Energies of Graphs and Matrices Duy Nguyen Texas Christian - - PowerPoint PPT Presentation

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Definitions Laplacian Energy Maximum energy The Integral Formula

Energies of Graphs and Matrices

Duy Nguyen Texas Christian University Parabola Talk October 6, 2010

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Definitions Laplacian Energy Maximum energy The Integral Formula

Summary

1 Definitions

Energy of Graph

2 Laplacian Energy

Laplacian Matrices Edge Deletion

3 Maximum energy 4 The Integral Formula

Integral Formula for Laplacian Energy

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Definitions Laplacian Energy Maximum energy The Integral Formula

Adjacency Matrix

Let G be a finite, undirected, simple graph with n vertices and m

• edges. Define the Adjacency matrix of G, as follows:

A(G)i,j =    1 if vi and vj are adjacent if vi and vj are not adjacent A(G) is a symmetric matrix whose eigenvalues λi are real and λ1 λ2 · · · λn.

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Definitions Laplacian Energy Maximum energy The Integral Formula

Energy of a Graph

1978, I. Gutman defined the energy of a graph E(G) to be the sum of the absolute values of the eigenvalues of its adjacency matrix. Concept originated in Chemistry H¨ uckel molecular orbital method uses π-electron energy to compute heat of combustion for hydrocarbons.

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Definitions Laplacian Energy Maximum energy The Integral Formula

Generalizations of Graph Energy

Two generalizations of the concept: Nikiforov(2007): The energy of a matrix A is the sum of its singular values (singular values = square roots of the eigenvalues of AA∗.) For any A ∈ Mm,n define the energy of A, E(A), E(A) =

m

• i=1

si(A). From above, we note that the usual energy of a graph G, E(G) = E(A(G)). Gutman and others: For a graph G on n vertices with associated matrix M, the energy of G is defined as: EM(G) =

n

• i=1

|µi − ¯ µ| where µi’s are the eigenvalues of M, and ¯ µ is the average of those eigenvalues.

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Definitions Laplacian Energy Maximum energy The Integral Formula

Definition of the Laplacian Matrix

Let n be the number of vertices, and m number of edges. Laplacian matrix L(G) = D(G) − A(G) where D(G) is the diagonal matrix of G with D(G)ii = degree of vi, and A(G) is the adjacency matrix. Laplacian matrix is symmetric, positive semidefinite, singular. Laplacian Energy LE(G) = n

i=1 |λi − 2m n | where λi are the

eigenvalues of the Laplacian matrix.

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Definitions Laplacian Energy Maximum energy The Integral Formula

Definition of Signless Laplacian Matrix

Signless Laplacian matrix L+(G) = D(G) + A(G) where D(G) is the degree matrix of G, and A(G) is the adjacency matrix. Signless Laplacian Energy LE+(G) = n

i=1 |λi − 2m n | where

λi are the eigenvalues of the signless Laplacian matrix

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Definitions Laplacian Energy Maximum energy The Integral Formula

Definitions:

Let A = [aij] be the n-by-n matrix with real entries. A is said to be symmetric if A = AT. Theorem: Symmetric matrices with real entries have real eigenvalues.

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Definitions Laplacian Energy Maximum energy The Integral Formula

Preliminaries on Energy of Graphs:

• 2m + n(n − 1)|detA|2/n E(G)

√ 2mn

• Only edges: 2√m E(G) 2m

Only vertices: 2√n − 1 E(G) n

2 (1 + √n)

Question: What is the maximal adjacency energy of graphs on n vertices and how to construct such graph? (Hard!)

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Definitions Laplacian Energy Maximum energy The Integral Formula

Finding Energy for Specific Graphs:

Laplacian Energy for complete graph Kn Lemma 1 If An×n is nonsingular, and if c and d are n × 1 columns, then det(A + cdT) = det(A)(1 + dTA−1c). Theorem Let L be the Laplacian matrix of the complete graph Kn, then

• 1. Charateristic polynomial of L is det(λI − L) = λ(λ − n)n−1
• 2. Laplacian Energy of Kn is LE(Kn) = 2(n − 1)

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Definitions Laplacian Energy Maximum energy The Integral Formula

Laplacian Energy Change - Induced Subgraph Deletion

H is an induced subgraph of G if the vertex set of H, V(H), is a subset of V(G) and the edge set of H, E(H) contains all edges in G that connect two vertices in V(H). ˜ H is the union of H and all other vertices of G (as isolated vertices).

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Definitions Laplacian Energy Maximum energy The Integral Formula

Ky Fan’s Inequality

n

• i=1

si(X) +

n

• i=1

si(Y)

n

• i=1

si(X + Y), where X, Y are n × n matrices. Theorem 1 [REU’09] Let H be an induced subgraph of a simple graph G. Suppose ˜ H denotes 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). Theorem 2[REU’09] the result in Theorem 1 also occurs for Signless Laplacian energy, LE+(G) − LE+(˜ H) LE+(G − E(H)) LE+(G) + LE+(˜ H).

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Definitions Laplacian Energy Maximum energy The Integral Formula

Proof of Theorem 1

Note that D(G) = D(˜ H) + D(G − E(H)). Since A(G) = A(H) XT X A(G − H)

• where X corresponds to the edges connecting H and G − H, we

have A(G) = A(H)

• +

XT X A(G − H)

• = A(˜

H) + A(G − E(H)). Therefore, L(G) = D(G) − A(G) = L(˜ H) + L(G − E(H))

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Definitions Laplacian Energy Maximum energy The Integral Formula

• Cont. Proof

Since m = |E(˜ H)| + |E(G − E(H))|, it results that L(G)−2m n I =

• L(˜

H) − 2|E(˜ H)| n I

• +
• L(G − E(H)) − 2|E(G − E(H)|

n I

• .

Hence, by Ky Fan’s inequality, we have LE(G) LE(˜ H) + LE(G − E(H))

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Definitions Laplacian Energy Maximum energy The Integral Formula

Lemma Suppose ˜ H consists of K2 and n − 2 isolated vertices. Then LE(˜ H) = 4(n−1)

n

. Corollary [REU’09] LE(G) − 4(n − 1) n LE(G − {e}) LE(G) + 4(n − 1) n . Proof. Apply Theorem 1 with H = K2 and ˜ H consists of K2 and (n − 2) isolated vertices.

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Definitions Laplacian Energy Maximum energy The Integral Formula

We can do better!

Theorem 3 LE(G) LE(˜ H) + LE(G − E(H)) 4m(1 − 1 n)

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Definitions Laplacian Energy Maximum energy The Integral Formula

Hyperenergetic graphs

Initial Conjecture (1978): Among graph with n vertices, the complete graph Kn has the maximum adjacency energy (equal to 2(n − 1)). Soon disproved by Chris Godsil. Definition A graph G having energy greater than the complete graph on the same number of vertices is called hyperenergetic. Gutman performed a useful experiment: Start with n-isolated vertices, add egdes one-by-one uniformly at random, until end up with Kn. Their main observation is: The expected energy of a random (n, m)−graph first increases, attain a maximum at some m, then decreases.

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Definitions Laplacian Energy Maximum energy The Integral Formula

Cont.

Figure: average energy vs. edges on n=30

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Definitions Laplacian Energy Maximum energy The Integral Formula

Maximum Laplacian Energy

A pineapple PApq is a graph abtained from the complete graph Kp by attaching q pendant vertices to the same vertex of Kp. Conjecture The maximum Laplacian energy among graphs on n vertices has a pineapple PA⌈ 2n+1

3

⌉,⌊ n−1

3 ⌋ Texas Christian University

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Definitions Laplacian Energy Maximum energy The Integral Formula

The Coulson Integral (1940)

Coulson Theorem If G is a graph on n vertices, then E(G) = 1 πp.v. +∞

−∞

• n − ixφ′(ix)

φ(ix)

• dx.

where φ is the characteristic polynomial of A(G). We have proved similar integral formulas for the Laplacian, Signless Laplacian, and Distance Energies.

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Definitions Laplacian Energy Maximum energy The Integral Formula

Integral Formula for Laplacian Energy

Theorem 6[REU’09] If G is a graph on n vertices and m edges, then LE(G) = 1 πp.v. +∞

−∞

• n − ixφ′

L(ix)

φL(ix)

• dx.

where φL is the characteristic polynomial of L(G) − 2m

n I.

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Definitions Laplacian Energy Maximum energy The Integral Formula

Conjecture

We can apply this integral formula for proving the following Conjecture LE(Pn) LE(Tn) LE(Sn)

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Definitions Laplacian Energy Maximum energy The Integral Formula

Bibliography

So, Robbiano, Abreu, Gutman, Applications of a theorem by Ky Fan in the theory of graph energy, Linear Algebra Appl. (2009) in press. Gutman, Kiani, Mirzakhah, Zhou, On incidence energy of a graph, Linear Algebra Appl. (2009) In press. Robbiano, Jimenez, Applications of a theorem by Ky Fan in the theory of Laplacian energy of graphs, (2008) in press. Gutman, Zhou, Laplacian Energy of a graph, Linear Algebra

• Appl. 414 (2006) 29-37.

Ramane, Revankar, Gutman, Rao, Acharya, Walikar, Bounds for the distance energy of a graph, Kragujevac J. Math. 31 (2008) 59-68. Day, So, Graph energy change due to edge deletion, Linear Algebra Appl. 428 (2008) 2070-2078. Day, So, Singular value inequality and graph energy change,

• Elec. J. of Linear Algebra, 16 (2007) 291-299.
• I. Gutman Algebraic Combinatorics and Applications, Springer,

Berlin, (2001), 198-207.

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Definitions Laplacian Energy Maximum energy The Integral Formula

Acknowledgements

This research was supported by the NSF through the grant DMS 08-51321. Texas Christian University for support through TCU Undergraduate Summer Research Grant Program (USRGP). I thank Professor Sivaram Narayan and Central Michigan

• University. Especially, I thank Professor Gilbert and Professor

Dou for writing me letters of recommendation. Furthermore, Professor Gilbert and Professor Prokhorenkov spent time correcting and helping my talk. The software newgraph 1.1.3 was provided by Dragan Stevanovic.

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