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. 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
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(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
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
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
The Coulson Integral Formula This formula for E(G) was obtained by Charles Coulson in 1940. 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
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
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
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
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
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
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. 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 L ij = 1 if v i v j di 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 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 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. (n LE(G) 2m ( ) ] 2m 2 n + 1) [2M n n where M = m + 1 2 (d i 2m n )2. 2 M LE(G) 2M 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 s i (X ) + s i (Y ) n s i (Z). 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) n LE(G e) LE(G) + 4(n 1). 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 NLE(G H) = r p + r NLE(G) + s n + s NLE(H) + p r p + r + n s n + s. 22 / 32
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 + 1 23 / 32
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
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
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
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
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
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
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
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 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 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 32 / 32