- Email: [email protected]
Matrix power inequalities and the number of walks in graphs
Discrete Applied Mathematics (
)
–
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Matrix power inequalities and the number of walks in graphs Hanjo Täubig ∗ , Jeremias Weihmann Institut für Informatik, Technische Universität München, Boltzmannstr. 3, D-85748 Garching, Germany
article
info
Article history: Received 3 December 2012 Received in revised form 30 July 2013 Accepted 1 October 2013 Available online xxxx Keywords: Inequalities Matrix power Sum of entries Adjacency matrix Number of walks Nonnegative matrix Hermitian matrix Spectral radius Largest eigenvalue
abstract We unify and generalize several inequalities for the number wk of walks of length k in graphs, and for the entry sum of matrix powers. First, we present a weighted sandwich theorem for Hermitian matrices which generalizes a matrix theorem by Marcus and Newman and which further generalizes our former unification of inequalities for the number of walks in undirected graphs by Lagarias et al. and by Dress and Gutman. The new inequality uses an arbitrary nonnegative weighting of the indices (vertices) which allows to apply the theorem to index (vertex) subsets (i.e., inequalities considering the number wk (S , S ) of walks of length k that start at a vertex of a given vertex subset S and that end within the same subset). We also deduce a stronger variation of the sandwich theorem for the case of positive-semidefinite Hermitian matrices which generalizes another inequality of Marcus and Newman. Further, we show a similar theorem for nonnegative symmetric matrices which is another unification and generalization of inequalities for walk numbers by Erdős and Simonovits, by Dress and Gutman, and by Ilić and Stevanović. In the last part, we generalize lower bounds for the spectral radius of adjacency matrices and upper bounds for the energy of graphs. © 2013 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Motivation In this work, we investigate powers of Hermitian matrices. We present inequalities relating entries of different powers of a matrix to each other. In the special case of an adjacency matrix, the entries of its kth power are the numbers of walks of length k between the vertices that correspond to the row/column indices. Similar to the number of present edges in a (sub)graph that is used to define the (statistical) density of this (sub)graph by dividing it through the number of possible edges in a complete graph on the same vertex set, the number of walks of length k induces a density of order k: the ratio of the number of k-walks to the maximum possible number of k-walks [20]. Applying our inequalities to this concept of density yields statements about the relation between densities of different orders. Another application of our results is found in symmetric models of computation, which exhibit undirected configuration graphs. One particular example for such a model is the symmetric Turing machine which was defined by Lewis and Papadimitriou [23] to characterize the complexity class Symmetric Logspace (SL) for which undirected s, t-connectivity (USTCON) is a complete problem. In this context, the number of computation paths consisting of k transitions equals the number of walks of length k in the corresponding configuration graph starting at the initial configuration. Assuming that the configuration graph is finite, it is also interesting to investigate the total number of different computation path segments of certain lengths starting at arbitrary vertices. Bounds could be given in terms of the number of configurations, total number of transitions,
∗
Corresponding author. Tel.: +49 89 289 17740; fax: +49 89 289 17707. E-mail addresses: [email protected] (H. Täubig), [email protected] (J. Weihmann).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.10.002
2
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
number of transitions incident to each configuration, and so on. Other bounds could take into account the number of computation path segments of other lengths. A more universal application of counting the number of walks is to exploit their relationship to the largest eigenvalue λ1 of adjacency matrices. To this end, we derive new lower bounds for λ1 in terms of the number of walks. In turn, λ1 can be used to bound other important graph measures. In [14], Hoffman obtained the bound 1 − λ1 /λn ≤ χ for the chromatic number χ , relating it to the ratio of λ1 to the smallest eigenvalue λn . Also, the clique number ω can be bounded using Wilf’s inequality [34] n/(n −λ1 ) ≤ ω. Another interesting application of λ1 considers the SIS model of disease spreading, in which a susceptible (S) individual is possibly infected by an already infected (I) neighbor, and subsequently may become cured again. If an infected individual infects a certain neighbor with probability β and is cured with probability δ , then the expected size of the infected part of the population reduces exponentially if β/δ < 1/λ1 , i.e., 1/λ1 is the epidemic threshold in this model (see [10,3]). Besides the spreading of viruses in biological and computer networks, this model can also be applied to rumor spreading and information broadcasting. More information on applications of graph spectra can be found in [27,6,4,33]. 1.2. Notation and basic facts Throughout the paper, we assume that N denotes the set of nonnegative integers and that [n] is the set {1, . . . , n}. Let A be an n × n-matrix with complex entries. We write sum(A) for the sum of the entries of A. For the kth power Ak of A, we use [k] [1] ai,j to denote the (i, j)-entry of Ak and define ai,j = ai,j for convenience. Let G = (V , E ) be an undirected graph having n vertices, m edges, and adjacency matrix A. We investigate directed walks, i.e., sequences of vertices, where each pair of consecutive vertices is connected by an edge. Nodes and edges can be used repeatedly in the same walk. The length k of a walk is counted in terms of edges. For k ∈ N and x, y ∈ V , let wk (x, y) denote the number of walks of length k that start at vertex x and end at vertex y. Since G is undirected, we have wk (x, y) = wk (y, x). For vertex subsets X , Y ⊆ V , wk (X , Y ) denotes the number of walks of length k starting at a vertex of X and ending at a vertex of Y . We write wk (x) = length k that start at node x (which is the same as the y∈V wk (x, y) for the number of walks of number of walks of length k that end at node x). Accordingly, wk = x∈V wk (x) denotes the total number of walks of length k.
For the adjacency matrix A of a graph G, we will frequently make use of the equalities wk = sum(Ak ) and wk (i, j) = ai,j . [k]
1.3. Related work 1.3.1. Inequalities for the number of walks First, we briefly review results for undirected graphs. Let a, b, c , k, ℓ, p ∈ N be nonnegative integers. Erdős and Simonovits (and actually Godsil) [9] noticed that the following inequality using the average degree d = 2m/n can be shown using results of Mulholland and Smith [28,29], Blakley and Roy [2], and London [24]:
w1 wk ≥ nd = n w0 k
k or
w1k ≤ w0k−1 wk .
(1)
Lagarias, Mazo, Shepp, and McKay [21,22] showed that
w2a+b · wb ≤ w0 · w2(a+b) ,
(2)
and presented counterexamples for wr · ws ≤ n · wr +s whenever r + s is odd and r , s ≥ 1. Dress and Gutman [8] reported the inequality
wa2+b ≤ w2a · w2b .
(3)
These inequalities were generalized by Täubig et al. [32] to the ‘‘sandwich theorem’’ (for nonnegative integers a, b, c ∈ N):
w2a+c · w2a+2b+c ≤ w2a · w2(a+b+c )
(4)
and the following inequality (for nonnegative integers k, ℓ, p ∈ N and k ≥ 2 or w2ℓ > 0): 1 w2kℓ+p ≤ w2k− ℓ · w2ℓ+pk .
(5)
For all graphs with w2ℓ > 0 (i.e., for graphs with at least one edge or for ℓ = 0), this is equivalent to (w2ℓ+p /w2ℓ ) ≤ w2ℓ+pk / w2ℓ and (w2ℓ+p /w2ℓ )k−1 ≤ w2ℓ+pk /w2ℓ+p . They also showed that similar inequalities are valid for closed walks (for all v ∈ V ): k
w2a+c (v, v) · w2a+2b+c (v, v) ≤ w2a (v, v) · w2(a+b+c ) (v, v)
(6)
and, for k ≥ 2 or w2ℓ (v, v) > 0,
w2ℓ+p (v, v)k ≤ w2ℓ+pk (v, v) · w2ℓ (v, v)k−1 .
(7)
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
3
Later, (4) turned out to be a special case of the following earlier result of Marcus and Newman [25] for Hermitian matrices A: sum A2a+c · sum A2a+2b+c ≤ sum A2a · sum A2(a+b+c ) .
(8)
In the same paper, Marcus and Newman showed the following inequality for positive-semidefinite symmetric matrices A:
sum Ak+1
2
≤ sum Ak · sum Ak+2 .
(9)
1.3.2. Lower bounds for the largest eigenvalue Collatz and Sinogowitz [5] proved that the average degree d = 2m/n ≤ λ1 is a lower bound for the largest eigenvalue of the adjacency matrix. Hofmeister [15,16] later showed that v∈V d2v /n ≤ λ21 . These bounds are equivalent to w1 /w0 ≤ λ1 and w2 /w0 ≤ λ21 . 2 other publications considered the sum of squares of walk numbers to obtain the lower bounds v∈V w2 (v) / Several 2 2 2 2 2 2 2 2 v∈V dv ≤ λ1 [35], v∈V w3 (v) / v∈V w2 (v) ≤ λ1 [17], and v∈V wk+1 (v) / v∈V wk (v) ≤ λ1 [18], but they did not mention the corresponding number of walks of the double length (w4 /w2 ≤ λ21 , w6 /w4 ≤ λ21 and w2k+2 /w2k ≤ λ21 ). These w results were generalized by Nikiforov [30] to wk+r ≤ λr1 for all r ≥ 1 and even numbers k ≥ 0.1 k
For the maximum degree ∆, Nosal [31] proved the lower bound
√ k wk (v, v) and λ1 ≥ maxv∈V ,wℓ (v)>0
λ1 ≥ maxv∈V
k
w2ℓ+k (v,v) . w2ℓ (v,v)
√ ∆ ≤ λ1 . This was generalized by Täubig et al. [32] to
For a survey of bounds of the largest eigenvalue, see [7]. 2. Inequalities for the entry sum of matrix powers
In this section, A denotes an n × n Hermitian matrix. In this case, the sum of all entries of A is a real number. Also the sum of all entries for any principal submatrix is a real number (in particular, this applies to each entry on the main diagonal). More generally, it is possible to define a scaling vector ⃗s ∈ Rn which assigns a real scaling factor si to each index i ∈ [n]. By multiplying rows and columns with their respective scaling factors, we obtain a Hermitian matrix again, for which the sum of all entries is a real number. Of course, the same applies to the powers of the matrix. Accordingly, we define the weighted sum, sum⃗s Ak = ⃗sT Ak⃗s.
This method allows us, for instance, to calculate the entry sum of a principal submatrix of Ak by using the characteristic vector of a subset S in place of ⃗s. A well-known property of any Hermitian matrix A is that all n eigenvalues λ1 ≥ · · · ≥ λn are real numbers. Further, A can be diagonalized by a unitary matrix U consisting of n orthonormal eigenvectors of A, i.e., A = UDU ∗ , where U ∗ is the conjugate transpose of U and D is the diagonal matrix containing the corresponding eigenvalues λi . Then we have a[xk,y] =
n
uxi u¯ yi λki ,
i=1
where c¯ denotes the complex conjugate of c ∈ C. n For any real scaling vector ⃗s ∈ Rn , we define Bi,⃗s = x=1 sx uxi for the weighted column sums of U, i.e., the ith entry of T k ∗ k k ∗ ⃗s U. We know that A = (UDU ) = UD U . Now, we use the following generalized definitions for entry sums of matrix powers. For index x ∈ [n], let rx[k],⃗s denote the weighted sum of the terms a[xk,y] over all y ∈ [n]: [k],⃗s
rx
=
n
sy a[xk,y]
=
n
y =1
sy
y =1
n
¯ λ =
uxi uyi ki
i =1
n
λ
uxi ki
i=1
n
sy u¯ yi
=
y=1
n
uxi B¯ i,⃗s λki .
i=1
Then, the total weighted sum of the entries is
k
sum⃗s A
=
n x =1
[k],⃗s
sx rx
=
n x =1
sx
n i=1
uxi B¯ i,⃗s λki =
n i=1
B¯ i,⃗s λki
n
sx uxi
x =1
=
n
Bi,⃗s B¯ i,⃗s λki .
i=1
2.1. The weighted sandwich theorem Theorem 1 (Weighted Sandwich Theorem). For all Hermitian matrices A, nonnegative integers a, b, c ∈ N, and scaling vectors
⃗s ∈ Rn , the following inequality holds: sum⃗s A2a+c · sum⃗s A2a+2b+c ≤ sum⃗s A2a · sum⃗s A2(a+b+c ) .
1 Note that in Nikiforov’s notation the values for k are odd, since he defines the length of a walk in terms of the number of nodes instead of edges.
4
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
Proof. Let B∗i,⃗s = Bi,⃗s B¯ i,⃗s , and consider the difference between both sides of the inequality: n
B∗i,⃗s λ2a i
n
2(a+b+c )
B∗j,⃗s λj
−
n
=
n n
n
2(a+b+c )
B∗i,⃗s B∗j,⃗s λ2a i λj
B∗j,⃗s λj2a+2b+c
j =1
i=1
j =1
i =1
+c B∗i,⃗s λ2a i
+c 2a+2b+c − λ2a λj i
i=1 j=1
=
n−1 n
2(a+b+c )
B∗i,⃗s B∗j,⃗s λ2a i λj
2(a+b+c ) +c 2a+2b+c +c 2a+2b+c − λ2a λj + λ2a − λ2a λi j λi i j
i=1 j=i+1
=
n−1 n
2(b+c )
2a λj B∗i,⃗s B∗j,⃗s λ2a i λj
+c +c − λci λ2b + λ2i (b+c ) − λcj λ2b j i
i=1 j=i+1 n−1
=
n
+c +c 2a λ2b − λ2b B∗i,⃗s B∗j,⃗s λ2a i λj j i
c λj − λci .
i=1 j=i+1
Note that the product of a complex number and its conjugate is a nonnegative real number. Therefore, each term within the 2b+c 2a − λi2b+c ) and (λcj − λci ) must have last line must be nonnegative, since B∗i,⃗s , B∗j,⃗s , λ2a i , and λj are all nonnegative, and (λj the same sign. Setting ⃗s to the characteristic vector of an index subset S ⊆ [n] gives a relation for the sum of entries restricted to the corresponding principal submatrix of the matrix power, where we denote the sum of the corresponding matrix entries by sum Ak [S , S ] . Corollary 2. For all Hermitian matrices A, nonnegative integers a, b, c ∈ N, and subsets S ⊆ [n], the following inequality holds: sum A2a+c [S , S ] · sum A2a+2b+c [S , S ] ≤ sum A2a [S , S ] · sum A2(a+b+c ) [S , S ] .
Note that, in general, sum Ak [S , S ] is different from sum A[S , S ]k , i.e., the entry sum of the kth power of the principal
submatrix. Applied to the adjacency matrix A of an undirected graph, sum Ak [S , S ] = wk (S , S ) is the number of walks of length k starting and ending at vertices of S, allowing all vertices of V as intermediate vertices. On the other hand, sum A[S , S ]k is the number of walks where all vertices have to be in S, i.e., the number of walks of length k in the subgraph induced by S. The corollary implies that
w2a+c (S , S ) · w2a+2b+c (S , S ) ≤ w2a (S , S ) · w2(a+b+c ) (S , S ). By setting S = [n] (⃗s = 1n ), we obtain (8) (the result of Marcus and Newman [25]) from Corollary 2. For adjacency matrices, this yields [32]
w2a+c · w2a+2b+c ≤ w2a · w2(a+b+c ) . In the case where S contains only one index v , we obtain a statement for the entry of v on the main diagonal (which corresponds to the number of closed walks from v to v in the case of adjacency matrices): 2a+c ] [2a+2b+c ] 2a] 2(a+b+c )] a[v,v · av,v ≤ a[v,v · a[v,v .
Now, we deduce a sandwich theorem for positive-semidefinite Hermitian matrices that generalizes (9). Theorem 3. For all positive-semidefinite Hermitian matrices A, integers a, b, c ∈ N, and weight vectors ⃗s ∈ Rn , the following inequality holds: sum⃗s Aa+b · sum⃗s Aa+b+c ≤ sum⃗s Aa · sum⃗s Aa+2b+c .
Proof. The proof is essentially the same as for Theorem 1, except that squares of eigenvalues are not required as they are nonnegative in the case of positive-semidefinite matrices. As in the proof of Theorem 1, we consider the difference between both sides of the inequality: n i=1
B∗i,⃗s λai
n j =1
B∗j,⃗s λaj +2b+c −
n i =1
B∗i,⃗s λai +b
n j=1
B∗j,⃗s λja+b+c =
n−1 n
B∗i,⃗s B∗j,⃗s λai λaj λjb+c − λib+c
b λj − λbi .
i=1 j=i+1
Again, B∗i,⃗s and B∗j,⃗s are nonnegative numbers. Furthermore, λai and λaj are nonnegative, and (λbj +c − λib+c ) and (λbj − λbi ) must have the same sign since λi , λj ≥ 0. Therefore, each term within the last line must be nonnegative.
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
5
As before, setting ⃗s to the characteristic vector of an index subset S ⊆ [n] yields the following result for principal submatrices. Corollary 4. For all positive-semidefinite Hermitian matrices A, numbers a, b, c ∈ N, and subsets S ⊆ [n], the following inequality holds: sum Aa+b [S , S ] · sum Aa+b+c [S , S ] ≤ sum Aa [S , S ] · sum Aa+2b+c [S , S ] .
2.2. Weighted generalization of the inequalities by Erdős and Simonovits, Dress and Gutman, and Ilić and Stevanović In [32], we verified that (5) was a generalization of (1) (the inequality of Erdős and Simonovits), of (3) (the inequality of Dress and Gutman), and of two inequalities by Ilić and Stevanović. Similar to the result of the previous subsection, we will now generalize (5) to the weighted case. Again, this can be used to obtain inequalities for principal submatrices for any index subset S ⊆ [n]. We will use the following theorem of Mulholland and Smith [28,29] (see also Blakley and Roy [2], as well as Blakley and Dixon [1]). Theorem 5 (Mulholland and Smith). For any positive integer k, nonnegative real nonzero n-vector v ⃗ , and nonnegative real symmetric nonzero n × n-matrix S, the following inequality holds:
⟨⃗v , S v⃗ ⟩k ≤ ⟨⃗v , v⃗ ⟩k−1 ⟨⃗v , S k v⃗ ⟩. Note that this inequality also holds if S is the zero matrix. Furthermore, it holds if k = 0 and v ⃗ is not the zero vector, or if k ≥ 2 in the case that v ⃗ is the zero vector. Let ⃗s be nonnegative and A nonnegative and symmetric. We apply Theorem 5 to the nonnegative symmetric matrix S = Ap and the nonnegative vector v ⃗ consisting of the values rx[ℓ],⃗s . This yields the following theorem. Theorem 6. For every nonnegative real symmetric matrix A, nonnegative weight vector ⃗s, and k, ℓ, p ∈ N, the following inequality holds if k ≥ 2 or sum⃗s A2ℓ > 0:
k−1 ≤ sum⃗s A2ℓ · sum⃗s A2ℓ+pk . For all matrices with sum⃗s A2ℓ > 0, this is equivalent to 2ℓ+pk 2ℓ+p k
sum⃗s A2ℓ+p
k
sum⃗s A
sum⃗s A2ℓ
≤
sum⃗s A
sum⃗s A2ℓ
sum⃗s A2ℓ+p
and
sum⃗s A2ℓ
k−1
sum⃗s A2ℓ+pk
≤
sum⃗s A2ℓ+p
.
Corollary 7. For real symmetric matrix A, subset S ⊆ [n], and k, ℓ, p ∈ N, the following inequality holds if each nonnegative k ≥ 2 or sum A2ℓ [S , S ] > 0:
sum A2ℓ+p [S , S ]
k
k−1 ≤ sum A2ℓ [S , S ] · sum A2ℓ+pk [S , S ] .
If the matrix is the adjacency matrix of a graph G = (V , E ) and ⃗s is the characteristic vector of a vertex subset S ∈ V , then rv[ℓ],⃗s is the vector of walks of length ℓ that start at vertex v and end at a vertex of this subset S. This way, each of the length-k walks from vertex x to vertex y is multiplied by rx[ℓ],⃗s and ry[ℓ],⃗s , i.e., the number of length-ℓ walks starting at a vertex of S and ending at x and the number of length-ℓ walks starting at y and ending at a vertex of S, respectively. This results in counting the walks of length k that are extended at the beginning and at the end by all possible walks of length ℓ, i.e., walks of length k + 2ℓ, that start and end at vertices of S (where the intermediate vertices may also come from V \ S). Corollary 8. For every graph G = (V , E ), vertex subset S ⊆ V , and k, ℓ, p ∈ N, the following inequality holds if k ≥ 2 or w2ℓ (S , S ) > 0:
w2ℓ+p (S , S )k ≤ w2ℓ (S , S )k−1 · w2ℓ+pk (S , S ). For all graphs with w2ℓ (S , S ) > 0, this is equivalent to
w2ℓ+p (S , S ) w2ℓ (S , S )
k ≤
w2ℓ+pk (S , S ) w2ℓ (S , S )
and
w2ℓ+p (S , S ) w2ℓ (S , S )
k−1 ≤
w2ℓ+pk (S , S ) . w2ℓ+p (S , S )
For ℓ = 0, we obtain an inequality which compares the average number of walks (per vertex) of lengths p and pk. Corollary 9. For every graph G = (V , E ), vertex subset S ⊆ V with |S | ≥ 1, and k, p ∈ N, the following inequalities hold:
wp (S , S )k ≤ |S |k−1 wpk (S , S )
and
wp (S , S ) |S |
k ≤
wpk (S , S ) . |S |
6
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
As a special case (ℓ = 0 and p = 1), we obtain w1 (S , S )k ≤ wk (S , S ) · w0 (S , S )k−1 , where w1 (S , S ) is the number of edges in the subgraph induced by S and w1 (S , S )/w0 (S , S ) = w1 (S , S )/|S | is the average degree in this subgraph. If the chosen subset S contains only a single vertex v , then we get a statement about closed walks using v [32]:
w2ℓ+p (v, v)k ≤ w2ℓ (v, v)k−1 · w2ℓ+pk (v, v). If the subset S includes all of the vertices, then we get the following result [32]: 1 (10) w2kℓ+p ≤ w2k− ℓ · w2ℓ+pk . Setting k = 2 leads to (3), i.e., the inequality of Dress and Gutman [8]. Furthermore, this inequality is a generalization of the
two inequalities M1 n
≥
4m2
M2
and
n2
m
≥
4m2 n2
that were proposed by Ilić and Stevanović [19] for the so-called Zagreb indices M1 = These bounds are equivalent to
w2 w2 ≥ 12 w0 w0
v∈V
d2v and M2 =
{x,y}∈E
dx dy .
w3 /2 w2 ≥ 12 . w1 /2 w0
and
Additionally, Corollary 8 implies the following special case (via (10) or Corollary 9), which is interesting in its own right since it compares the average number of walks (per vertex) of lengths p and pk: wpk ≤ nk−1 wpk and As a special case (ℓ = 0 and p = 1), we get w ≤ wk · w Erdős and Simonovits [9].
k−1 , 0
k 1
which is (by
w1 w0
=
2m n
wp k
wpk
(k, p ∈ N). n ¯ = d) precisely (1), i.e., the inequality of n
≤
3. Bounds for the largest eigenvalue In the following, we consider powers of an adjacency matrix A. The Perron–Frobenius theorem guarantees that the spectral radius equals the largest eigenvalue. Hence, [λ1 (A)]k = λ1 (Ak ). The Rayleigh–Ritz theorem implies that
⃗xT A⃗x . ∥⃗x∦=0 ⃗ x xT ⃗ For a vertex subset S ⊆ V and a vertex v ∈ V , let wℓ (S , v) = s∈S wℓ (s, v) = s∈S wℓ (v, s) = wℓ (v, S ) be the number of walks of length ℓ from v to any vertex in S (or vice versa). Let w ⃗ ℓ (S ) denote the vector with entries wℓ (S , v) for all v ∈ V . Then we observe the following for any subset S ⊆ V with wℓ (S ) > 0: λ1 (A) = max
[λ1 (A)]k = λ1 (Ak ) ≥
w ⃗ ℓ (S )T Ak w ⃗ ℓ (S ) w2ℓ+k (S , S ) = . T w ⃗ ℓ (S ) w ⃗ ℓ (S ) w2ℓ (S , S )
Theorem 10. For any graph G = (V , E ), the spectral radius λ1 of the adjacency matrix satisfies the following inequality:
λ1 ≥
max
S ⊆V ,wℓ (S )>0
k
w2ℓ+k (S , S ) . w2ℓ (S , S ) √
The case ℓ = 0 and S = {v}√ corresponds to the form λ1 ≥ maxv∈V k wk (v, v), i.e., Theorem 10 is an even more general form of the lower bound λ1 ≥ ∆ of Nosal [31]. We now show that the new inequality for the spectral radius yields better bounds with increasing walk lengths if we restrict the walk lengths to even numbers. Correspondingly, we define a family of lower bounds in the case when w2ℓ (S , S ) > 0:
Fk,ℓ (S ) =
2k
w2k+2ℓ (S , S ) . w2ℓ (S , S )
Lemma 11. For k, ℓ, x, y ∈ N with k ≥ 1, the following inequality holds: max Fk+x,ℓ+y (S ) ≥ max Fk,ℓ (S ). S ⊆V
S ⊆V
Proof. Let w2ℓ (S , S ) > 0. To show that maxS ⊆V Fk+x,ℓ+y (S ) ≥ maxS ⊆V Fk,ℓ (S ), it is sufficient to verify that Fk+x,ℓ+y (S ) ≥ Fk,ℓ (S ) for all S ⊆ V . First, we show monotonicity in k, i.e.,
k+1
w2(k+1)+2ℓ (S , S ) = Fk2+1,ℓ ≥ Fk2,ℓ = w2ℓ (S , S )
k
w2k+2ℓ (S , S ) . w2ℓ (S , S )
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
7
For the base case k = 1, it is sufficient to show that
w2(1+1)+2ℓ (S , S ) ≥ w2ℓ (S , S )
w2+2ℓ (S , S ) w2ℓ (S , S )
2
.
This inequality is equivalent to w4+2ℓ (S , S ) · w2ℓ (S , S ) ≥ w2+2ℓ (S , S )2 , which follows from Corollary 2. Now we need to verify that
w2(k+2)+2ℓ (S , S ) w2ℓ (S , S )
w2(k+1)+2ℓ (S , S ) w2(k+1)+2ℓ (S , S ) ≥ w2ℓ (S , S ) w2ℓ (S , S )
w2k+2ℓ (S , S ) . w2ℓ (S , S )
This inequality is equivalent to
w2(k+2)+2ℓ (S , S ) · w2k+2ℓ (S , S ) ≥ w2(k+1)+2ℓ (S , S )2 , which again follows from Corollary 2. Next, we show monotonicity in ℓ, i.e.,
w2k+2(ℓ+1) (S , S ) = Fk2,ℓ+1 ≥ Fk2,ℓ = w2(ℓ+1) (S , S )
k
k
w2k+2ℓ (S , S ) . w2ℓ (S , S )
This is equivalent to
w2k+2(ℓ+1) (S , S ) · w2ℓ (S , S ) ≥ w2k+2ℓ (S , S ) · w2(ℓ+1) (S , S ), which follows from Corollary 2. Note that if a subset S has an ℓ0 such that w2ℓ0 (S , S ) > 0, then we have w2ℓ (S , S ) > 0 for all ℓ ≥ ℓ0 . Theorem 6 directly implies the additional monotonicity result
p
w2ℓ+p (S , S ) ≤ w2ℓ (S , S )
pk
w2ℓ+pk (S , S ) w2ℓ (S , S )
for our new bound as well as for the special cases S = {v} (closed walks) and S = V (all walks, i.e., Nikiforov’s bound w2ℓ+p ≤ λp1 ). In contrast to Lemma 11, these inequalities provide monotonicity statements also for certain odd walk lengths. w 2ℓ
An additional application for the spectral radius lower bounds is new upper bounds for the graph energy, which has direct applications for instance in theoretical chemistry. The total π -electron energy Eπ plays a central role in the Hückel theory of theoretical chemistry. In the case that all molecular orbitals are occupied by two electrons, this energy can be n/2 n defined as Eπ = 2 i=1 λi ; see [12,13]. For bipartite graphs, this is equal to i=1 |λi |, since the spectrum is symmetric and n the total sum of eigenvalues is zero. This motivated the definition of graph energy as E (G) = i=1 |λi |. The first bounds for this quantity were given by McClelland [26]:
2m + n(n − 1)|det A|2/n ≤ E (G) ≤
√
2mn.
Later, several other bounds were published [11]. A more recent result is the following [18]. The energy of a connected graph G with n ≥ 2 vertices is bounded by
wk+1 (v)2 wk+1 (v)2 v∈V v∈ V . E (G) ≤ + (n − 1) 2m − wk (v)2 wk (v)2 v∈V
v∈V
We note that this corresponds to E (G) ≤
w2k+2 + w2k
w2k+2 (n − 1) 2m − . w2k
We now deduce a generalized upper bound for the graph energy, using our lower bound for the spectral radius. Since
λ1 ≥ 0, the definition of the graph energy can be written as n n λ2i ≤ λ1 + (n − 1) 2m − λ21 , E (G) = λ1 + |λi | ≤ λ1 + (n − 1) i =2
where the inequalities follow from
i=2
t
k=1
ak
2
≤t·
t
k=1
a2k and
n
i=1
λ2i = 2m.
8
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
√ Since the function f (x) = x + (n − 1)(2m − x2 ) has derivative f ′ (x) = 1 − √ n−1x , and is therefore monotonically 2m−x2 √ √ decreasing in the interval 2m/n ≤ x ≤ 2m, we have √ w2 ( S , S ) w1 (S , S ) |E (G[S ])| ≥ = . 2m ≥ λ1 ≥ Fk,ℓ (S ) ≥ F1,0 (S ) = w0 ( S , S ) w0 (S , S ) |S | Thus, we have f (λ1 ) ≤ f Fk,ℓ (S ) for every set S with average degree d(G[S ]) ≥ d = 2m/n of the induced subgraph G[S ].
For each such set S, this implies that
2k w2k+2ℓ (S , S ) k w2k+2ℓ (S , S ) + (n − 1) 2m − . w2ℓ (S , S ) w2ℓ (S , S ) w (S ,S ) (Corollary 2), the same applies if w1 (S ,S ) = d(G[S ]) ≥ 2m . n 0
E (G) ≤ f (λ1 ) ≤ f Fk,ℓ (S ) ≤
w (S ,S )
w (S ,S )2
Since w2 (S ,S ) ≥ w1 (S ,S )2 0 0 Acknowledgment
We are grateful to Ernst W. Mayr for valuable comments. References [1] G.R. Blakley, R.D. Dixon, Hölder type inequalities in cones, Journal of Mathematical Analysis and Applications 14 (1966) 1–4. [2] G.R. Blakley, P. Roy, A Hölder type inequality for symmetric matrices with nonnegative entries, Proceedings of the American Mathematical Society 16 (1965) 1244–1245. [3] D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec, C. Faloutsos, Epidemic thresholds in real networks, ACM Transactions on Information and System Security 10 (2008) 13:1–13:26. [4] S.M. Cioabă, Some applications of eigenvalues of graphs, in: M. Dehmer (Ed.), Structural Analysis of Complex Networks, Birkhäuser, 2011, pp. 357–379. [5] L. Collatz, U. Sinogowitz, Spektren endlicher Grafen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 21 (1957) 63–77. [6] D.M. Cvetković, I. Gutman (Eds.), Applications of Graph Spectra, in: Zbornik Radova, vol. 13(21), Mathematical Institute SANU, Belgrade, 2009. [7] D.M. Cvetković, P. Rowlinson, The largest eigenvalue of a graph: a survey, Linear and Multilinear Algebra 28 (1990) 3–33. [8] A. Dress, I. Gutman, The number of walks in a graph, Applied Mathematics Letters 16 (2003) 797–801. [9] P. Erdős, M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982) 275–288. [10] A. Ganesh, L. Massoulié, D. Towsley, The effect of network topology on the spread of epidemics, in: Proceedings of the 24th Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM’05, vol. 2, pp. 1455–1466. [11] I. Gutman, The energy of a graph: old and new results, in: Proceedings of the Euroconference Algebraic Combinatorics and Applications, ALCOMA’99, Springer, 2001, pp. 196–211. [12] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π -electron energy of alternant hydrocarbons, Chemical Physics Letters 17 (1972) 535–538. [13] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. The loop rule, Chemical Physics Letters 20 (1973) 257–260. [14] A.J. Hoffman, On eigenvalues and colorings of graphs, in: B. Harris (Ed.), Graph Theory and its Applications, Academic Press, 1970, pp. 79–91. [15] M. Hofmeister, Spectral radius and degree sequence, Mathematische Nachrichten 139 (1988) 37–44. [16] M. Hofmeister, A note on almost regular graphs, Mathematische Nachrichten 166 (1994) 259–262. [17] Y. Hong, X.-D. Zhang, Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees, Discrete Mathematics 296 (2005) 187–197. [18] Y. Hou, Z. Tang, C. Woo, On the spectral radius, k-degree and the upper bound of energy in a graph, MATCH: Communications in Mathematical and in Computer Chemistry 57 (2007) 341–350. [19] A. Ilić, D. Stevanović, On comparing Zagreb indices, MATCH: Communications in Mathematical and in Computer Chemistry 62 (2009) 681–687. [20] S. Kosub, Local density, in: U. Brandes, T. Erlebach (Eds.), Network Analysis—Methodological Foundations, in: Lecture Notes in Computer Science, vol. 3418, Springer, 2005, pp. 112–142. [21] J.C. Lagarias, J.E. Mazo, L.A. Shepp, B.D. McKay, An inequality for walks in a graph, SIAM Review 25 (1983) 403. [22] J.C. Lagarias, J.E. Mazo, L.A. Shepp, B.D. McKay, An inequality for walks in a graph, SIAM Review 26 (1984) 580–582. [23] H.R. Lewis, C.H. Papadimitriou, Symmetric space-bounded computation, Theoretical Computer Science 19 (1982) 161–187. [24] D. London, Inequalities in quadratic forms, Duke Mathematical Journal 33 (1966) 511–522. [25] M. Marcus, M. Newman, The sum of the elements of the powers of a matrix, Pacific Journal of Mathematics 12 (1962) 627–635. [26] B.J. McClelland, Properties of the latent roots of a matrix: the estimation of π -electron energies, The Journal of Chemical Physics 54 (1971) 640–643. [27] B. Mohar, S. Poljak, Eigenvalues in combinatorial optimization, in: Combinatorial and Graph–Theoretical Problems in Linear Algebra, in: The IMA Volumes in Mathematics and its Applications, vol. 50, Springer, 1993, pp. 107–151. [28] H.P. Mulholland, C.A.B. Smith, An inequality arising in genetical theory, The American Mathematical Monthly 66 (1959) 673–683. [29] H.P. Mulholland, C.A.B. Smith, Corrections: an inequality arising in genetical theory, The American Mathematical Monthly 67 (1960) 161. [30] V. Nikiforov, Walks and the spectral radius of graphs, Linear Algebra and its Applications 418 (2006) 257–268. [31] E. Nosal, Eigenvalues of Graphs, Master’s Thesis, University of Calgary, 1970. [32] H. Täubig, J. Weihmann, S. Kosub, R. Hemmecke, E.W. Mayr, Inequalities for the number of walks in graphs, Algorithmica 66 (2013) 804–828. [33] P. Van Mieghem, Graph Spectra for Complex Networks, Cambridge University Press, 2011. [34] H.S. Wilf, Spectral bounds for the clique and independence numbers of graphs, Journal of Combinatorial Theory, Series B 40 (1986) 113–117. [35] A. Yu, M. Lu, F. Tian, On the spectral radius of graphs, Linear Algebra and its Applications 387 (2004) 41–49.
)
–
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Matrix power inequalities and the number of walks in graphs Hanjo Täubig ∗ , Jeremias Weihmann Institut für Informatik, Technische Universität München, Boltzmannstr. 3, D-85748 Garching, Germany
article
info
Article history: Received 3 December 2012 Received in revised form 30 July 2013 Accepted 1 October 2013 Available online xxxx Keywords: Inequalities Matrix power Sum of entries Adjacency matrix Number of walks Nonnegative matrix Hermitian matrix Spectral radius Largest eigenvalue
abstract We unify and generalize several inequalities for the number wk of walks of length k in graphs, and for the entry sum of matrix powers. First, we present a weighted sandwich theorem for Hermitian matrices which generalizes a matrix theorem by Marcus and Newman and which further generalizes our former unification of inequalities for the number of walks in undirected graphs by Lagarias et al. and by Dress and Gutman. The new inequality uses an arbitrary nonnegative weighting of the indices (vertices) which allows to apply the theorem to index (vertex) subsets (i.e., inequalities considering the number wk (S , S ) of walks of length k that start at a vertex of a given vertex subset S and that end within the same subset). We also deduce a stronger variation of the sandwich theorem for the case of positive-semidefinite Hermitian matrices which generalizes another inequality of Marcus and Newman. Further, we show a similar theorem for nonnegative symmetric matrices which is another unification and generalization of inequalities for walk numbers by Erdős and Simonovits, by Dress and Gutman, and by Ilić and Stevanović. In the last part, we generalize lower bounds for the spectral radius of adjacency matrices and upper bounds for the energy of graphs. © 2013 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Motivation In this work, we investigate powers of Hermitian matrices. We present inequalities relating entries of different powers of a matrix to each other. In the special case of an adjacency matrix, the entries of its kth power are the numbers of walks of length k between the vertices that correspond to the row/column indices. Similar to the number of present edges in a (sub)graph that is used to define the (statistical) density of this (sub)graph by dividing it through the number of possible edges in a complete graph on the same vertex set, the number of walks of length k induces a density of order k: the ratio of the number of k-walks to the maximum possible number of k-walks [20]. Applying our inequalities to this concept of density yields statements about the relation between densities of different orders. Another application of our results is found in symmetric models of computation, which exhibit undirected configuration graphs. One particular example for such a model is the symmetric Turing machine which was defined by Lewis and Papadimitriou [23] to characterize the complexity class Symmetric Logspace (SL) for which undirected s, t-connectivity (USTCON) is a complete problem. In this context, the number of computation paths consisting of k transitions equals the number of walks of length k in the corresponding configuration graph starting at the initial configuration. Assuming that the configuration graph is finite, it is also interesting to investigate the total number of different computation path segments of certain lengths starting at arbitrary vertices. Bounds could be given in terms of the number of configurations, total number of transitions,
∗
Corresponding author. Tel.: +49 89 289 17740; fax: +49 89 289 17707. E-mail addresses: [email protected] (H. Täubig), [email protected] (J. Weihmann).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.10.002
2
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
number of transitions incident to each configuration, and so on. Other bounds could take into account the number of computation path segments of other lengths. A more universal application of counting the number of walks is to exploit their relationship to the largest eigenvalue λ1 of adjacency matrices. To this end, we derive new lower bounds for λ1 in terms of the number of walks. In turn, λ1 can be used to bound other important graph measures. In [14], Hoffman obtained the bound 1 − λ1 /λn ≤ χ for the chromatic number χ , relating it to the ratio of λ1 to the smallest eigenvalue λn . Also, the clique number ω can be bounded using Wilf’s inequality [34] n/(n −λ1 ) ≤ ω. Another interesting application of λ1 considers the SIS model of disease spreading, in which a susceptible (S) individual is possibly infected by an already infected (I) neighbor, and subsequently may become cured again. If an infected individual infects a certain neighbor with probability β and is cured with probability δ , then the expected size of the infected part of the population reduces exponentially if β/δ < 1/λ1 , i.e., 1/λ1 is the epidemic threshold in this model (see [10,3]). Besides the spreading of viruses in biological and computer networks, this model can also be applied to rumor spreading and information broadcasting. More information on applications of graph spectra can be found in [27,6,4,33]. 1.2. Notation and basic facts Throughout the paper, we assume that N denotes the set of nonnegative integers and that [n] is the set {1, . . . , n}. Let A be an n × n-matrix with complex entries. We write sum(A) for the sum of the entries of A. For the kth power Ak of A, we use [k] [1] ai,j to denote the (i, j)-entry of Ak and define ai,j = ai,j for convenience. Let G = (V , E ) be an undirected graph having n vertices, m edges, and adjacency matrix A. We investigate directed walks, i.e., sequences of vertices, where each pair of consecutive vertices is connected by an edge. Nodes and edges can be used repeatedly in the same walk. The length k of a walk is counted in terms of edges. For k ∈ N and x, y ∈ V , let wk (x, y) denote the number of walks of length k that start at vertex x and end at vertex y. Since G is undirected, we have wk (x, y) = wk (y, x). For vertex subsets X , Y ⊆ V , wk (X , Y ) denotes the number of walks of length k starting at a vertex of X and ending at a vertex of Y . We write wk (x) = length k that start at node x (which is the same as the y∈V wk (x, y) for the number of walks of number of walks of length k that end at node x). Accordingly, wk = x∈V wk (x) denotes the total number of walks of length k.
For the adjacency matrix A of a graph G, we will frequently make use of the equalities wk = sum(Ak ) and wk (i, j) = ai,j . [k]
1.3. Related work 1.3.1. Inequalities for the number of walks First, we briefly review results for undirected graphs. Let a, b, c , k, ℓ, p ∈ N be nonnegative integers. Erdős and Simonovits (and actually Godsil) [9] noticed that the following inequality using the average degree d = 2m/n can be shown using results of Mulholland and Smith [28,29], Blakley and Roy [2], and London [24]:
w1 wk ≥ nd = n w0 k
k or
w1k ≤ w0k−1 wk .
(1)
Lagarias, Mazo, Shepp, and McKay [21,22] showed that
w2a+b · wb ≤ w0 · w2(a+b) ,
(2)
and presented counterexamples for wr · ws ≤ n · wr +s whenever r + s is odd and r , s ≥ 1. Dress and Gutman [8] reported the inequality
wa2+b ≤ w2a · w2b .
(3)
These inequalities were generalized by Täubig et al. [32] to the ‘‘sandwich theorem’’ (for nonnegative integers a, b, c ∈ N):
w2a+c · w2a+2b+c ≤ w2a · w2(a+b+c )
(4)
and the following inequality (for nonnegative integers k, ℓ, p ∈ N and k ≥ 2 or w2ℓ > 0): 1 w2kℓ+p ≤ w2k− ℓ · w2ℓ+pk .
(5)
For all graphs with w2ℓ > 0 (i.e., for graphs with at least one edge or for ℓ = 0), this is equivalent to (w2ℓ+p /w2ℓ ) ≤ w2ℓ+pk / w2ℓ and (w2ℓ+p /w2ℓ )k−1 ≤ w2ℓ+pk /w2ℓ+p . They also showed that similar inequalities are valid for closed walks (for all v ∈ V ): k
w2a+c (v, v) · w2a+2b+c (v, v) ≤ w2a (v, v) · w2(a+b+c ) (v, v)
(6)
and, for k ≥ 2 or w2ℓ (v, v) > 0,
w2ℓ+p (v, v)k ≤ w2ℓ+pk (v, v) · w2ℓ (v, v)k−1 .
(7)
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
3
Later, (4) turned out to be a special case of the following earlier result of Marcus and Newman [25] for Hermitian matrices A: sum A2a+c · sum A2a+2b+c ≤ sum A2a · sum A2(a+b+c ) .
(8)
In the same paper, Marcus and Newman showed the following inequality for positive-semidefinite symmetric matrices A:
sum Ak+1
2
≤ sum Ak · sum Ak+2 .
(9)
1.3.2. Lower bounds for the largest eigenvalue Collatz and Sinogowitz [5] proved that the average degree d = 2m/n ≤ λ1 is a lower bound for the largest eigenvalue of the adjacency matrix. Hofmeister [15,16] later showed that v∈V d2v /n ≤ λ21 . These bounds are equivalent to w1 /w0 ≤ λ1 and w2 /w0 ≤ λ21 . 2 other publications considered the sum of squares of walk numbers to obtain the lower bounds v∈V w2 (v) / Several 2 2 2 2 2 2 2 2 v∈V dv ≤ λ1 [35], v∈V w3 (v) / v∈V w2 (v) ≤ λ1 [17], and v∈V wk+1 (v) / v∈V wk (v) ≤ λ1 [18], but they did not mention the corresponding number of walks of the double length (w4 /w2 ≤ λ21 , w6 /w4 ≤ λ21 and w2k+2 /w2k ≤ λ21 ). These w results were generalized by Nikiforov [30] to wk+r ≤ λr1 for all r ≥ 1 and even numbers k ≥ 0.1 k
For the maximum degree ∆, Nosal [31] proved the lower bound
√ k wk (v, v) and λ1 ≥ maxv∈V ,wℓ (v)>0
λ1 ≥ maxv∈V
k
w2ℓ+k (v,v) . w2ℓ (v,v)
√ ∆ ≤ λ1 . This was generalized by Täubig et al. [32] to
For a survey of bounds of the largest eigenvalue, see [7]. 2. Inequalities for the entry sum of matrix powers
In this section, A denotes an n × n Hermitian matrix. In this case, the sum of all entries of A is a real number. Also the sum of all entries for any principal submatrix is a real number (in particular, this applies to each entry on the main diagonal). More generally, it is possible to define a scaling vector ⃗s ∈ Rn which assigns a real scaling factor si to each index i ∈ [n]. By multiplying rows and columns with their respective scaling factors, we obtain a Hermitian matrix again, for which the sum of all entries is a real number. Of course, the same applies to the powers of the matrix. Accordingly, we define the weighted sum, sum⃗s Ak = ⃗sT Ak⃗s.
This method allows us, for instance, to calculate the entry sum of a principal submatrix of Ak by using the characteristic vector of a subset S in place of ⃗s. A well-known property of any Hermitian matrix A is that all n eigenvalues λ1 ≥ · · · ≥ λn are real numbers. Further, A can be diagonalized by a unitary matrix U consisting of n orthonormal eigenvectors of A, i.e., A = UDU ∗ , where U ∗ is the conjugate transpose of U and D is the diagonal matrix containing the corresponding eigenvalues λi . Then we have a[xk,y] =
n
uxi u¯ yi λki ,
i=1
where c¯ denotes the complex conjugate of c ∈ C. n For any real scaling vector ⃗s ∈ Rn , we define Bi,⃗s = x=1 sx uxi for the weighted column sums of U, i.e., the ith entry of T k ∗ k k ∗ ⃗s U. We know that A = (UDU ) = UD U . Now, we use the following generalized definitions for entry sums of matrix powers. For index x ∈ [n], let rx[k],⃗s denote the weighted sum of the terms a[xk,y] over all y ∈ [n]: [k],⃗s
rx
=
n
sy a[xk,y]
=
n
y =1
sy
y =1
n
¯ λ =
uxi uyi ki
i =1
n
λ
uxi ki
i=1
n
sy u¯ yi
=
y=1
n
uxi B¯ i,⃗s λki .
i=1
Then, the total weighted sum of the entries is
k
sum⃗s A
=
n x =1
[k],⃗s
sx rx
=
n x =1
sx
n i=1
uxi B¯ i,⃗s λki =
n i=1
B¯ i,⃗s λki
n
sx uxi
x =1
=
n
Bi,⃗s B¯ i,⃗s λki .
i=1
2.1. The weighted sandwich theorem Theorem 1 (Weighted Sandwich Theorem). For all Hermitian matrices A, nonnegative integers a, b, c ∈ N, and scaling vectors
⃗s ∈ Rn , the following inequality holds: sum⃗s A2a+c · sum⃗s A2a+2b+c ≤ sum⃗s A2a · sum⃗s A2(a+b+c ) .
1 Note that in Nikiforov’s notation the values for k are odd, since he defines the length of a walk in terms of the number of nodes instead of edges.
4
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
Proof. Let B∗i,⃗s = Bi,⃗s B¯ i,⃗s , and consider the difference between both sides of the inequality: n
B∗i,⃗s λ2a i
n
2(a+b+c )
B∗j,⃗s λj
−
n
=
n n
n
2(a+b+c )
B∗i,⃗s B∗j,⃗s λ2a i λj
B∗j,⃗s λj2a+2b+c
j =1
i=1
j =1
i =1
+c B∗i,⃗s λ2a i
+c 2a+2b+c − λ2a λj i
i=1 j=1
=
n−1 n
2(a+b+c )
B∗i,⃗s B∗j,⃗s λ2a i λj
2(a+b+c ) +c 2a+2b+c +c 2a+2b+c − λ2a λj + λ2a − λ2a λi j λi i j
i=1 j=i+1
=
n−1 n
2(b+c )
2a λj B∗i,⃗s B∗j,⃗s λ2a i λj
+c +c − λci λ2b + λ2i (b+c ) − λcj λ2b j i
i=1 j=i+1 n−1
=
n
+c +c 2a λ2b − λ2b B∗i,⃗s B∗j,⃗s λ2a i λj j i
c λj − λci .
i=1 j=i+1
Note that the product of a complex number and its conjugate is a nonnegative real number. Therefore, each term within the 2b+c 2a − λi2b+c ) and (λcj − λci ) must have last line must be nonnegative, since B∗i,⃗s , B∗j,⃗s , λ2a i , and λj are all nonnegative, and (λj the same sign. Setting ⃗s to the characteristic vector of an index subset S ⊆ [n] gives a relation for the sum of entries restricted to the corresponding principal submatrix of the matrix power, where we denote the sum of the corresponding matrix entries by sum Ak [S , S ] . Corollary 2. For all Hermitian matrices A, nonnegative integers a, b, c ∈ N, and subsets S ⊆ [n], the following inequality holds: sum A2a+c [S , S ] · sum A2a+2b+c [S , S ] ≤ sum A2a [S , S ] · sum A2(a+b+c ) [S , S ] .
Note that, in general, sum Ak [S , S ] is different from sum A[S , S ]k , i.e., the entry sum of the kth power of the principal
submatrix. Applied to the adjacency matrix A of an undirected graph, sum Ak [S , S ] = wk (S , S ) is the number of walks of length k starting and ending at vertices of S, allowing all vertices of V as intermediate vertices. On the other hand, sum A[S , S ]k is the number of walks where all vertices have to be in S, i.e., the number of walks of length k in the subgraph induced by S. The corollary implies that
w2a+c (S , S ) · w2a+2b+c (S , S ) ≤ w2a (S , S ) · w2(a+b+c ) (S , S ). By setting S = [n] (⃗s = 1n ), we obtain (8) (the result of Marcus and Newman [25]) from Corollary 2. For adjacency matrices, this yields [32]
w2a+c · w2a+2b+c ≤ w2a · w2(a+b+c ) . In the case where S contains only one index v , we obtain a statement for the entry of v on the main diagonal (which corresponds to the number of closed walks from v to v in the case of adjacency matrices): 2a+c ] [2a+2b+c ] 2a] 2(a+b+c )] a[v,v · av,v ≤ a[v,v · a[v,v .
Now, we deduce a sandwich theorem for positive-semidefinite Hermitian matrices that generalizes (9). Theorem 3. For all positive-semidefinite Hermitian matrices A, integers a, b, c ∈ N, and weight vectors ⃗s ∈ Rn , the following inequality holds: sum⃗s Aa+b · sum⃗s Aa+b+c ≤ sum⃗s Aa · sum⃗s Aa+2b+c .
Proof. The proof is essentially the same as for Theorem 1, except that squares of eigenvalues are not required as they are nonnegative in the case of positive-semidefinite matrices. As in the proof of Theorem 1, we consider the difference between both sides of the inequality: n i=1
B∗i,⃗s λai
n j =1
B∗j,⃗s λaj +2b+c −
n i =1
B∗i,⃗s λai +b
n j=1
B∗j,⃗s λja+b+c =
n−1 n
B∗i,⃗s B∗j,⃗s λai λaj λjb+c − λib+c
b λj − λbi .
i=1 j=i+1
Again, B∗i,⃗s and B∗j,⃗s are nonnegative numbers. Furthermore, λai and λaj are nonnegative, and (λbj +c − λib+c ) and (λbj − λbi ) must have the same sign since λi , λj ≥ 0. Therefore, each term within the last line must be nonnegative.
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
5
As before, setting ⃗s to the characteristic vector of an index subset S ⊆ [n] yields the following result for principal submatrices. Corollary 4. For all positive-semidefinite Hermitian matrices A, numbers a, b, c ∈ N, and subsets S ⊆ [n], the following inequality holds: sum Aa+b [S , S ] · sum Aa+b+c [S , S ] ≤ sum Aa [S , S ] · sum Aa+2b+c [S , S ] .
2.2. Weighted generalization of the inequalities by Erdős and Simonovits, Dress and Gutman, and Ilić and Stevanović In [32], we verified that (5) was a generalization of (1) (the inequality of Erdős and Simonovits), of (3) (the inequality of Dress and Gutman), and of two inequalities by Ilić and Stevanović. Similar to the result of the previous subsection, we will now generalize (5) to the weighted case. Again, this can be used to obtain inequalities for principal submatrices for any index subset S ⊆ [n]. We will use the following theorem of Mulholland and Smith [28,29] (see also Blakley and Roy [2], as well as Blakley and Dixon [1]). Theorem 5 (Mulholland and Smith). For any positive integer k, nonnegative real nonzero n-vector v ⃗ , and nonnegative real symmetric nonzero n × n-matrix S, the following inequality holds:
⟨⃗v , S v⃗ ⟩k ≤ ⟨⃗v , v⃗ ⟩k−1 ⟨⃗v , S k v⃗ ⟩. Note that this inequality also holds if S is the zero matrix. Furthermore, it holds if k = 0 and v ⃗ is not the zero vector, or if k ≥ 2 in the case that v ⃗ is the zero vector. Let ⃗s be nonnegative and A nonnegative and symmetric. We apply Theorem 5 to the nonnegative symmetric matrix S = Ap and the nonnegative vector v ⃗ consisting of the values rx[ℓ],⃗s . This yields the following theorem. Theorem 6. For every nonnegative real symmetric matrix A, nonnegative weight vector ⃗s, and k, ℓ, p ∈ N, the following inequality holds if k ≥ 2 or sum⃗s A2ℓ > 0:
k−1 ≤ sum⃗s A2ℓ · sum⃗s A2ℓ+pk . For all matrices with sum⃗s A2ℓ > 0, this is equivalent to 2ℓ+pk 2ℓ+p k
sum⃗s A2ℓ+p
k
sum⃗s A
sum⃗s A2ℓ
≤
sum⃗s A
sum⃗s A2ℓ
sum⃗s A2ℓ+p
and
sum⃗s A2ℓ
k−1
sum⃗s A2ℓ+pk
≤
sum⃗s A2ℓ+p
.
Corollary 7. For real symmetric matrix A, subset S ⊆ [n], and k, ℓ, p ∈ N, the following inequality holds if each nonnegative k ≥ 2 or sum A2ℓ [S , S ] > 0:
sum A2ℓ+p [S , S ]
k
k−1 ≤ sum A2ℓ [S , S ] · sum A2ℓ+pk [S , S ] .
If the matrix is the adjacency matrix of a graph G = (V , E ) and ⃗s is the characteristic vector of a vertex subset S ∈ V , then rv[ℓ],⃗s is the vector of walks of length ℓ that start at vertex v and end at a vertex of this subset S. This way, each of the length-k walks from vertex x to vertex y is multiplied by rx[ℓ],⃗s and ry[ℓ],⃗s , i.e., the number of length-ℓ walks starting at a vertex of S and ending at x and the number of length-ℓ walks starting at y and ending at a vertex of S, respectively. This results in counting the walks of length k that are extended at the beginning and at the end by all possible walks of length ℓ, i.e., walks of length k + 2ℓ, that start and end at vertices of S (where the intermediate vertices may also come from V \ S). Corollary 8. For every graph G = (V , E ), vertex subset S ⊆ V , and k, ℓ, p ∈ N, the following inequality holds if k ≥ 2 or w2ℓ (S , S ) > 0:
w2ℓ+p (S , S )k ≤ w2ℓ (S , S )k−1 · w2ℓ+pk (S , S ). For all graphs with w2ℓ (S , S ) > 0, this is equivalent to
w2ℓ+p (S , S ) w2ℓ (S , S )
k ≤
w2ℓ+pk (S , S ) w2ℓ (S , S )
and
w2ℓ+p (S , S ) w2ℓ (S , S )
k−1 ≤
w2ℓ+pk (S , S ) . w2ℓ+p (S , S )
For ℓ = 0, we obtain an inequality which compares the average number of walks (per vertex) of lengths p and pk. Corollary 9. For every graph G = (V , E ), vertex subset S ⊆ V with |S | ≥ 1, and k, p ∈ N, the following inequalities hold:
wp (S , S )k ≤ |S |k−1 wpk (S , S )
and
wp (S , S ) |S |
k ≤
wpk (S , S ) . |S |
6
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
As a special case (ℓ = 0 and p = 1), we obtain w1 (S , S )k ≤ wk (S , S ) · w0 (S , S )k−1 , where w1 (S , S ) is the number of edges in the subgraph induced by S and w1 (S , S )/w0 (S , S ) = w1 (S , S )/|S | is the average degree in this subgraph. If the chosen subset S contains only a single vertex v , then we get a statement about closed walks using v [32]:
w2ℓ+p (v, v)k ≤ w2ℓ (v, v)k−1 · w2ℓ+pk (v, v). If the subset S includes all of the vertices, then we get the following result [32]: 1 (10) w2kℓ+p ≤ w2k− ℓ · w2ℓ+pk . Setting k = 2 leads to (3), i.e., the inequality of Dress and Gutman [8]. Furthermore, this inequality is a generalization of the
two inequalities M1 n
≥
4m2
M2
and
n2
m
≥
4m2 n2
that were proposed by Ilić and Stevanović [19] for the so-called Zagreb indices M1 = These bounds are equivalent to
w2 w2 ≥ 12 w0 w0
v∈V
d2v and M2 =
{x,y}∈E
dx dy .
w3 /2 w2 ≥ 12 . w1 /2 w0
and
Additionally, Corollary 8 implies the following special case (via (10) or Corollary 9), which is interesting in its own right since it compares the average number of walks (per vertex) of lengths p and pk: wpk ≤ nk−1 wpk and As a special case (ℓ = 0 and p = 1), we get w ≤ wk · w Erdős and Simonovits [9].
k−1 , 0
k 1
which is (by
w1 w0
=
2m n
wp k
wpk
(k, p ∈ N). n ¯ = d) precisely (1), i.e., the inequality of n
≤
3. Bounds for the largest eigenvalue In the following, we consider powers of an adjacency matrix A. The Perron–Frobenius theorem guarantees that the spectral radius equals the largest eigenvalue. Hence, [λ1 (A)]k = λ1 (Ak ). The Rayleigh–Ritz theorem implies that
⃗xT A⃗x . ∥⃗x∦=0 ⃗ x xT ⃗ For a vertex subset S ⊆ V and a vertex v ∈ V , let wℓ (S , v) = s∈S wℓ (s, v) = s∈S wℓ (v, s) = wℓ (v, S ) be the number of walks of length ℓ from v to any vertex in S (or vice versa). Let w ⃗ ℓ (S ) denote the vector with entries wℓ (S , v) for all v ∈ V . Then we observe the following for any subset S ⊆ V with wℓ (S ) > 0: λ1 (A) = max
[λ1 (A)]k = λ1 (Ak ) ≥
w ⃗ ℓ (S )T Ak w ⃗ ℓ (S ) w2ℓ+k (S , S ) = . T w ⃗ ℓ (S ) w ⃗ ℓ (S ) w2ℓ (S , S )
Theorem 10. For any graph G = (V , E ), the spectral radius λ1 of the adjacency matrix satisfies the following inequality:
λ1 ≥
max
S ⊆V ,wℓ (S )>0
k
w2ℓ+k (S , S ) . w2ℓ (S , S ) √
The case ℓ = 0 and S = {v}√ corresponds to the form λ1 ≥ maxv∈V k wk (v, v), i.e., Theorem 10 is an even more general form of the lower bound λ1 ≥ ∆ of Nosal [31]. We now show that the new inequality for the spectral radius yields better bounds with increasing walk lengths if we restrict the walk lengths to even numbers. Correspondingly, we define a family of lower bounds in the case when w2ℓ (S , S ) > 0:
Fk,ℓ (S ) =
2k
w2k+2ℓ (S , S ) . w2ℓ (S , S )
Lemma 11. For k, ℓ, x, y ∈ N with k ≥ 1, the following inequality holds: max Fk+x,ℓ+y (S ) ≥ max Fk,ℓ (S ). S ⊆V
S ⊆V
Proof. Let w2ℓ (S , S ) > 0. To show that maxS ⊆V Fk+x,ℓ+y (S ) ≥ maxS ⊆V Fk,ℓ (S ), it is sufficient to verify that Fk+x,ℓ+y (S ) ≥ Fk,ℓ (S ) for all S ⊆ V . First, we show monotonicity in k, i.e.,
k+1
w2(k+1)+2ℓ (S , S ) = Fk2+1,ℓ ≥ Fk2,ℓ = w2ℓ (S , S )
k
w2k+2ℓ (S , S ) . w2ℓ (S , S )
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
7
For the base case k = 1, it is sufficient to show that
w2(1+1)+2ℓ (S , S ) ≥ w2ℓ (S , S )
w2+2ℓ (S , S ) w2ℓ (S , S )
2
.
This inequality is equivalent to w4+2ℓ (S , S ) · w2ℓ (S , S ) ≥ w2+2ℓ (S , S )2 , which follows from Corollary 2. Now we need to verify that
w2(k+2)+2ℓ (S , S ) w2ℓ (S , S )
w2(k+1)+2ℓ (S , S ) w2(k+1)+2ℓ (S , S ) ≥ w2ℓ (S , S ) w2ℓ (S , S )
w2k+2ℓ (S , S ) . w2ℓ (S , S )
This inequality is equivalent to
w2(k+2)+2ℓ (S , S ) · w2k+2ℓ (S , S ) ≥ w2(k+1)+2ℓ (S , S )2 , which again follows from Corollary 2. Next, we show monotonicity in ℓ, i.e.,
w2k+2(ℓ+1) (S , S ) = Fk2,ℓ+1 ≥ Fk2,ℓ = w2(ℓ+1) (S , S )
k
k
w2k+2ℓ (S , S ) . w2ℓ (S , S )
This is equivalent to
w2k+2(ℓ+1) (S , S ) · w2ℓ (S , S ) ≥ w2k+2ℓ (S , S ) · w2(ℓ+1) (S , S ), which follows from Corollary 2. Note that if a subset S has an ℓ0 such that w2ℓ0 (S , S ) > 0, then we have w2ℓ (S , S ) > 0 for all ℓ ≥ ℓ0 . Theorem 6 directly implies the additional monotonicity result
p
w2ℓ+p (S , S ) ≤ w2ℓ (S , S )
pk
w2ℓ+pk (S , S ) w2ℓ (S , S )
for our new bound as well as for the special cases S = {v} (closed walks) and S = V (all walks, i.e., Nikiforov’s bound w2ℓ+p ≤ λp1 ). In contrast to Lemma 11, these inequalities provide monotonicity statements also for certain odd walk lengths. w 2ℓ
An additional application for the spectral radius lower bounds is new upper bounds for the graph energy, which has direct applications for instance in theoretical chemistry. The total π -electron energy Eπ plays a central role in the Hückel theory of theoretical chemistry. In the case that all molecular orbitals are occupied by two electrons, this energy can be n/2 n defined as Eπ = 2 i=1 λi ; see [12,13]. For bipartite graphs, this is equal to i=1 |λi |, since the spectrum is symmetric and n the total sum of eigenvalues is zero. This motivated the definition of graph energy as E (G) = i=1 |λi |. The first bounds for this quantity were given by McClelland [26]:
2m + n(n − 1)|det A|2/n ≤ E (G) ≤
√
2mn.
Later, several other bounds were published [11]. A more recent result is the following [18]. The energy of a connected graph G with n ≥ 2 vertices is bounded by
wk+1 (v)2 wk+1 (v)2 v∈V v∈ V . E (G) ≤ + (n − 1) 2m − wk (v)2 wk (v)2 v∈V
v∈V
We note that this corresponds to E (G) ≤
w2k+2 + w2k
w2k+2 (n − 1) 2m − . w2k
We now deduce a generalized upper bound for the graph energy, using our lower bound for the spectral radius. Since
λ1 ≥ 0, the definition of the graph energy can be written as n n λ2i ≤ λ1 + (n − 1) 2m − λ21 , E (G) = λ1 + |λi | ≤ λ1 + (n − 1) i =2
where the inequalities follow from
i=2
t
k=1
ak
2
≤t·
t
k=1
a2k and
n
i=1
λ2i = 2m.
8
H. Täubig, J. Weihmann / Discrete Applied Mathematics (
)
–
√ Since the function f (x) = x + (n − 1)(2m − x2 ) has derivative f ′ (x) = 1 − √ n−1x , and is therefore monotonically 2m−x2 √ √ decreasing in the interval 2m/n ≤ x ≤ 2m, we have √ w2 ( S , S ) w1 (S , S ) |E (G[S ])| ≥ = . 2m ≥ λ1 ≥ Fk,ℓ (S ) ≥ F1,0 (S ) = w0 ( S , S ) w0 (S , S ) |S | Thus, we have f (λ1 ) ≤ f Fk,ℓ (S ) for every set S with average degree d(G[S ]) ≥ d = 2m/n of the induced subgraph G[S ].
For each such set S, this implies that
2k w2k+2ℓ (S , S ) k w2k+2ℓ (S , S ) + (n − 1) 2m − . w2ℓ (S , S ) w2ℓ (S , S ) w (S ,S ) (Corollary 2), the same applies if w1 (S ,S ) = d(G[S ]) ≥ 2m . n 0
E (G) ≤ f (λ1 ) ≤ f Fk,ℓ (S ) ≤
w (S ,S )
w (S ,S )2
Since w2 (S ,S ) ≥ w1 (S ,S )2 0 0 Acknowledgment
We are grateful to Ernst W. Mayr for valuable comments. References [1] G.R. Blakley, R.D. Dixon, Hölder type inequalities in cones, Journal of Mathematical Analysis and Applications 14 (1966) 1–4. [2] G.R. Blakley, P. Roy, A Hölder type inequality for symmetric matrices with nonnegative entries, Proceedings of the American Mathematical Society 16 (1965) 1244–1245. [3] D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec, C. Faloutsos, Epidemic thresholds in real networks, ACM Transactions on Information and System Security 10 (2008) 13:1–13:26. [4] S.M. Cioabă, Some applications of eigenvalues of graphs, in: M. Dehmer (Ed.), Structural Analysis of Complex Networks, Birkhäuser, 2011, pp. 357–379. [5] L. Collatz, U. Sinogowitz, Spektren endlicher Grafen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 21 (1957) 63–77. [6] D.M. Cvetković, I. Gutman (Eds.), Applications of Graph Spectra, in: Zbornik Radova, vol. 13(21), Mathematical Institute SANU, Belgrade, 2009. [7] D.M. Cvetković, P. Rowlinson, The largest eigenvalue of a graph: a survey, Linear and Multilinear Algebra 28 (1990) 3–33. [8] A. Dress, I. Gutman, The number of walks in a graph, Applied Mathematics Letters 16 (2003) 797–801. [9] P. Erdős, M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982) 275–288. [10] A. Ganesh, L. Massoulié, D. Towsley, The effect of network topology on the spread of epidemics, in: Proceedings of the 24th Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM’05, vol. 2, pp. 1455–1466. [11] I. Gutman, The energy of a graph: old and new results, in: Proceedings of the Euroconference Algebraic Combinatorics and Applications, ALCOMA’99, Springer, 2001, pp. 196–211. [12] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π -electron energy of alternant hydrocarbons, Chemical Physics Letters 17 (1972) 535–538. [13] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. The loop rule, Chemical Physics Letters 20 (1973) 257–260. [14] A.J. Hoffman, On eigenvalues and colorings of graphs, in: B. Harris (Ed.), Graph Theory and its Applications, Academic Press, 1970, pp. 79–91. [15] M. Hofmeister, Spectral radius and degree sequence, Mathematische Nachrichten 139 (1988) 37–44. [16] M. Hofmeister, A note on almost regular graphs, Mathematische Nachrichten 166 (1994) 259–262. [17] Y. Hong, X.-D. Zhang, Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees, Discrete Mathematics 296 (2005) 187–197. [18] Y. Hou, Z. Tang, C. Woo, On the spectral radius, k-degree and the upper bound of energy in a graph, MATCH: Communications in Mathematical and in Computer Chemistry 57 (2007) 341–350. [19] A. Ilić, D. Stevanović, On comparing Zagreb indices, MATCH: Communications in Mathematical and in Computer Chemistry 62 (2009) 681–687. [20] S. Kosub, Local density, in: U. Brandes, T. Erlebach (Eds.), Network Analysis—Methodological Foundations, in: Lecture Notes in Computer Science, vol. 3418, Springer, 2005, pp. 112–142. [21] J.C. Lagarias, J.E. Mazo, L.A. Shepp, B.D. McKay, An inequality for walks in a graph, SIAM Review 25 (1983) 403. [22] J.C. Lagarias, J.E. Mazo, L.A. Shepp, B.D. McKay, An inequality for walks in a graph, SIAM Review 26 (1984) 580–582. [23] H.R. Lewis, C.H. Papadimitriou, Symmetric space-bounded computation, Theoretical Computer Science 19 (1982) 161–187. [24] D. London, Inequalities in quadratic forms, Duke Mathematical Journal 33 (1966) 511–522. [25] M. Marcus, M. Newman, The sum of the elements of the powers of a matrix, Pacific Journal of Mathematics 12 (1962) 627–635. [26] B.J. McClelland, Properties of the latent roots of a matrix: the estimation of π -electron energies, The Journal of Chemical Physics 54 (1971) 640–643. [27] B. Mohar, S. Poljak, Eigenvalues in combinatorial optimization, in: Combinatorial and Graph–Theoretical Problems in Linear Algebra, in: The IMA Volumes in Mathematics and its Applications, vol. 50, Springer, 1993, pp. 107–151. [28] H.P. Mulholland, C.A.B. Smith, An inequality arising in genetical theory, The American Mathematical Monthly 66 (1959) 673–683. [29] H.P. Mulholland, C.A.B. Smith, Corrections: an inequality arising in genetical theory, The American Mathematical Monthly 67 (1960) 161. [30] V. Nikiforov, Walks and the spectral radius of graphs, Linear Algebra and its Applications 418 (2006) 257–268. [31] E. Nosal, Eigenvalues of Graphs, Master’s Thesis, University of Calgary, 1970. [32] H. Täubig, J. Weihmann, S. Kosub, R. Hemmecke, E.W. Mayr, Inequalities for the number of walks in graphs, Algorithmica 66 (2013) 804–828. [33] P. Van Mieghem, Graph Spectra for Complex Networks, Cambridge University Press, 2011. [34] H.S. Wilf, Spectral bounds for the clique and independence numbers of graphs, Journal of Combinatorial Theory, Series B 40 (1986) 113–117. [35] A. Yu, M. Lu, F. Tian, On the spectral radius of graphs, Linear Algebra and its Applications 387 (2004) 41–49.