a cluster (C, S) of order k and degree s and V (C) = {v1 , v2 , · · · , vk }, and G be the graph obtained ) edges among vertices v1 , v2 , · · · , vk . from G by adding t (1 ≤ t ≤ k(k−1) 2 Then E(G) < E(G ). Let mG (λ) denote the multiplicity of λ as an eigenvalue of the graph G. In the following, we give some properties of eigenvalues of the graph G∗s . Theorem 2.2. Let Gs (s ≥ 2) be the graph defined in Lemma 2.2 and λ1 (H) ≥ λ2 (H) ≥ · · · ≥ λt (H) be the eigenvalues of the graph H. Then mGs (λk (H)) ≥ (s − 1), k = 1, 2, · · · , t. Proof. Let X be a eigenvector of H1 corresponding to λk (H1 ), (1 ≤ k ≤ t) and Yj (2 ≤ j ≤ s) be a valuation of Gs defined by ⎧ ⎪ ⎨ Y (v1i ) = X(v1i ), i = 1, 2, · · · , t; Y (vji ) = −X(v1i ), i = 1, 2, · · · , t; ⎪ ⎩ Y (u) = 0, u = v11 , v12 , · · · v1t ; vj1 , vj2 , · · · vjt . Then Yj (2 ≤ j ≤ s) is an eigenvector of Gs corresponding to λk (H), (1 ≤ k ≤ t). The results follows. 9
References [1] R. A. Brualdi, Energy of a graph, in: Notes for AIM Workshop on Spectra of Families of Matrices Described by Graphs, Digraphs, and Sign Patterns, 2006 [2] D. M. Cardoso, O. Rojo, Edge perturbation on graphs with clusters: Adjacency, Laplacian and signless Laplacian eigenvalues, Linear Algebra and its Applications, 512 (2017), 113-128. [3] D. M. Cvetkovi´c, M. Doob and H. Sachs, Spectra of Graphs-Theory and Application, Deutscher Verlag der Wissenschaften - Academic Press, Berlin-New York, 1980; second edition 1982; third edition, Johann Ambrosius Barth Verlag, Heidelberg-Leipzig, 1995. [4] R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge University Press, 1986. [5] P. Lancaster, M. Tismenetsky, The Theory of Matrices with Applications. 2d ed., Academic Press, New York, 1985.
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