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

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1 Energy of Graphs Sivaram K. Narayan Central Michigan University Presented at CMU on October 10, / 32

2 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

3 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. Two graphs G and G are isomorphic if and only if there exists a permutation matrix P of order n such that PAP T = A. 3 / 32

4 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

5 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 a k λ n k. Then for k 1, a k = S L k ( 1) ω(s) 2 c(s) 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

6 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: i. ii. iii. n λ i = 0 n λ 2 i = 2m λ i λ j = m i<j 6 / 32

7 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. The term originates from Quantum Chemistry. In Hückel 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

8 The Coulson Integral Formula This formula for E(G) was obtained by Charles Coulson in Theorem E(G) = 1 π = 1 π [ ] n ıxφ (G, ıx) dx φ(g, ıx) [n x ddx ] 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 (the principal value of the integral). t F (x)dx = lim F (x)dx t t 8 / 32

9 Bounds for E(G) Theorem (McClelland,1971) If G is a graph with n vertices, m edges and adjacency matrix A, then Proof. 2m + n(n 1) det A 2 n E(G) 2mn. ( n ) 2 By Cauchy-Schwarz inequality, λ i n n λ i 2 = 2mn. 9 / 32

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

11 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 λ i λ j 2m + 2 i<j λ i λ j = 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 mk 2 and isolated vertices. 11 / 32

12 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 = 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. 12 / 32

13 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). 13 / 32

14 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

15 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. For adjacency matrix A, E A (G) = E(G) since tr(a) = / 32

16 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 L ij = 1 if v i v j di d j 0 otherwise. 16 / 32

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

18 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. (n LE(G) 2m ( ) ] 2m 2 n + 1) [2M n n where M = m (d i 2m n )2. 2 M LE(G) 2M 18 / 32

19 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

20 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 s i (X ) + s i (Y ) n s i (Z). 20 / 32

21 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 21 / 32

22 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 NLE(G H) = r p + r NLE(G) + s n + s NLE(H) + p r p + r + n s n + s. 22 / 32

23 Shadow Graph Let G be a graph with vertex set V = {v 1, v 2,..., v n }. Define the shadow graph S(G) of G to be the graph with vertex set and edge set Theorem (Hubbard, N, Woods) V {u 1, u 2,..., u n } E(G) {{u i, v j } : {v i, v j } E(G)}. E(S p (G)) = 4p + 1E(G) for any graph G NLE(S p (G)) = 2p + 1 NLE(G) for any regular graph G. p / 32

24 Distance energy Let G be a connected graph with vertex set V (G) = {v 1, v 2,..., v n }. The distance matrix D(G) of G is defined so that the (i, j) entry is equal to d G (v i, v j ) where distance is the length of the shortest path between the vertices v i and v j. The distance energy DE(G) is defined as DE(G) = n µ i where µ 1 µ 2... µ n are the eigenvalues of D. (Note that D is a real symmetric matrix with trace zero.) 24 / 32

25 An upper bound for DE(G) The distance degree D i of v i is D i := n d ij. j=1 The second distance degree T i of v i is T i = n d ij D j. Theorem (G. Indulal) n DE(G) n T 2 i Di 2 + (n 1) S j=1 n n T 2 i Di 2 where S is the sum of the squares of entries in the distance matrix. 25 / 32

26 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 or v is an isolated vertex. n 2 (n 1)(n + 1) 6 DE(P n ) n 3 (n 1)(n + 1) 6 DE(S n ) DE(P n ) 26 / 32

27 Energy of a matrix Let A M m,n (C) and let s 1 (A) s 2 (A)... s m (A) be the singular values of A. Define the energy of A as E(G) = E(A(G)). Theorem (V. Nikiforov) E(A) = m s j. If m n, A is an m n nonnegative matrix with maximum entry α and A 1 := a ij nα, then i,j E(A) A 1 mn + (m 1) j=1 ( ) tr(aa ) A 2 1 n(m + m) α. mn 2 27 / 32

28 Incidence energy For a graph G with vertex set {v 1,..., v n } and edge set {e 1, e 2,..., e m }, the (i, j)th entry of the incidence matrix I(G) is 1 if v i is incident with e j and 0 otherwise. I(G) is a vertex-edge incidence matrix. If the singular values of I(G) are σ 1, σ 2,..., σ n then define incidence energy as IE(G) = n σ i. I(G)I(G) T = D(G) + A(G) = L + (G) called the signless Laplacian of G. Therefore IE(G) = n µ + i where µ + 1,..., µ+ n are the eigenvalues of the signless Laplacian matrix. 28 / 32

29 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. 29 / 32

30 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 + (p 1)(m m2 pq 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). 30 / 32

31 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), V. Nikiforov The energy of graphs and matrices J. Math. Anal.Appl. 326(2007), I. Gutman The energy of graphs: Old and New Results, Algebraic Combinatorics and Applications Springer, Berlin 2001, J.H. Koolen, V. Moulton Maximal energy graphs Adv. Appl. Math.26, 2001, / 32

32 J.H. Koolen, V. Moulton Maximal energy bipartite graphs Graphs Combin., 19 (2003), G. Indulal Sharp bounds on the distance spectral radius and the distance energy of graphs Linear Alg. Appln., 430 (2009), W.H. Haemers Strongly regular graphs with maximal energy Linear Alg. Appln., 429 (2008), I. Gutman, B. Zhou Laplacian energy of a graph Linear Alg. Appln., 414 (2006), / 32

Energies of Graphs and Matrices

Energies of Graphs and Matrices Duy Nguyen T Parabola Talk October 6, 2010 Summary 1 Definitions Energy of Graph 2 Laplacian Energy Laplacian Matrices Edge Deletion 3 Maximum energy 4 The Integral Formula