Energy of Graphs

Sivaram K. NarayanCentral Michigan University

Presented at CMU on October 10, 2015

1 / 32

Graphs

I We will consider simple graphs (no loops, no multiple edges).

I Let V = {v1, v2, . . . , vn} denote the set of vertices. The edgeset consists of unordered pairs of vertices. We assume G hasm edges.

I Two vertices vi and vj are said to be adjacent if there is anedge {vi , vj} joining them. We denote this by vi ∼ vj .

2 / 32

Adjacency matrix

I Given a graph G with vertex set {v1, v2, . . . , vn} we define theadjacency matrix A = [aij ] of G as follows:

aij =

{1 if vi ∼ vj

0 if vi 6∼ vj

I A is a symmetric (0,1) matrix of trace zero.

I Two graphs G and G′

are isomorphic if and only if there existsa permutation matrix P of order n such that PAPT = A

′.

3 / 32

Adjacency matrix

I A sequence of ` successively adjacent edges is called a walk oflength ` and is denoted by b0, b1, b2, . . . , b`−1, b`. Thevertices b0 and b` are the end points of the walk.

I Let us form

A2 =

[n∑

t=1

aitatj

], i , j = 1, 2, . . . , n.

The element in the (i , j) position of A2 equals the number ofwalks of length 2 with vi and vj as endpoints. The diagonalentries denote the number of closed walks of length 2.

4 / 32

Characteristic polynomial of G

I The polynomial φ(G , λ) = det(λI − A) is called thecharacteristic polynomial of G . The collection of the neigenvalues of A is called the spectrum of G .

Theorem (Sachs Theorem)

Let G be a graph with characteristic polynomial

φ(G , λ) =n∑

k=0

akλn−k . Then for k ≥ 1,

ak =∑S∈Lk

(−1)ω(S)2c(S)

where Lk denotes the set of Sachs subgraphs of G with k vertices,that is, the subgraphs S in which every component is either a K2

or a cycle; ω(S) is the number of connected components of S , andc(S) is the number of cycles contained in S . In addition, a0 = 1.

5 / 32

Spectrum of G

I Since A is symmetric, the spectrum of G consists of n realnumbers λ1 ≥ λ2 ≥ . . . ≥ λn−1 ≥ λn.

I Because λ1 ≥ |λi |, i = 2, 3, . . . , n the eigenvalue λ1 is calledthe spectral radius of G .

I The following relations are easy to establish:

i.n∑

i=1

λi = 0

ii.n∑

i=1

λ2i = 2m

iii.∑i<j

λiλj = −m

6 / 32

Energy of G

The following graph parameter was introduced by Ivan Gutman.

DefinitionIf G is an n-vertex graph and λ1, . . . , λn are its eigenvalues, thenthe energy of G is

E (G ) =n∑

i=1

|λi |.

I The term originates from Quantum Chemistry. In Huckelmolecular orbital theory, the Hamiltonian operator is related tothe adjacency matrix of a pertinently constructed graph. Thetotal π−electron energy has an expression similar to E (G ).

7 / 32

The Coulson Integral Formula

This formula for E (G ) was obtained by Charles Coulson in 1940.

Theorem

E (G ) =1

π

∫ ∞−∞

[n − ıxφ

′(G , ıx)

φ(G , ıx)

]dx

=1

π

∫ ∞−∞

[n − x

d

dxlnφ(G , ıx)

]dx

where G is a graph, φ(G , x) is the characteristic polynomial of G ,φ′(G , x) is its first derivative and∫ ∞

−∞F (x)dx = lim

t→∞

∫ t

−tF (x)dx

(the principal value of the integral).

8 / 32

Bounds for E (G )

Theorem (McClelland,1971)

If G is a graph with n vertices, m edges and adjacency matrix A,then √

2m + n(n − 1)| detA|2n ≤ E (G ) ≤

√2mn.

Proof.

By Cauchy-Schwarz inequality,

(n∑

i=1|λi |)2

≤ nn∑

i=1|λi |2 = 2mn.

9 / 32

Bounds for E (G ), cont’d

Proof cont’d.

Observe that

(n∑

i=1|λi |)2

=n∑

i=1λ2i + 2

∑i<j|λi ||λj |.

Using AM-GM inequality we get

2

n(n − 1)

∑i<j

|λi ||λj | ≥ (∏i<j

|λi ||λj |)2

n(n−1) = (n∏

i=1

|λi |n−1)2

n(n−1)

= (n∏

i=1

|λi |)2n = | det(A)|

2n .

Hence E (G )2 ≥ 2m + n(n − 1)| det(A)|2n .

10 / 32

Bounds for E (G ), cont’d

Corollary

If detA 6= 0, then E (G ) ≥√

2m + n(n − 1) ≥ n.

Also, E (G )2 = 2m + 2∑i<j

|λi ||λj | ≥ 2m + 2|∑i<j

λiλj |

= 2m + 2|−m| = 4m.

Proposition

If G is a graph containing m edges, then 2√m ≤ E (G ) ≤ 2m.

Moreover, E (G ) = 2√m holds if and only if G is a complete

bipartite graph plus arbitrarily many isolated vertices andE (G ) = 2m holds if and only if G is mK2 and isolated vertices.

11 / 32

Strongly regular graphs

DefinitionA graph G is said to be strongly regular with parameters(n, k, λ, µ) whenever G has n vertices, is regular of degree k , everypair of adjacent vertices has λ common neighbors, and every pairof distinct nonadjacent vertices has µ common neighbors.

I In terms of the adjacency matrix A, the definition translatesinto:

A2 = kI + λA + µ(J − A− I)

where J is the all-ones matrix and J − A− I is the adjancencymatrix of the complement of G .

12 / 32

Maximal Energy Graphs

Theorem (Koolen and Moulton)

The energy of a graph G on n vertices is at most n(1 +√n)/2.

Equality holds if and only if G is a strongly regular graph withparameters

(n, (n +√n)/2, (n + 2

√n)/4, (n + 2

√n)/4).

13 / 32

Graphs with extremal energies

I One of the fundamental questions in the study of graphenergy is which graphs from a given class have minimal ormaximal energy.

I Among tree graphs on n vertices the star has minimal energyand the path has maximal energy.

I Equienergetic graphs: non-isomorphic graphs that have thesame energy.

I The smallest pair of equienergetic, noncospectral connectedgraphs of the same order are C5 and W1,4.

14 / 32

Generalization of E (G )

I For a graph G on n vertices, let M be a matrix associatedwith G . Let µ1, . . . , µn be the eigenvalues of M and let

µ =tr(M)

n

be the average of µ1, . . . , µn. The M-energy of G is thendefined as

EM(G ) :=n∑

i=1

|µi − µ|.

I For adjacency matrix A, EA(G ) = E (G ) since tr(A) = 0.

15 / 32

Laplacian matrix

I The classical Laplacian matrix of a graph G on n vertices isdefined as

L(G ) = D(G )− A(G )

where D(G ) = diag(deg(v1), . . . , deg(vn)) and A(G ) is theadjacency matrix.

I The normalized Laplacian matrix, L(G ), of a graph G (withno isolated vertices) is given by

Lij =

1 if i = j

− 1√didj

if vi ∼ vj

0 otherwise.

16 / 32

Laplacian Energy

DefinitionLet µ1, . . . , µn be the eigenvalues of L(G ). Then the Laplacianenergy LE (G ), is defined as

LE (G ) :=n∑

i=1

∣∣∣∣µi − 2m

n

∣∣∣∣ .DefinitionLet µ1, . . . , µn be the eigenvalues of the normalized Laplacianmatrix L(G ). The normalized Laplacian energy NLE (G ), is definedas

NLE (G ) :=n∑

i=1

|µi − 1|.

17 / 32

Remarks on LE (G )

I If the graph G consists of components G1 and G2, thenE (G ) = E (G1) + E (G2).

I If the graph G consists of components G1 and G2, and if G1

and G2 have equal average vertex degrees, thenLE (G ) = LE (G1) + LE (G2). Otherwise, the equality need nothold.

I LE (G ) ≥ E (G ) holds for bipartite graphs.

I LE (G ) ≤ 2m

n+

√√√√(n − 1)

[2M −

(2m

n

)2]

where M = m + 12

n∑i=1

(di − 2mn )2.

I 2√M ≤ LE (G ) ≤ 2M

18 / 32

Edge deletion

I Let H be a subgraph of G . We denote by G −H the subgraphof G obtained by removing the vertices of H. We denote byG − E (H) the subgraph of G obtained by deleting all edges ofH but retaining all vertices of H.

I Theorem (L. Buggy, A. Culiuc, K. McCall, N, D. Nguyen)

Let H be an induced subgraph of a graph G . Suppose H is theunion of H and vertices of G − H as isolated vertices. Then

LE (G )− LE (H) ≤ LE (G − E (H)) ≤ LE (G ) + LE (H)

where E (H) is the edge set of H.

19 / 32

Singular values of a matrix

I The singular values s1(A) ≥ s2(A) ≥ . . . sm(A) of a m × nmatrix A are the square roots of the eigenvalues of AA∗.

I Note that if A ∈ Mn is a Hermitian (or real symmetric) matrixwith eigenvalues µ1, . . . , µn then the singular values of A arethe moduli of µi .

I Proof of the previous theorem uses the following Ky Fan’sinequality for singular values.

Theorem (Ky Fan)

Let X ,Y , and Z be in Mn(C) such that X + Y = Z . Then

n∑i=1

si (X ) +n∑

i=1

si (Y ) ≥n∑

i=1

si (Z ).

20 / 32

Edge deletion

Corollary

Suppose H is a single edge e of G and H consists of e and n − 2isolated vertices. Then

LE (G )− 4(n − 1)

n≤ LE (G − e) ≤ LE (G ) +

4(n − 1)

n.

21 / 32

Join of Graphs

I The join of graph G with graph H, denoted G ∨ H is thegraph obtained from the disjoint union of G and H by addingthe edges {{x , y} : x ∈ V (G ), y ∈ V (H)}.

Theorem (A. Hubbard, N, C. Woods)

Let G be a r -regular graph on n vertices and H be s-regular graphon p vertices. Then

NLE (G ∨ H) =r

p + rNLE (G ) +

s

n + sNLE (H) +

p − r

p + r+

n − s

n + s.

22 / 32

Shadow Graph

I Let G be a graph with vertex set V = {v1, v2, . . . , vn}. Definethe shadow graph S(G ) of G to be the graph with vertex set

V ∪ {u1, u2, . . . , un}

and edge set

E (G ) ∪ {{ui , vj} : {vi , vj} ∈ E (G )}.

Theorem (Hubbard, N, Woods)

E (Sp(G )) =√

4p + 1E (G ) for any graph G

NLE (Sp(G )) =2p + 1

p + 1NLE (G ) for any regular graph G.

23 / 32

Distance energy

I Let G be a connected graph with vertex setV (G ) = {v1, v2, . . . , vn}. The distance matrix D(G ) of G isdefined so that the (i , j) entry is equal to dG (vi , vj) wheredistance is the length of the shortest path between thevertices vi and vj .

I The distance energy DE (G ) is defined as

DE (G ) =n∑

i=1

|µi |

where µ1 ≥ µ2 ≥ . . . ≥ µn are the eigenvalues of D. (Notethat D is a real symmetric matrix with trace zero.)

24 / 32

An upper bound for DE (G )

I The distance degree Di of vi is Di :=n∑

j=1dij .

I The second distance degree Ti of vi is Ti =n∑

j=1dijDj .

I Theorem (G. Indulal)

DE (G ) ≤

√√√√√√√n∑

i=1T 2i

n∑i=1

D2i

+ (n − 1)

√√√√√√√S −

n∑i=1

T 2i

n∑i=1

D2i

where S is the sum of the squares of entries in the distance matrix.

25 / 32

Vertex sum of G and H

I Let G and H are two graphs with u ∈ V (G ) and v ∈ V (H).We define the vertex sum of G and H, denoted G ◦ H, to bethe graph obtained by identifying the vertices u and v .

I Theorem (Buggy, Culiuc, McCall, N, Nguyen)

DE (G ◦ H) ≤ DE (G ) + DE (H) and equality holds if and only if uor v is an isolated vertex.

I

√n2(n − 1)(n + 1)

6≤ DE (Pn) ≤

√n3(n − 1)(n + 1)

6

I DE (Sn) ≤ DE (Pn)

26 / 32

Energy of a matrix

I Let A ∈ Mm,n(C) and let s1(A) ≥ s2(A) ≥ . . . ≥ sm(A) be thesingular values of A. Define the energy of A as

E(A) =m∑j=1

sj .

I E (G ) = E(A(G )).

Theorem (V. Nikiforov)

If m ≤ n, A is an m × n nonnegative matrix with maximum entryα and ||A||1 :=

∑i ,j|aij | ≥ nα, then

E(A) ≤ ||A||1√mn

+

√(m − 1)

(tr(AA∗)−

||A||21mn

)≤ α√n(m +

√m)

2.

27 / 32

Incidence energy

I For a graph G with vertex set {v1, . . . , vn} and edge set{e1, e2, . . . , em}, the (i , j)th entry of the incidence matrixI(G ) is 1 if vi is incident with ej and 0 otherwise. I(G ) is avertex-edge incidence matrix.

I If the singular values of I(G ) are σ1, σ2, . . . , σn then defineincidence energy as

IE (G ) =n∑

i=1

σi .

I I(G )I(G )T = D(G ) + A(G ) = L+(G ) called the signless

Laplacian of G . Therefore IE (G ) =n∑

i=1

õ+i where

µ+1 , . . . , µ+n are the eigenvalues of the signless Laplacian

matrix.

28 / 32

BIBD

I A balanced incomplete block design BIBD(v , b, r , k , λ) is apair (V ,B) where V is a v -set of points, B is a collection ofk subsets of V called blocks such that any pair of distinctpoints occur in exactly λ blocks. Here b is the number ofblocks and r is the number of blocks containing each point.

I The incidence matrix of a BIBD is a (0,1)-matrix whose rowsand columns are indexed by the points and the blocks,respectively, and the entry (p,B) is 1 if and only if p ∈ B.

29 / 32

Energy of (0,1) matrices

Theorem (H. Kharaghani and B. Tayfeh-Rezaie)

Let M be a p × q (0,1) matrix with m ones, where m ≥ q ≥ p.Then

E(M) ≤ m√pq

+

√(p − 1)(m − m2

pq).

The equality is attained if and only if M is the incidence matrix ofa BIBD.

Theorem (H. Kharaghani and B. Tayfeh-Rezaie)

Let G be a (p, q)-bipartite graph. Then E (G ) ≤ (√p + 1)

√pq.

The equality is attained if and only if G is the incidence graph of a

BIBD(p, q, q(p +√p)/2p, (p +

√p)/2, q(p + 2

√p)/4p).

30 / 32

References

Xueliang Li, Yongtang Shi and Ivan Gutman

Graph Energy

Springer, New York 2010

H. Kharaghani, B. Tafyeh-Rezaie

On the Energy of (0,1) matrices

Linear Algebra and its Applications 429(2008), 2046-2051

V. Nikiforov

The energy of graphs and matrices

J. Math. Anal.Appl. 326(2007), 1472-1475

I. Gutman

The energy of graphs: Old and New Results, Algebraic Combinatorics andApplications

Springer, Berlin 2001, 196-211

J.H. Koolen, V. Moulton

Maximal energy graphs

Adv. Appl. Math.26, 2001, 47-52

31 / 32

J.H. Koolen, V. Moulton

Maximal energy bipartite graphs

Graphs Combin., 19 (2003), 131-135

G. Indulal

Sharp bounds on the distance spectral radius and the distance energy ofgraphs

Linear Alg. Appln., 430 (2009), 106-113

W.H. Haemers

Strongly regular graphs with maximal energy

Linear Alg. Appln., 429 (2008), 2719-2723

I. Gutman, B. Zhou

Laplacian energy of a graph

Linear Alg. Appln., 414 (2006), 29-37

32 / 32