Energy of Graphs Sivaram K. Narayan Central Michigan University - - PowerPoint PPT Presentation

Energy of Graphs

Sivaram K. Narayan Central Michigan University Presented at CMU on October 10, 2015

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Graphs

◮ We will consider simple graphs (no loops, no multiple edges). ◮ Let V = {v1, v2, . . . , vn} denote the set of vertices. The edge

set consists of unordered pairs of vertices. We assume G has m edges.

◮ Two vertices vi and vj are said to be adjacent if there is an

edge {vi, vj} joining them. We denote this by vi ∼ vj.

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Adjacency matrix

◮ Given a graph G with vertex set {v1, v2, . . . , vn} we define the

adjacency matrix A = [aij] of G as follows: aij =

• 1

if vi ∼ vj if vi ∼ vj

◮ A is a symmetric (0,1) matrix of trace zero. ◮ Two graphs G and G

′ are isomorphic if and only if there exists

a permutation matrix P of order n such that PAPT = A

′.

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Adjacency matrix

◮ A sequence of ℓ successively adjacent edges is called a walk of

length ℓ and is denoted by b0, b1, b2, . . . , bℓ−1, bℓ. The vertices b0 and bℓ are the end points of the walk.

◮ 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 of walks of length 2 with vi and vj as endpoints. The diagonal entries denote the number of closed walks of length 2.

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Characteristic polynomial of G

◮ The polynomial φ(G, λ) = det(λI − A) is called the

characteristic polynomial of G. The collection of the n eigenvalues 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

• r a cycle; ω(S) is the number of connected components of S, and

c(S) is the number of cycles contained in S. In addition, a0 = 1.

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Spectrum of G

◮ Since A is symmetric, the spectrum of G consists of n real

numbers λ1 ≥ λ2 ≥ . . . ≥ λn−1 ≥ λn.

◮ Because λ1 ≥ |λi|, i = 2, 3, . . . , n the eigenvalue λ1 is called

the spectral radius of G.

◮ The following relations are easy to establish:

i.

n

• i=1

λi = 0 ii.

n

• i=1

λ2

i = 2m

iii.

i<j

λiλj = −m

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Energy of G

The following graph parameter was introduced by Ivan Gutman.

Definition

If G is an n-vertex graph and λ1, . . . , λn are its eigenvalues, then the energy of G is E(G) =

n

• i=1

|λi|.

◮ The term originates from Quantum Chemistry. In H¨

uckel molecular orbital theory, the Hamiltonian operator is related to the adjacency matrix of a pertinently constructed graph. The total π−electron energy has an expression similar to E(G).

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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

dx ln φ(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

−t

F(x)dx (the principal value of the integral).

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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)| det A|

2 n ≤ E(G) ≤

√ 2mn.

Proof.

By Cauchy-Schwarz inequality, n

• i=1

|λi| 2 ≤ n

n

• i=1

|λi|2 = 2mn.

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Bounds for E(G), cont’d

Proof cont’d.

Observe that n

• i=1

|λi| 2 =

n

• i=1

λ2

i + 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|)

2 n = | det(A)| 2 n .

Hence E(G)2 ≥ 2m + n(n − 1)| det(A)|

2 n .

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Bounds for E(G), cont’d

Corollary

If det A = 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 and E(G) = 2m holds if and only if G is mK2 and isolated vertices.

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Strongly regular graphs

Definition

A graph G is said to be strongly regular with parameters (n, k, λ, µ) whenever G has n vertices, is regular of degree k, every pair of adjacent vertices has λ common neighbors, and every pair

• f distinct nonadjacent vertices has µ common neighbors.

◮ In terms of the adjacency matrix A, the definition translates

into: A2 = kI + λA + µ(J − A − I) where J is the all-ones matrix and J − A − I is the adjancency matrix of the complement of G.

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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 with parameters (n, (n + √n)/2, (n + 2√n)/4, (n + 2√n)/4).

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Graphs with extremal energies

◮ One of the fundamental questions in the study of graph

energy is which graphs from a given class have minimal or maximal energy.

◮ Among tree graphs on n vertices the star has minimal energy

and the path has maximal energy.

◮ Equienergetic graphs: non-isomorphic graphs that have the

same energy.

◮ The smallest pair of equienergetic, noncospectral connected

graphs of the same order are C5 and W1,4.

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Generalization of E(G)

◮ For a graph G on n vertices, let M be a matrix associated

with 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 then defined as EM(G) :=

n

• i=1

|µi − µ|.

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

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Laplacian matrix

◮ The classical Laplacian matrix of a graph G on n vertices is

defined as L(G) = D(G) − A(G) where D(G) = diag(deg(v1), . . . , deg(vn)) and A(G) is the adjacency matrix.

◮ The normalized Laplacian matrix, L(G), of a graph G (with

no isolated vertices) is given by Lij =        1 if i = j −

1

√

didj

if vi ∼ vj

• therwise.

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Laplacian Energy

Definition

Let µ1, . . . , µn be the eigenvalues of L(G). Then the Laplacian energy LE(G), is defined as LE(G) :=

n

• i=1
• µi − 2m

n

• .

Definition

Let µ1, . . . , µn be the eigenvalues of the normalized Laplacian matrix L(G). The normalized Laplacian energy NLE(G), is defined as NLE(G) :=

n

• i=1

|µi − 1|.

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Remarks on LE(G)

◮ If the graph G consists of components G1 and G2, then

E(G) = E(G1) + E(G2).

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

and G2 have equal average vertex degrees, then LE(G) = LE(G1) + LE(G2). Otherwise, the equality need not hold.

◮ LE(G) ≥ E(G) holds for bipartite graphs. ◮ LE(G) ≤ 2m

n +

• (n − 1)
• 2M −

2m n 2 where M = m + 1

2 n

• i=1

(di − 2m

n )2. ◮ 2

√ M ≤ LE(G) ≤ 2M

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Edge deletion

◮ Let H be a subgraph of G. We denote by G − H the subgraph

• f G obtained by removing the vertices of H. We denote by

G − E(H) the subgraph of G obtained by deleting all edges of H but retaining all vertices of H.

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

Let H be an induced subgraph of a graph G. Suppose H is the union 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.

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Singular values of a matrix

◮ The singular values s1(A) ≥ s2(A) ≥ . . . sm(A) of a m × n

matrix A are the square roots of the eigenvalues of AA∗.

◮ Note that if A ∈ Mn is a Hermitian (or real symmetric) matrix

with eigenvalues µ1, . . . , µn then the singular values of A are the moduli of µi.

◮ Proof of the previous theorem uses the following Ky Fan’s

inequality 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).

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Edge deletion

Corollary

Suppose H is a single edge e of G and H consists of e and n − 2 isolated vertices. Then LE(G) − 4(n − 1) n ≤ LE(G − e) ≤ LE(G) + 4(n − 1) n .

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Join of Graphs

◮ The join of graph G with graph H, denoted G ∨ H is the

graph obtained from the disjoint union of G and H by adding the 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 graph

• n p vertices. Then

NLE(G ∨ H) = r p + r NLE(G) + s n + s NLE(H) + p − r p + r + n − s n + s .

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Shadow Graph

◮ Let G be a graph with vertex set V = {v1, v2, . . . , vn}. Define

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

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Distance energy

◮ Let G be a connected graph with vertex set

V (G) = {v1, v2, . . . , vn}. The distance matrix D(G) of G is defined so that the (i, j) entry is equal to dG(vi, vj) where distance is the length of the shortest path between the vertices vi and vj.

◮ The distance energy DE(G) is defined as

DE(G) =

n

• i=1

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

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An upper bound for DE(G)

◮ The distance degree Di of vi is Di := n

• j=1

dij.

◮ The second distance degree Ti of vi is Ti = n

• j=1

dijDj.

◮ Theorem (G. Indulal)

DE(G) ≤

• n
• i=1

T 2

i n

• i=1

D2

i

+ (n − 1)

• S −

n

• i=1

T 2

i n

• i=1

D2

i

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

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Vertex sum of G and H

◮ 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 be the graph obtained by identifying the vertices u and v.

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

DE(G ◦ H) ≤ DE(G) + DE(H) and equality holds if and only if u

• r v is an isolated vertex.

◮

• n2(n − 1)(n + 1)

6 ≤ DE(Pn) ≤

• n3(n − 1)(n + 1)

6

◮ DE(Sn) ≤ DE(Pn)

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Energy of a matrix

◮ Let A ∈ Mm,n(C) and let s1(A) ≥ s2(A) ≥ . . . ≥ sm(A) be the

singular values of A. Define the energy of A as E(A) =

m

• j=1

sj.

◮ E(G) = E(A(G)).

Theorem (V. Nikiforov)

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

i,j

|aij| ≥ nα, then E(A) ≤ ||A||1 √mn +

• (m − 1)
• tr(AA∗) − ||A||2

1

mn

• ≤ α

√n(m + √m) 2 .

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Incidence energy

◮ For a graph G with vertex set {v1, . . . , vn} and edge set

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

◮ If the singular values of I(G) are σ1, σ2, . . . , σn then define

incidence energy as IE(G) =

n

• i=1

σi.

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

Laplacian of G. Therefore IE(G) =

n

• i=1
• µ+

i where

µ+

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

matrix.

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BIBD

◮ A balanced incomplete block design BIBD(v, b, r, k, λ) is a

pair (V , B) where V is a v-set of points, B is a collection of k subsets of V called blocks such that any pair of distinct points occur in exactly λ blocks. Here b is the number of blocks and r is the number of blocks containing each point.

◮ The incidence matrix of a BIBD is a (0,1)-matrix whose rows

and columns are indexed by the points and the blocks, respectively, and the entry (p, B) is 1 if and only if p ∈ B.

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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 of a 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).

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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 and Applications Springer, Berlin 2001, 196-211 J.H. Koolen, V. Moulton Maximal energy graphs

• Adv. Appl. Math.26, 2001, 47-52

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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 of graphs 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

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