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 1 / 32

Graphs ◮ We will consider simple graphs (no loops, no multiple edges). ◮ Let V = { v 1 , v 2 , . . . , v n } denote the set of vertices. The edge set consists of unordered pairs of vertices. We assume G has m edges. ◮ Two vertices v i and v j are said to be adjacent if there is an edge { v i , v j } joining them. We denote this by v i ∼ v j . 2 / 32

Adjacency matrix ◮ Given a graph G with vertex set { v 1 , v 2 , . . . , v n } we define the adjacency matrix A = [ a ij ] of G as follows: � 1 if v i ∼ v j a ij = 0 if v i �∼ v j ◮ A is a symmetric (0,1) matrix of trace zero. ′ are isomorphic if and only if there exists ◮ Two graphs G and G a permutation matrix P of order n such that PAP T = A ′ . 3 / 32

Adjacency matrix ◮ A sequence of ℓ successively adjacent edges is called a walk of length ℓ and is denoted by b 0 , b 1 , b 2 , . . . , b ℓ − 1 , b ℓ . The vertices b 0 and b ℓ are the end points of the walk. ◮ Let us form � n � � A 2 = a it a tj , i , j = 1 , 2 , . . . , n . t =1 The element in the ( i , j ) position of A 2 equals the number of walks of length 2 with v i and v j as endpoints. The diagonal entries denote the number of closed walks of length 2. 4 / 32

Characteristic polynomial of G ◮ The polynomial φ ( G , λ ) = det( λ I − 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 � n a k λ n − k . Then for k ≥ 1 , φ ( G , λ ) = k =0 � ( − 1) ω ( S ) 2 c ( S ) a k = S ∈ L k where L k denotes the set of Sachs subgraphs of G with k vertices, that is, the subgraphs S in which every component is either a K 2 or 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, a 0 = 1 . 5 / 32

Spectrum of G ◮ Since A is symmetric, the spectrum of G consists of n real numbers λ 1 ≥ λ 2 ≥ . . . ≥ λ n − 1 ≥ λ n . ◮ Because λ 1 ≥ | λ i | , i = 2 , 3 , . . . , n the eigenvalue λ 1 is called the spectral radius of G . ◮ The following relations are easy to establish: � n i. λ i = 0 i =1 � n λ 2 ii. i = 2 m i =1 iii. � λ i λ j = − m i < j 6 / 32

Energy of G The following graph parameter was introduced by Ivan Gutman. Definition If G is an n -vertex graph and λ 1 , . . . , λ n are its eigenvalues, then the energy of G is n � E ( G ) = | λ i | . i =1 ◮ 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 ). 7 / 32

The Coulson Integral Formula This formula for E ( G ) was obtained by Charles Coulson in 1940. Theorem � � � ∞ ′ ( G , ˙ E ( G ) = 1 n − ˙ ı x φ ı x ) dx π φ ( G , ˙ ı x ) −∞ � � � ∞ = 1 n − x d dx ln φ ( G , ˙ ı x ) dx π −∞ where G is a graph, φ ( G , x ) is the characteristic polynomial of G , ′ ( G , x ) is its first derivative and φ � ∞ � t F ( x ) dx = lim F ( x ) dx t →∞ − t −∞ (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 � √ 2 n ≤ E ( G ) ≤ 2 m + n ( n − 1) | det A | 2 mn . Proof. � n � 2 � � n | λ i | 2 = 2 mn . By Cauchy-Schwarz inequality, | λ i | ≤ n i =1 i =1 9 / 32

Bounds for E ( G ), cont’d Proof cont’d. � n � 2 � � n i + 2 � λ 2 Observe that | λ i | = | λ i || λ j | . i =1 i =1 i < j Using AM-GM inequality we get n � � � 2 2 2 n ( n − 1) = ( | λ i | n − 1 ) | λ i || λ j | ≥ ( | λ i || λ j | ) n ( n − 1) n ( n − 1) i < j i < j i =1 n � 2 2 n = | det( A ) | n . = ( | λ i | ) i =1 Hence E ( G ) 2 ≥ 2 m + n ( n − 1) | det( A ) | 2 n . 10 / 32

Bounds for E ( G ), cont’d Corollary � If det A � = 0 , then E ( G ) ≥ 2 m + n ( n − 1) ≥ n. � � Also, E ( G ) 2 = 2 m + 2 | λ i || λ j | ≥ 2 m + 2 | λ i λ j | i < j i < j = 2 m + 2 |− m | = 4 m . Proposition If G is a graph containing m edges, then 2 √ m ≤ E ( G ) ≤ 2 m. Moreover, E ( G ) = 2 √ m holds if and only if G is a complete bipartite graph plus arbitrarily many isolated vertices and E ( G ) = 2 m holds if and only if G is mK 2 and isolated vertices. 11 / 32

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 of distinct nonadjacent vertices has µ common neighbors. ◮ In terms of the adjacency matrix A , the definition translates into: A 2 = k I + λ A + µ ( J − A − I ) where J is the all-ones matrix and J − A − I is the adjancency matrix 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 with parameters ( n + √ n ) / 2 , ( n + 2 √ n ) / 4 , ( n + 2 √ n ) / 4) . ( n , 13 / 32

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 C 5 and W 1 , 4 . 14 / 32

Generalization of E ( G ) ◮ For a graph G on n vertices, let M be a matrix associated with G . Let µ 1 , . . . , µ n be the eigenvalues of M and let µ = tr ( M ) n be the average of µ 1 , . . . , µ n . The M - energy of G is then defined as n � E M ( G ) := | µ i − µ | . i =1 ◮ For adjacency matrix A , E A ( G ) = E ( G ) since tr ( A ) = 0. 15 / 32

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 ( v 1 ) , . . . , deg ( v n )) and A ( G ) is the adjacency matrix. ◮ The normalized Laplacian matrix , L ( G ), of a graph G (with no isolated vertices) is given by   1 if i = j   1 √ − if v i ∼ v j L ij = d i d j    0 otherwise . 16 / 32

Laplacian Energy Definition Let µ 1 , . . . , µ n be the eigenvalues of L ( G ). Then the Laplacian energy LE ( G ), is defined as � � n � � � � µ i − 2 m � � LE ( G ) := � . n i =1 Definition Let µ 1 , . . . , µ n be the eigenvalues of the normalized Laplacian matrix L ( G ). The normalized Laplacian energy NLE ( G ), is defined as n � NLE ( G ) := | µ i − 1 | . i =1 17 / 32

Remarks on LE ( G ) ◮ If the graph G consists of components G 1 and G 2 , then E ( G ) = E ( G 1 ) + E ( G 2 ). ◮ If the graph G consists of components G 1 and G 2 , and if G 1 and G 2 have equal average vertex degrees, then LE ( G ) = LE ( G 1 ) + LE ( G 2 ). Otherwise, the equality need not hold. ◮ LE ( G ) ≥ E ( G ) holds for bipartite graphs. � � � 2 � � � 2 m � ◮ LE ( G ) ≤ 2 m � ( n − 1) n + 2 M − n � n M = m + 1 ( d i − 2 m n ) 2 . where 2 i =1 √ ◮ 2 M ≤ LE ( G ) ≤ 2 M 18 / 32

Edge deletion ◮ Let H be a subgraph of G . We denote by G − H the subgraph of 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. 19 / 32

Singular values of a matrix ◮ The singular values s 1 ( A ) ≥ s 2 ( A ) ≥ . . . s m ( A ) of a m × n matrix A are the square roots of the eigenvalues of AA ∗ . ◮ Note that if A ∈ M n is a Hermitian (or real symmetric) matrix with eigenvalues µ 1 , . . . , µ n then the singular values of A are the moduli of µ i . ◮ 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 M n ( C ) such that X + Y = Z. Then n n n � � � s i ( X ) + s i ( Y ) ≥ s i ( Z ) . i =1 i =1 i =1 20 / 32

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) ≤ LE ( G − e ) ≤ LE ( G ) + 4( n − 1) . n n 21 / 32

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 on p vertices. Then r n + s NLE ( H ) + p − r s p + r + n − s NLE ( G ∨ H ) = p + r NLE ( G ) + n + s . 22 / 32