Relations between degrees, conjugate degrees and graph energies

Accepted Manuscript Relations between degrees, conjugate degrees and graph energies

Kinkar Ch. Das, Seyed Ahmad Mojallal, Ivan Gutman

PII: DOI: Reference:

S0024-3795(16)30525-0 http://dx.doi.org/10.1016/j.laa.2016.11.009 LAA 13926

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Linear Algebra and its Applications

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Please cite this article in press as: K.Ch. Das et al., Relations between degrees, conjugate degrees and graph energies, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2016.11.009

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Relations between Degrees, Conjugate Degrees and Graph Energies Kinkar Ch. Das1 , Seyed Ahmad Mojallal1 and Ivan Gutman2,3,∗ 1

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea, e-mail: [email protected] , ahmad [email protected] 2

3

Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia, e-mail: [email protected]

State University of Novi Pazar, Novi Pazar, Serbia

Received: June 12, 2016 —————————————————————— ——————————————– Abstract

Let G be a simple graph of order n with maximum degree Δ and minimum degree δ. Let (d) = (d1 , d2 , . . . , dn ) and (d∗ ) = (d∗1 , d∗2 , . . . , d∗n ) be the sequences of degrees and conjugate n √ n    di and π ∗ = d∗i , and prove that π ∗ ≤ LEL ≤ IE ≤ π degrees of G. We define π = i=1

i=1

where LEL and IE are, respectively, the Laplacian–energy–like invariant and the incidence √ energy of G. Moreover, we prove that π − π ∗ > ( δ/2)(n − Δ) for a certain class of graphs. Finally, we compare the energy of G and π, and present an upper bound for the Laplacian energy in terms of degree sequence. AMS classification: 05C50, 05C07 Keywords: Degree sequence; Conjugate degree sequence; Energy (of graph); Laplacian energy; Laplacian–energy–like invariant; Incidence energy

1

Introduction

Throughout this paper we are concerned with simple graphs. Let G = (V, E) be such a graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G), where |V (G)| = n and |E(G)| = m. Let di be the degree of vertex vi for i = 1, 2, . . . , n such that d1 ≥ d2 ≥ · · · ≥ dn . The maximum and minimum vertex degrees are denoted by Δ and δ, respectively. The conjugate of a degree sequence (d) = (d1 , d2 , . . . , dn ) is the sequence (d∗ ) = (d∗1 , d∗2 , . . . , d∗n )

1

where d∗i = |{j : dj ≥ i}| [23]. One can easily see that d∗n = 0. As usual, by Kn and Pn we denote the complete graph and the path graph of order n. The adjacency matrix A(G) of G is the matrix whose (i, j)-entry is 1 if vi vj ∈ E(G) and 0 otherwise. Its eigenvalues are denoted by λ1 ≥ λ2 ≥ · · · ≥ λn−1 ≥ λn . The energy of the graph G is defined as n 

E(G) =

|λi | .

i=1

For its basic properties and applications, including lower and upper bounds, see [5,7,9,17,18]. Nikiforov extended the concept of graph energy to of any matrix [24, 25]. For any (not necessarily square) matrix B whose singular values are s1 , s2 , . . . , sn , the energy E(B) of B is defined as E(B) =

n 

si .

i=1

The Laplacian matrix of G is L(G) = D(G) − A(G), where D(G) is the diagonal matrix of vertex degrees. The Laplacian matrix has nonnegative eigenvalues such that n ≥ μ1 ≥ μ2 ≥ · · · ≥ μn = 0. In 2006, the Laplacian energy LE(G) of the graph G was defined as [15]  n    2m   LE = LE(G) = (1) μi − n  . i=1

For its basic properties, including various lower and upper bounds, see [6, 7, 15]. In 2008, Liu and Liu considered a new Laplacian–spectrum–based graph invariant [20] LEL = LEL(G) =

n  √

μi

k=1

and named it Laplacian–energy–like invariant. The motivation for introducing LEL was in its analogy to the earlier much studied graph energy and Laplacian energy. For details on LEL see the review [19], the recent papers [4, 8, 26] and the references cited therein. The signless Laplacian matrix of G is Q(G) = D(G) + A(G). Let q1 , q2 , . . . , qn be the eigenvalues of Q(G). The incidence energy of the graph G (which originally has been defined in a fully different manner, namely as the Nikiforov–type energy of the vertex–edge incidence matrix [16]), satisfies the relation [13] IE(G) =

n  √

qi .

i=1

The mathematical theory of incidence energy is nowadays elaborated in considerable detail, see [3, 13, 14, 16] and the references cited therein.

2

It is well known that n 

qi =

i=1

n 

μi =

n 

i=1

λ2i =

i=1

n 

d∗i =

i=1

n 

di = 2m .

i=1

Akbari et al. [1] proved that IE(G) ≥ LEL(G)

n  √

that is,

qi ≥

i=1

n  √

μi .

(2)

k=1

This result served as a motivation for seeking for analogous relations between pairs of the following invariants: n  √ i=1

qi ,

n  √

μi ,

i=1

n 

|λi | ,

i=1

n  

d∗i ,

i=1

n  

di .

i=1

In what follows, we use the notation n   di π(G) =

and

n   π (G) = d∗i . ∗

i=1

(3)

i=1

The rest of the paper is structured as follows. In Section 2, we state some previously known results, needed for the subsequent considerations. In Section 3, we introduce two definitions and one theorem from the theory of majorization. In Section 4, we obtain a relation between π ∗ (G), LEL(G), IE(G) and π(G). Moreover, we find a lower bound on π(G) − π ∗ (G). In Section 5, we compare graph energy and π(G) of graph G, and we present an upper bound for Laplacian energy of graph G in terms of degree sequence.

2

Preliminaries

In this section, we list some known results that are needed in the next sections. First we recall that the subdivision graph S(G) is the graph obtained by inserting a new vertex into every edge of G. The subdivision graph of any graph is bipartite. In [14], the following simple relation between the energy of a subdivision graph S(G) and the incidence energy of the parent graph G was established: E(S(G)) = 2 IE(G) . The following result is known from matrix theory.

3

(4)

Lemma 2.1. [10] Let A and B be two real square matrices of order n and let C = A + B. Then E(C) ≤ E(A) + E(B) . For some recent applications of this result by Ky Fan see [28] The set of graphs {G1 , G2 , . . . , Gk } forms an edge partition of the graph G if |V (Gi )| = n for i = 1, 2, . . . , k, if the edge sets E(Gi ) are non-empty and mutually disjoint, and if k i=1 E(Gi ) = E(G). By Lemma 2.1, one immediately arrives at the following result. Lemma 2.2. Let the set {G1 , G2 , . . . , Gk } be an edge partition of graph G. Then E(G) ≤

k 

E(Gi ) .

i=1

There is a nice relation between the summation of t largest eigenvalues and the summation of t largest diagonal entries of a real symmetric matrix. Lemma 2.3. [27] Let C be a real symmetric matrix of order n with eigenvalues ρ1 ≥ ρ2 ≥ · · · ≥ ρn and diagonal elements c1 ≥ c2 ≥ · · · ≥ cn . Then t  i=1

ρi ≥

t 

ci , 1 ≤ t ≤ n .

i=1

In 1994, Merris discovered a relation between the Laplacian eigenvalues of a threshold graph and the conjugate of its degree sequence [22]. Later Bai [2] proved a related, more general result, that settled a conjecture of Grone and Merris [11, 12]. Lemma 2.4. [22] Let G be a threshold graph on n vertices with conjugate degree sequence (d∗ ) = (d∗1 , d∗2 , . . . , d∗n ). Then μi = d∗i holds for 1 ≤ i ≤ n. Lemma 2.5. [2] Let G be a graph with degree sequence (d) and conjugate degree sequence (d∗ ). Then k  i=1

μi ≤

k 

d∗i

, 1≤k ≤n−1

and

i=1

n  i=1

4

μi =

n  i=1

d∗i = 2 m .

3

Majorization Theory

In this section, we re-state two definitions and a theorem from the theory of majorization. Throughout this section, we consider vectors x = (x1 , . . . , xn ) ∈ Rn , whose components are aranged in a non-increasing manner: x1 ≥ x2 ≥ · · · ≥ xn . Definition 3.1. [21] For x, y ∈ Rn , with components arranged as specified above, we write x ≺ y if k  i=1

xi ≤

k 

yi , k = 1, 2, . . . , n − 1

and

i=1

n 

xi =

i=1

n 

yi .

i=1

When x ≺ y, then x is said to be majorized by y or that y majorizes x. Definition 3.2. [21] For x, y ∈ Rn , with components arranged as specified above, we write x ≺W y if

k 

xi ≤

i=1

and x ≺

W

k 

yi , k = 1, 2, . . . , n

i=1

y if k 

xn+1−i ≥

i=1

k 

yn+1−i , k = 1, 2, . . . , n .

i=1

In either case, x is said to be weakly majorized by y (or that y weakly majorizes x). More specifically, if x ≺W y , then x is said to be weakly submajorized by y. If x ≺

W

y , then x is

said to be weakly supermajorized by y . Alternatively, we say weakly majorized from below or weakly majorized from above, respectively. Theorem 3.3. [21] For all convex functions g, x ≺ y ⇒ (g(x1 ), . . . , g(xn )) ≺W (g(y1 ), . . . , g(yn )) and for all concave functions g, x ≺ y ⇒ (g(x1 ), . . . , g(xn )) ≺

4

W

(g(y1 ), . . . , g(yn )) .

Relations between graph invariants

In this section we present some relations between π ∗ , LEL, IE, and π of a graph G. One of our main results is the following:

5

Theorem 4.1. Let G be a graph. Then π ∗ (G) ≤ LEL(G) ≤ IE(G) ≤ π(G) .

(5)

Moreover, if G is a threshold graph, then π ∗ (G) = LEL(G).     Proof. By Lemma 2.5 and Definition 3.1, μ1 , μ2 , . . . , μn ≺ d∗1 , d∗2 , . . . , d∗n . By Theorem √ 3.3 and Definition 3.2 with considering the concave function g(x) = x, k  √

μn+1−i ≥

i=1

k 

d∗n+1−i , k = 1, 2, . . . , n

i=1

implying that n  √ i=1

μn+1−i ≥

n 

d∗n+1−i

n  √

i.e.,

i=1

μi ≥

i=1

n   d∗i i=1

that is, LEL(G) ≥ π ∗ (G) . This proves the left inequality in (5). Let S(G) be the subdivision graph of graph G such that |V (S(G))| = n + m and |E(S(G))| = 2 m. We may assume that G1 , G2 , . . . , Gn are spanning subgraphs of S(G) such that

 E(Gi ) = vi x : x ∈ NG (vi ) , i = 1, 2, . . . , n .

Since {G1 , G2 , . . . , Gn } is an edge partition of G, by Lemma 2.2, E(S(G)) ≤ E(G1 ) + E(G2 ) + · · · + E(Gn ) . On the other hand, E(Gi ) = 2



(6)

di

as each subgraph Gi is isomorphic to K1,di ∪ (m + n − di − 1) K1 . Bearing in mind Eq. (6), we get E(S(G)) ≤ 2

n  

di

i=1

which in view of Eqs. (3) and (4) yields IE(G) ≤ π(G) . This completes the proof for the right inequality in (5). Together with the relation (2), the first part of the proof is done. The second part of the proof follows directly from Lemma 2.4.

6

Remark 4.2. Similarly as in the proof of Theorem 4.1, one can easily see that n 

(d∗i )α ≤

i=1

n 

(μi )α ≤

i=1

n 

(qi )α

i=1

holds for 0 < α ≤ 1. From the inequalities (5), we see that the Laplacian–energy–like invariant LEL and incidence energy IE are always located between π ∗ and π, that is, π ∗ ≤ π. We now give a stronger result of this kind, which holds for a particular class Λn of graphs. Let Λn be the set of all graphs G of order n for which either d∗i (G) > di (G)

for i = 1, . . . , Δ

(7)

or there exists a positive number p (1 ≤ p ≤ Δ − 1), such that d∗i (G) > di (G) for i = 1, 2, . . . , p

d∗i (G) ≤ di (G) for i = p + 1, p + 2, . . . , Δ .

and

(8) One can easily check that all regular graphs of order n (for which (7) holds) and all threshold graphs of order n (for which (8) holds, except for Kn ) belong to Λn . However, Λn embraces many more graphs. Let |Gn | be the number of connected graphs of order n and let |Λn | denote the number of connected graphs of order n in Λn . By a computer–aided calculation we obtained the following data:

n 3 4 5 6 7 8

|Gn | 2 6 21 112 853 11117

|Λn | 2 6 21 109 833 10865

These results suggest that also in the general case, Λn may contain the majority of connected graphs of order n. We are now ready to present a lower bound on π(G) − π ∗ (G) for G ∈ Λn .

7

Theorem 4.3. Let G ∈ Λn be a graph of order n with maximum degree Δ and minimum degree δ. Then

√ ∗

π(G) − π (G) >

δ (n − Δ) . 2

Proof. In order to prove this result, suppose that k = Δ. We have d∗1 ≥ d∗2 ≥ · · · ≥ d∗k .   √ √ Since G ∈ Λn , first we assume that (7) holds. Then we have dk + d∗k ≤ di + d∗i , i = 1, 2, . . . , k. Now, k  n     ∗ π(G) − π (G) = ( di − di ) + di ∗

i=1

=

i=k+1

k n    (di − d∗i )  ∗+ √ di di + di i=1 i=k+1

≥ −√

n k    1  ∗ (d∗i − di ) + di dk + dk i=1 i=k+1

= −√

n n    1  ∗ di + di dk + dk i=k+1 i=k+1

>



n  di di − √ 2 dk i=k+1

as

k 

d∗i = 2m

i=1

as d∗k > dk .

Since f (x) =

√

x x− √ , x ≤ dk+1 2 dk

is an increasing function, we have f (x) ≥ f (δ). Therefore, π(G) − π ∗ (G) >



√

δ δ− √ 2 dk

(n − k)

√ ≥

δ (n − Δ) 2

as dk ≥ δ .

Next we assume that (8) holds. Therefore d∗i > di , i = 1, 2, . . . , p and di ≥ d∗i , i = p + 1, p + 2, . . . , k. From this, we get 

and

di +

  d∗i ≥ dp + d∗p , i = 1, 2, . . . , p

   di + d∗i ≤ dp + d∗p , i = p + 1, p + 2, . . . , k .

8

Now, k  n     ( di − d∗i ) + di

π(G) − π ∗ (G) =

i=1

= −

≥ −

i=k+1

p k n     (d∗ − di ) (di − d∗i )  ∗+ √ i  ∗+ √ di di + di di + di i=1 i=p+1 i=k+1 p  i=1



=

k n    (d∗ − d ) (d − d∗i )  i i +  i  + di dp + d∗p i=p+1 dp + d∗p i=k+1

k n    1 (di − d∗i ) + di  ∗ dp + dp i=1 i=k+1

1 = −  dp + d∗p n 

=



n 

>

i=k+1

Similarly as before,



∗

π(G) − π (G) >

√

k 

d∗i

n 

 di

+

i=k+1

di  dp + d∗p

 di di −  2 dp

δ δ−  2 dp

− 2m +

i=1

 di − 

i=k+1





 as

n  

di

i=k+1 k 

d∗i = 2 m

i=1





as d∗p > dp .

√ (n − k) ≥

δ (n − Δ) 2

as dp ≥ δ ,

which completes the proof.

We now define π1 (G) =

    d1 + 1 + d2 + · · · + dn−1 + dn − 1 .

Theorem 4.4. Let G be a graph with degree sequence (d) = (d1 , . . . , dn ). Then LEL(G) ≤ π1 (G) ≤ π(G) . Proof. It is well known that [11, 12] k  i=1

μi ≥

k 

di + 1 , k = 1, 2, . . . , n − 1

i=1

that is, 

   μ1 , μ2 , . . . , μn d1 + 1, d2 , . . . , dn − 1 .

9

(9)

By Theorem 3.3 and Definition 3.2 with concave function g(x) =

√

x, we get LEL(G) ≤

π1 (G), which is the first inequality in (9). One can easily see that 

   d1 + 1, d2 , . . . , dn − 1 d1 , d2 , . . . , dn .

Again by Theorem 3.3 and Definition 3.2 with concave function g(x) =

√

x, one arrives at

π1 (G) ≤ π(G), which completes (9).

5

Estimating graph energies by degree sequences

We now apply Nikiforov’s extended definition of energy [24, 25] to a real symmetric matrix B and the diagonal entries of B 2 . Lemma 5.1. Let B be a real symmetric matrix of order n and let b1 , b2 , . . . , bn be the diagonal entries of the matrix B 2 . Then E(B) ≤

n  

bi .

i=1

Proof. Let the eigenvalues of B be denoted by θ1 , θ2 , . . . , θn . Then the eigenvalues of B 2 are θ12 , θ22 , . . . , θn2 . Without loss of generality, we may assume that θ12 ≥ θ22 ≥ · · · ≥ θn2 and b1 ≥ b2 ≥ · · · ≥ bn . Then by Lemma 2.3, k 

θi2 ≥

i=1

Moreover,

k 

bi , k = 1, 2, . . . , n − 1 .

i=1 n 

θi2 =

i=1

n 

bi = trace(B 2 ) .

i=1

In view of Definition 3.1, we have    θ12 , θ22 , . . . , θn2 b1 , b2 , . . . , bn .



By Theorem 3.3 and Definition 3.2 with considering the concave function g(x) = the above, we get

k  i=1

2 θn+1−i ≤

k  

bn+1−i , k = 1, 2, . . . , n

i=1

10

√

x, from

that is,

n 

2 θn+1−i ≤

i=1

n  

bn+1−i

i=1

that is, E(B) =

n 

|θi | ≤

i=1

n  

bi .

i=1

We now compare E and π. Corollary 5.2. Let G be a graph. Then E(G) ≤ π(G) .

(10)

Proof. Let B = A2 (G), where A(G) is the adjacency matrix of graph G. Then (10) follows by Lemma 5.1. Jooyandeh et al. [16] showed that there exist graphs whose incidence energy is less than or equal to the graph energy. Moreover, they conjectured that E(G) < IE(G) would hold for almost every graph G. Now, by the following example we show that the graph energy E(G) can be (i) between IE(G) and π(G), or (ii) between LEL(G) and IE(G), or (iii) between π ∗ (G) and LEL(G), or (iv) less than π ∗ (G).

G

G

2

1

Fig. 1. The graphs considered in Example 5.3.

Example 5.3. Let G1 and G2 be the graphs depicted in Figure 1. Numerical results (rounded to three decimal places), pertaining to E(G), π ∗ (G), LEL(G), IE(G), and π(G) for the graphs K7 , G1 , G2 , and P7 are listed in the following table.

11

E(G) 12 8.693 8.828 8.054

G K7 G1 G2 P7

π ∗ (G) 15.874 7.613 6.881 4.881

LEL(G) 15.874 9.235 8.513 7.875

IE(G) 16.880 9.633 8.911 7.875

π(G) 17.146 10.292 9.656 9.071

As immediately seen: E(K7 ) < π ∗ (K7 )

,

π ∗ (G1 ) < E(G1 ) < LEL(G1 )

LEL(G2 ) < E(G2 ) < IE(G2 )

,

IE(P7 ) < E(P7 ) < π(P7 ) .

In view of the above examples, we may ask the following questions: Problem 5.4. (1) Characterize all (connected) graphs for which E(G) ≤ π ∗ (G). (2) Characterize all (connected) graphs for which π ∗ (G) ≤ E(G) ≤ LEL(G). (3) Characterize all (connected) graphs for which LEL(G) ≤ E(G) ≤ IE(G). (4) Characterize all (connected) graphs for which IE(G) ≤ E(G) ≤ π(G).

Using Lemma 5.1, we now offer an upper bound for the Laplacian energy of the graph G in terms of n, m, and degree sequence. Theorem 5.5. Let G be a graph of order n with m edges and degree sequence d1 , d2 , . . . , dn . Then LE(G) ≤

n 



i=1

2m di − n

2 + di .

(11)

Proof. Let M = L(G) − 2m n In , where L(G) and In are the Laplacian matrix of graph G and the identity matrix of order n, respectively. Then the eigenvalues of M are μi −2m/n , i = 1, 2, . . . , n. Now,



2

M =

2m L− In n

2

= L2 −

4m 4m2 L + 2 In . n n

Therefore the i-th diagonal entry of M 2 is 2

(M )ii =



4m 4m2 L − L + 2 In n n 2

12

= (L2 )ii − ii

4m 4m2 Lii + 2 (In )ii n n

=

d2i

4m 4 m2 + di − di + 2 = n n



2m di − n

2 + di .

Then, bearing in mind the definition of Laplacian energy, Eq. (1), by Lemma 5.1, 

n  2m 2 di − LE(G) = E(M ) ≤ + di . n i=1

This completes the proof. It is well known that for regular graphs, E = LE and di = 2m/n for all i = 1, 2, . . . , n. Remark 5.6. In the case of regular graphs 

n n    2m 2 di − + di = di = π(G) n i=1

i=1

and therefore the inequalities in (10) and (11) coincide.

Acknowledgement. The first author is supported by the National Research Foundation funded by the Korean government with Grant no. 2013R1A1A2009341.

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