University of Lethbridge Research Repository OPUS http://opus.uleth.ca Theses Arts and Science, Faculty of 2010 Energy of graphs and digraphs Jahanbakht, Nafiseh Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science, c2010 http://hdl.handle.net/10133/2489 Downloaded from University of Lethbridge Research Repository, OPUS ENERGY OF GRAPHS AND DIGRAPHS NAFISEH JAHANBAKHT Mathematics, Shahid Beheshti University, 2002 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE Department of Mathematics and Computer Science University of Lethbridge LETHBRIDGE, ALBERTA, CANADA c Nafiseh Jahanbakht, 2010 This thesis is dedicated to Dr. Kourosh Tavakoli. iii Abstract The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. The concept is related to the energy of a class of molecules in chemistry and was first brought to mathematics by Gutman in 1978 ([8]). In this thesis, we do a comprehensive study on the energy of graphs and digraphs. In Chapter 3, we review some existing upper and lower bounds for the energy of a graph. We come up with some new results in this chapter. A graph with n vertices is hyper-energetic if its energy is greater than 2n 2. Some classes of graphs are proved to be hyper-energetic. We find − a new class of hyper-energetic graphs which is introduced and proved to be hyper-energetic in Section 3.3. The energy of a digraph is the sum of the absolute values of the real part of the eigenvalues of its adjacency matrix. In Chapter 4, we study the energy of digraphs in a way that Pe˜na and Rada in [19] have defined. Some known upper and lower bounds for the energy of digraphs are re- viewed. In Section 4.5, we bring examples of some classes of digraphs in which we find their energy. Keywords. Energy of a graph, hyper-energetic graph, energy of a di- graph. iv Acknowledgments I am deeply indebted to my advisor, Dr. Hadi Kharaghani, whose com- passionate supervision was invaluable. It was a great honor for me to work under his supervision. I would also like to thank Dr. Wolf Holzmann, Dr. Amir Akbary, and Dr. Mark Walton for their helps and supports. I would like to thank Lily Liu for her new ideas that helped me write Section 3.3. I would like to warmly thank my father who was my first teacher and showed me the joy of learning, and my mother for her endless kind- ness. They have always been my endless source of inspiration. I would also like to thank my spouse, Kourosh Tavakoli, to whom this thesis is dedicated, for all his encouragements and supports at the time I was com- pletely disappointed. I would like to thank my sister, Nahal, whose recovery is the best gift from God. The last but not the least is my brother, Mohammad Mehdi, whom I would like to deeply thank for all his supports from the first day I started my program at the University of Lethbridge. v Contents Approval/Signature Page ii Dedication iii Abstract iv Acknowledgments v Table of Contents vi List of Figures vii 1 Introduction 1 2 Preliminaries 8 2.1 BasicDefinitions ..................... 8 2.2 Eigenvaluesofagraph . 11 2.3 StronglyRegularGraphs . 18 3 Energy of Graphs 22 3.1 Minimalenergygraphs . 26 3.2 MaximalEnergyGraphs . 29 3.3 Hyper-energeticGraphs. 37 3.3.1 GeneralizationandMainResult . 48 4 Energy of Digraphs 55 4.1 Introduction........................ 55 4.2 CoefficientsTheorem . 55 4.3 Integralrepresentationoftheenergy . 60 4.4 Upperandlowerbounds . 63 4.4.1 Upperboundfortheenergyofdigraphs . 63 4.4.2 Lowerboundfortheenergyofdigraphs . 69 4.5 Energyofsomedigraphs . 71 Bibliography 78 vi List of Figures 1.1 G2 ............................. 3 1.2 C4 ............................. 5 2.1 Pseudoregulargraphs. 10 2.2 Paleygraphoforder13 . 18 3.1 Contour Γ ......................... 23 3.2 Ahyper-energeticgraphon8vertices . 38 3.3 G2 ............................. 42 3.4 Anexampleoflocalstronglyregulargraphs . 49 4.1 Digraph........................... 58 4.2 Contour Γ ......................... 61 4.3 P7 : Paleydigraphoforder7 . 72 4.4 Digraphs for matrices D1 and D2 ............. 74 vii Chapter 1 Introduction The concept of the energy of a graph was introduced three decades ago by Ivan Gutman [8]. This notion is related to the total electron energy of a class of organic molecules in computational chemistry. The total energy of the so-called π-electrons is calculated by the formula n Eπ = ∑ λ j (1.1) j=1 | | where n is the number of the molecular orbital energy levels and λ js are eigenvalues of the adjacency matrix of the so-called molecular or H¨uckel graph. Although in chemistry the expression (1.1) is valid only for the class of “H¨uckel graphs”, the right-hand side of (1.1) is well-defined for any class of graphs in mathematics. This motivated Gutman to define the energy of a graph. Definition 1.0.1 [8] Let G be a graph, the energy of G, denoted by E(G), is the sum of the absolute values of the eigenvalues of G, i.e. if λ1,...,λn n are the eigenvalues of G, then E(G) = ∑ λi . i=1 | | For one example, by using the eigenvalues found in page 14, the energy of a complete graph of order n is computed as E(Kn) = n 1 + 1 (n 1) = 2n 2 . | − | | − | − − 1 There are some bounds on the energy of a graph. In this thesis, we mention some of the most well-known bounds. For a graph G of order n with m edges, McClelland ([17]) in early 70’s, gave the following general bounds on its energy where A is the adjacency matrix of G. 2 2m + n(n 1) det(A) n E(G) √2mn. − | | ≤ ≤ q A lower bound for the energy of a graph only in terms of its number of edges is E(G) 2√m with equality if and only if G is a complete ≥ bipartite graph plus some isolated vertices. In terms of the number of vertices the lower bound is E(G) 2√n 1 with equality if and only G ≥ − is the star K1,n 1. − For the upper bound, there is a well-known result due to Koolen and Moulton ([14]) which is an improvement on the McCelland bound. For a graph G with n vertices and m edges where 2m n, they proved ≥ 2m 2m 2 E(G) + (n 1) 2m n v n ≤ u − " − # u t n with equality if and only if G is Kn, 2 K2, or a strongly regular graph (2m (2m/n)2) (SRG) with two eigenvalues having absolute value −(n 1) . − Next, if we consider the left hand side of the aboveq inequality as a function of m, it is maximized when m =(n2 + n√n)/4. By substituting 2 this amount in the above formula we find n(1 + √n) E(G) . (1.2) ≤ 2 Koolen and Moulton ([14]) proved that (1.2) is also valid for 2m < n and that the equality holds if and only if G is an SRG with parameters (n,(n + √n)/2,(n + 2√n)/4,(n + 2√n)/4). They also conjectured that for a given ε > 0 there exists a graph G of n order n such that for almost all n 1, E(G) (1 ε) (√n + 1) which ≥ ≥ − 2 was later proved in [18] by Nikiforov. There was a conjecture in 1978 that between graphs of order n, the complete graph Kn has the maximum energy. Although it was rejected and it was shown that there exist subgraphs of Kn with energy greater than that of Kn, it was an introduction for defining hyper-energetic graphs. A graph G with n vertices is hyper-energetic if E(G) > 2n 2. Some − classes of graphs have shown to be hyper-energetic. Figure 1.1: G2 In this thesis, we introduce a new class of graphs which we prove that they are hyper-energetic. Our first example in this class is a 4-regular 3 graph with 13 vertices that we call G2 (see Figure 1.1). In general case, we construct Gm as follows. Consider 2m 2 copies − of K2m and m copies of K2 and one copy of K1. Add edges to make it 2m-regular by adding 2m edges from the single vertex K1 to vertices of m copies of K2. Then add one edge from each vertex of K2m to the m copies of K2 (see Figure 3.4 in Section 3.3.1). Gm isa2m-regular graph with n =(2m 1)2m+1 vertices. We found − 2m 3 that the characteristic polynomial of Gm is (x 2m)(x (2m 1)) − − − − (x + 1)(2m 3)(2m 1)(x2 (2m 1))m(x2 + 2x (2m 3))m and from that − − − − − − we can find the energy of Gm. Then we prove that for m > 2, graph Gm is hyper-energetic. If we want to generalize the concept of energy for the case of di- graphs, we should be reminded that the adjacency matrix is not symmet- ric and the eigenvalues might be complex numbers. Pe˜na and Rada in [19] proposed the following definition for the energy of digraphs. Definition 1.0.2 Let G be a digraph, the energy of G, denoted by E(G), is the sum of the absolute values of the real part of the eigenvalues of G. In fact, Pe˜na and Rada proved the Coulson integral formula for the case of digraphs ([19]) and that was the motivation for the Definition 1.0.2.