Discrete Applied Mathematics ( ) –

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Discrete Applied Mathematics

journal homepage: www.elsevier.com/locate/dam

Matrix power inequalities and the number of walks in graphsHanjo Täubig ∗, Jeremias WeihmannInstitut für Informatik, Technische Universität München, Boltzmannstr. 3, D-85748 Garching, Germany

a r t i c l e i n f o

Article history:Received 3 December 2012Received in revised form 30 July 2013Accepted 1 October 2013Available online xxxx

Keywords:InequalitiesMatrix powerSum of entriesAdjacency matrixNumber of walksNonnegative matrixHermitian matrixSpectral radiusLargest eigenvalue

a b s t r a c t

We unify and generalize several inequalities for the number wk of walks of length k ingraphs, and for the entry sum of matrix powers.

First, we present a weighted sandwich theorem for Hermitian matrices which gener-alizes a matrix theorem by Marcus and Newman and which further generalizes our for-mer unification of inequalities for the number of walks in undirected graphs by Lagariaset al. and by Dress and Gutman. The new inequality uses an arbitrary nonnegative weight-ing of the indices (vertices) which allows to apply the theorem to index (vertex) subsets(i.e., inequalities considering the numberwk(S, S) of walks of length k that start at a vertexof a given vertex subset S and that end within the same subset). We also deduce a strongervariation of the sandwich theorem for the case of positive-semidefinite Hermitianmatriceswhich generalizes another inequality of Marcus and Newman.

Further, we show a similar theorem for nonnegative symmetric matrices which is an-other unification and generalization of inequalities for walk numbers by Erdős and Si-monovits, by Dress and Gutman, and by Ilić and Stevanović.

In the last part,we generalize lower bounds for the spectral radius of adjacencymatricesand 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 powersof a matrix to each other. In the special case of an adjacency matrix, the entries of its kth power are the numbers of walksof 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 possibleedges in a complete graph on the same vertex set, the number of walks of length k induces a density of order k: the ratioof the number of k-walks to the maximum possible number of k-walks [20]. Applying our inequalities to this concept ofdensity 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 configurationgraphs. One particular example for such amodel is the symmetric Turingmachinewhichwas defined by Lewis and Papadim-itriou [23] to characterize the complexity class Symmetric Logspace (SL) for which undirected s, t-connectivity (USTCON) isa complete problem. In this context, the number of computation paths consisting of k transitions equals the number of walksof length k in the corresponding configuration graph starting at the initial configuration. Assuming that the configurationgraph is finite, it is also interesting to investigate the total number of different computation path segments of certain lengthsstarting 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: taeubig@in.tum.de (H. Täubig), weihmann@in.tum.de (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 com-putation path segments of other lengths.

A more universal application of counting the number of walks is to exploit their relationship to the largest eigenvalue λ1of adjacency matrices. To this end, we derive new lower bounds for λ1 in terms of the number of walks. In turn, λ1 can beused to bound other important graph measures. In [14], Hoffman obtained the bound 1 − λ1/λn ≤ χ for the chromaticnumber χ , relating it to the ratio of λ1 to the smallest eigenvalue λn. Also, the clique number ω can be bounded usingWilf’sinequality [34] n/(n−λ1) ≤ ω. Another interesting application ofλ1 considers the SISmodel of disease spreading, inwhich asusceptible (S) individual is possibly infected by an already infected (I) neighbor, and subsequentlymay become cured again.If an infected individual infects a certain neighbor with probability β and is cured with probability δ, then the expected sizeof 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 rumorspreading 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 thatN denotes the set of nonnegative integers and that [n] is the set {1, . . . , n}. Let Abe 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 usea[k]i,j to denote the (i, j)-entry of Ak and define ai,j = a[1]

i,j for convenience.Let G = (V , E) be an undirected graph having n vertices,m edges, and adjacencymatrix A. We investigate directedwalks,

i.e., sequences of vertices, where each pair of consecutive vertices is connected by an edge. Nodes and edges can be usedrepeatedly 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) denotethe number of walks of length k that start at vertex x and end at vertex y. Since G is undirected, we havewk(x, y) = wk(y, x).For vertex subsets X, Y ⊆ V ,wk(X, Y ) denotes the number ofwalks of length k starting at a vertex of X and ending at a vertexof Y . We write wk(x) =

y∈V wk(x, y) for the number of walks of length k that start at node x (which is the same as the

number ofwalks of length k that end at node x). Accordingly,wk =

x∈V wk(x)denotes the total number ofwalks 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) = a[k]

i,j .

1.3. Related work

1.3.1. Inequalities for the number of walksFirst, we briefly review results for undirected graphs. Let a, b, c, k, ℓ, p ∈ N be nonnegative integers. Erdős and Si-

monovits (and actually Godsil) [9] noticed that the following inequality using the average degree d = 2m/n can be shownusing results of Mulholland and Smith [28,29], Blakley and Roy [2], and London [24]:

wk ≥ ndk= n

w1

w0

k

or wk1 ≤ wk−1

0 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] reportedthe inequality

w2a+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):

wk2ℓ+p ≤ wk−1

2ℓ · w2ℓ+pk. (5)

For all graphswithw2ℓ > 0 (i.e., for graphswith at least one edge or for ℓ = 0), this is equivalent to (w2ℓ+p/w2ℓ)k≤ 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 ):

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 Hermitianmatrices A:

sumA2a+c

· sumA2a+2b+c

≤ sumA2a

· sumA2(a+b+c) . (8)

In the same paper, Marcus and Newman showed the following inequality for positive-semidefinite symmetric matrices A:sum

Ak+12

≤ sumAk

· sumAk+2 . (9)

1.3.2. Lower bounds for the largest eigenvalueCollatz 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.

Several other publications considered the sum of squares of walk numbers to obtain the lower bounds

v∈V w2(v)2/v∈V d2v ≤ λ2

1 [35],

v∈V w3(v)2/

v∈V w2(v)2 ≤ λ21 [17], and

v∈V wk+1(v)2/

v∈V wk(v)2 ≤ λ2

1 [18], but they did notmention the corresponding number of walks of the double length (w4/w2 ≤ λ2

1, w6/w4 ≤ λ21 and w2k+2/w2k ≤ λ2

1). Theseresults were generalized by Nikiforov [30] to wk+r

wk≤ λr

1 for all r ≥ 1 and even numbers k ≥ 0.1

For the maximum degree ∆, Nosal [31] proved the lower bound√

∆ ≤ λ1. This was generalized by Täubig et al. [32] to

λ1 ≥ maxv∈Vk√

wk(v, v) and λ1 ≥ maxv∈V ,wℓ(v)>0k

w2ℓ+k(v,v)

w2ℓ(v,v).

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×nHermitian matrix. In this case, the sum of all entries of A is a real number. Also the sumof 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]. Bymultiplying rows and columnswith their respective scaling factors,we obtain aHermitianmatrix again, forwhich the sumofall entries is a real number. Of course, the same applies to the powers of thematrix. Accordingly,we define theweighted sum,

sumsAk

= sTAks.

This method allows us, for instance, to calculate the entry sum of a principal submatrix of Ak by using the characteristicvector 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, Acan be diagonalized by a unitary matrix U consisting of n orthonormal eigenvectors of A, i.e., A = UDU∗, where U∗ is theconjugate transpose of U and D is the diagonal matrix containing the corresponding eigenvalues λi. Then we have

a[k]x,y =

ni=1

uxiuyiλki ,

where c denotes the complex conjugate of c ∈ C.For any real scaling vector s ∈ Rn, we define Bi,s =

nx=1 sxuxi for the weighted column sums of U , i.e., the ith entry of

sTU . We know that Ak= (UDU∗)k = UDkU∗. Now, we use the following generalized definitions for entry sums of matrix

powers. For index x ∈ [n], let r [k],sx denote the weighted sum of the terms a[k]

x,y over all y ∈ [n]:

r [k],sx =

ny=1

sya[k]x,y =

ny=1

syn

i=1

uxiuyiλki =

ni=1

uxiλ

ki

ny=1

syuyi

=

ni=1

uxiBi,sλki .

Then, the total weighted sum of the entries is

sumsAk

=

nx=1

sxr [k],sx =

nx=1

sxn

i=1

uxiBi,sλki =

ni=1

Bi,sλ

ki

nx=1

sxuxi

=

ni=1

Bi,sBi,sλki .

2.1. The weighted sandwich theorem

Theorem 1 (Weighted Sandwich Theorem). For all Hermitian matrices A, nonnegative integers a, b, c ∈ N, and scaling vectorss ∈ Rn, the following inequality holds:

sumsA2a+c

· sumsA2a+2b+c

≤ sumsA2a

· sumsA2(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,sBi,s, and consider the difference between both sides of the inequality:

ni=1

B∗

i,sλ2ai

nj=1

B∗

j,sλ2(a+b+c)j −

ni=1

B∗

i,sλ2a+ci

nj=1

B∗

j,sλ2a+2b+cj

=

ni=1

nj=1

B∗

i,sB∗

j,s

λ2ai λ

2(a+b+c)j − λ2a+c

i λ2a+2b+cj

=

n−1i=1

nj=i+1

B∗

i,sB∗

j,s

λ2ai λ

2(a+b+c)j − λ2a+c

i λ2a+2b+cj + λ2a

j λ2(a+b+c)i − λ2a+c

j λ2a+2b+ci

=

n−1i=1

nj=i+1

B∗

i,sB∗

j,sλ2ai λ2a

j

λ2(b+c)j − λc

i λ2b+cj + λ

2(b+c)i − λc

j λ2b+ci

=

n−1i=1

nj=i+1

B∗

i,sB∗

j,sλ2ai λ2a

j

λ2b+cj − λ2b+c

i

λcj − λc

i

.

Note that the product of a complex number and its conjugate is a nonnegative real number. Therefore, each termwithin thelast line must be nonnegative, since B∗

i,s, B∗

j,s, λ2ai , and λ2a

j are all nonnegative, and (λ2b+cj − λ2b+c

i ) and (λcj − λc

i ) must havethe 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 cor-responding principal submatrix of the matrix power, where we denote the sum of the corresponding matrix entries by sumAk

[S, S].

Corollary 2. For all Hermitian matrices A, nonnegative integers a, b, c ∈ N, and subsets S ⊆ [n], the following inequality holds:

sumA2a+c

[S, S]· sum

A2a+2b+c

[S, S]

≤ sumA2a

[S, S]· sum

A2(a+b+c)

[S, S].

Note that, in general, sumAk

[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, sumAk

[S, S]

= wk(S, S) is the number of walksof length k starting and ending at vertices of S, allowing all vertices of V as intermediate vertices. On the other hand, sumA[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 adjacencymatrices, 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 (whichcorresponds to the number of closed walks from v to v in the case of adjacency matrices):

a[2a+c]v,v · a[2a+2b+c]

v,v ≤ a[2a]v,v · a[2(a+b+c)]

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 followinginequality holds:

sumsAa+b

· sumsAa+b+c

≤ sumsAa

· sumsAa+2b+c.

Proof. The proof is essentially the same as for Theorem 1, except that squares of eigenvalues are not required as they arenonnegative in the case of positive-semidefinite matrices.

As in the proof of Theorem 1, we consider the difference between both sides of the inequality:

ni=1

B∗

i,sλai

nj=1

B∗

j,sλa+2b+cj −

ni=1

B∗

i,sλa+bi

nj=1

B∗

j,sλa+b+cj =

n−1i=1

nj=i+1

B∗

i,sB∗

j,sλai λ

aj

λb+cj − λb+c

i

λbj − λb

i

.

Again, B∗

i,s and B∗

j,s are nonnegative numbers. Furthermore, λai and λa

j are nonnegative, and (λb+cj −λb+c

i ) and (λbj −λb

i ) musthave 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 sub-matrices.

Corollary 4. For all positive-semidefinite Hermitian matrices A, numbers a, b, c ∈ N, and subsets S ⊆ [n], the followinginequality holds:

sumAa+b

[S, S]· sum

Aa+b+c

[S, S]

≤ sumAa

[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 inequalityof Dress and Gutman), and of two inequalities by Ilić and Stevanović. Similar to the result of the previous subsection, wewill now generalize (5) to the weighted case. Again, this can be used to obtain inequalities for principal submatrices for anyindex subset S ⊆ [n]. We will use the following theorem of Mulholland and Smith [28,29] (see also Blakley and Roy [2], aswell as Blakley and Dixon [1]).

Theorem 5 (Mulholland and Smith). For any positive integer k, nonnegative real nonzero n-vector v, and nonnegative realsymmetric nonzero n × n-matrix S, the following inequality holds:

⟨v, Sv⟩k≤ ⟨v, v⟩

k−1⟨v, Skv⟩.

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, orif k ≥ 2 in the case that v is the zero vector. Let s be nonnegative and A nonnegative and symmetric. We apply Theorem 5to the nonnegative symmetric matrix S = Ap and the nonnegative vector v consisting of the values r [ℓ],s

x . This yields thefollowing theorem.

Theorem 6. For every nonnegative real symmetric matrix A, nonnegative weight vector s, and k, ℓ, p ∈ N, the followinginequality holds if k ≥ 2 or sums

A2ℓ

> 0:sums

A2ℓ+pk

≤sums

A2ℓk−1

· sumsA2ℓ+pk .

For all matrices with sumsA2ℓ

> 0, this is equivalent tosums

A2ℓ+p

sums

A2ℓ k

≤sums

A2ℓ+pk

sums

A2ℓ and

sums

A2ℓ+p

sums

A2ℓ k−1

≤sums

A2ℓ+pk

sums

A2ℓ+p

.

Corollary 7. For each nonnegative real symmetric matrix A, subset S ⊆ [n], and k, ℓ, p ∈ N, the following inequality holds ifk ≥ 2 or sum

A2ℓ

[S, S]

> 0:sum

A2ℓ+p

[S, S]k

≤sum

A2ℓ

[S, S]k−1

· sumA2ℓ+pk

[S, S].

If the matrix is the adjacencymatrix of a graph G = (V , E) and s is the characteristic vector of a vertex subset S ∈ V , thenr [ℓ],sv 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-kwalks from vertex x to vertex y is multiplied by r [ℓ],s

x and r [ℓ],sy , 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 countingthe 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 lengthk + 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 orw2ℓ(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 tow2ℓ+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−1wpk(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 edgesin 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]:

wk2ℓ+p ≤ wk−1

2ℓ · w2ℓ+pk. (10)Setting k = 2 leads to (3), i.e., the inequality of Dress and Gutman [8]. Furthermore, this inequality is a generalization of thetwo inequalities

M1

n≥

4m2

n2and

M2

m≥

4m2

n2

that were proposed by Ilić and Stevanović [19] for the so-called Zagreb indices M1 =

v∈V d2v and M2 =

{x,y}∈E dxdy.These bounds are equivalent to

w2

w0≥

w21

w20

andw3/2w1/2

≥w2

1

w20.

Additionally, Corollary 8 implies the following special case (via (10) or Corollary 9), which is interesting in its own rightsince it compares the average number of walks (per vertex) of lengths p and pk: wk

p ≤ nk−1wpk andwp

n

k≤

wpkn (k, p ∈ N).

As a special case (ℓ = 0 and p = 1), we get wk1 ≤ wk · wk−1

0 , which is (by w1w0

=2mn = d) precisely (1), i.e., the inequality of

Erdős and Simonovits [9].

3. Bounds for the largest eigenvalue

In the following, we consider powers of an adjacency matrix A. The Perron–Frobenius theorem guarantees that thespectral radius equals the largest eigenvalue. Hence, [λ1(A)]k = λ1(Ak). The Rayleigh–Ritz theorem implies that

λ1(A) = max∥x∥=0

xTAxxT x

.

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 numberof walks of length ℓ from v to any vertex in S (or vice versa). Let wℓ(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)]k = λ1(Ak) ≥wℓ(S)TAkwℓ(S)wℓ(S)T wℓ(S)

=w2ℓ+k(S, 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 ≥ maxS⊆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∈Vk√

wk(v, v), i.e., Theorem 10 is an even more generalform 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 ifwe restrict the walk lengths to even numbers. Correspondingly, we define a family of lower bounds in the case whenw2ℓ(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:

maxS⊆V

Fk+x,ℓ+y(S) ≥ maxS⊆V

Fk,ℓ(S).

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)

w2ℓ(S, S)= F 2

k+1,ℓ ≥ F 2k,ℓ =

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 toverify that

w2(k+2)+2ℓ(S, S)w2ℓ(S, S)

w2(k+1)+2ℓ(S, S)

w2ℓ(S, S)≥

w2(k+1)+2ℓ(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.,

k

w2k+2(ℓ+1)(S, S)w2(ℓ+1)(S, S)

= F 2k,ℓ+1 ≥ F 2

k,ℓ =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 forall 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 boundw2ℓ+pw2ℓ

≤ λp1). In contrast to Lemma 11, these inequalities provide monotonicity statements also for certain oddwalk lengths.

An additional application for the spectral radius lower bounds is new upper bounds for the graph energy, which hasdirect applications for instance in theoretical chemistry. The total π-electron energy Eπ plays a central role in the Hückeltheory of theoretical chemistry. In the case that all molecular orbitals are occupied by two electrons, this energy can bedefined as Eπ = 2

n/2i=1 λi; see [12,13]. For bipartite graphs, this is equal to

ni=1 |λi|, since the spectrum is symmetric and

the total sum of eigenvalues is zero. This motivated the definition of graph energy as E(G) =n

i=1 |λi|. The first bounds forthis 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 connectedgraph G with n ≥ 2 vertices is bounded by

E(G) ≤

v∈V

wk+1(v)2v∈V

wk(v)2+

(n − 1)

2m −

v∈V

wk+1(v)2v∈V

wk(v)2

.

We note that this corresponds to

E(G) ≤

w2k+2

w2k+

(n − 1)

2m −

w2k+2

w2k

.

We now deduce a generalized upper bound for the graph energy, using our lower bound for the spectral radius. Sinceλ1 ≥ 0, the definition of the graph energy can be written as

E(G) = λ1 +

ni=2

|λi| ≤ λ1 +

(n − 1)n

i=2

λ2i ≤ λ1 +

(n − 1)

2m − λ2

1

,

where the inequalities follow fromt

k=1 ak2

≤ t ·t

k=1 a2k and

ni=1 λ2

i = 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√2m−x2

, and is therefore monotonically

decreasing in the interval√2m/n ≤ x ≤

√2m, we have

√2m ≥ λ1 ≥ Fk,ℓ(S) ≥ F1,0(S) =

w2(S, S)w0(S, S)

≥

w1(S, S)w0(S, S)

=

|E(G[S])|

|S|.

Thus, we have f (λ1) ≤ fFk,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

E(G) ≤ f (λ1) ≤ fFk,ℓ(S)

≤

2k

w2k+2ℓ(S, S)w2ℓ(S, S)

+

(n − 1)

2m −

k

w2k+2ℓ(S, S)w2ℓ(S, S)

.

Since w2(S,S)w0(S,S)

≥w1(S,S)2

w0(S,S)2(Corollary 2), the same applies if w1(S,S)

w0(S,S)= d(G[S]) ≥

2mn .

Acknowledgment

We are grateful to Ernst W. Mayr for valuable comments.

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