Algebraic Combinatorics and Applications

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1 A. Betten A. Kohnert R. Laue A. Wassermann Editors Algebraic Combinatorics and Applications Springer A

2 The Energy of a Graph: Old and New Results 197 The Energy of a Graph: Old and New Results Ivan Gutman Faculty of Science, University of Kragujevac, P. 0. Box 60 YU Kragujevac, Yugoslavia gutmanoknez. uis.kg. ac. yu Abstract Let G be a graph possessing n vertices and m edges. The energy of G, denoted by E = E(G), is the sum of the absolute values of the eigenvalues of G. The connection between E and the total electron energy of a class of organic molecules is briefly outlined. Some (known) fundamental mathematical results on E are presented: the relation between E(G) and the characteristic polynomial of G, lower ahd upper bounds for E, especially those depending on n and m, graphs extremal with respect to E, n-vertex graphs for which E(G) > E(K,). The characterization of the n-vertex graph(s) with maximal value of E is an open problem. 1 Introduction In this article we are concerned with schlicht graphs, namely graphs without multiple, weighted or directed edges and without loops. Let G be such a graph. The number of its vertices and edges is denoted by n and m, respectively. Its vertices are labeled by vl, vz,...; vn. The adjacency matrix A = A(G) of G is a square matrix of order n whose (i, j)-entry is defined as if the vertices vi and vj are adjacent Aij = {' 0 otherwise. The eigenvalues of A(G) are said to be the eigenvalues of the graph G. A graph on n vertices has n eigenvalues (not all of which need to be distinct); these will be denoted by XI, xz,..., xn and labeled in a non-decreasing manner: xl > x2 >... > xn. The collection of all n eigenvalues of G forms the spectrum of G. The characteristic polynomial of A(G) is said to be the characteristic polynomial of the graph G. Thus, the characteristic polynomial of G is a monic polynomial of degree n defined via where I stands for the unit matrix of order n. The graph eigenvalues are just the zeros of the characteristic polynomial. The graph eigenvalues are necessarily real-valued numbers. Spectral properties of graphs (including properties of the characteristic polynomial) have been extensively studied; for a detailed survey see [I]. 2 A Quantum-Chemical Excursion Within the Huckel molecular orbital (HMO) theory (for details see [2,3]), the total energy of the so-called T-electrons is calculated from the molecular orbital energy levels El, Ez,..., En by means of the formula where gj is the number of T-electrons in the jth molecular orbital. On the other hand, the HMO Hamiltonian operator H is related to the adjacency mai trix A of a pertinently constructed graph, the so-called molecular or Huckel graph, via H=~I+PA (1) where the parameters a and p are assumed to be constants; for details see [4,5,6]. Consequently, E, =a+px, and therefore n En =nea+p)gjx, j= 1 with n, denoting the number of r-electrons in the underlying molecule. The right-hand side summation in (2) uniquely depends on the molecular graph, provided the occupation numbers gj are known. If one restricts the consideration to molecules in their ground states, then Eq. (2) becomes The non-trivial part of the HMO total T-electron energy is the expression in square brackets in Eqs. (3). This fact is usually stressed by using so-called p-units, i.e. by formally setting a = 0, P = 1. The form of the expression (3) is rather awkward and is not suitable for mathematical analysis. However, in the almost ubiquitously encountered special case when we get gj = 2 whenever xj > 0 & gj = 0 whenever xj < 0 En = nea+2/3)xj + with C indicating summation over all positive-valued graph eigenvalues. + (2)

4 200 Ivan Gutman The Energy of a Graph: Old and New Results 201 dependence of the total n-electron energy of a molecule (as computed within the HMO model) on the structure of this molecule. More on this matter can be found elsewhere [5,10,11,12]. Corollary 4.2. If G1 and G2 are two graphs with equal number of vertices, then Figurel. For some T > XI, the contour r goes along the y-axis from point (0, T) to point (0, -T) and then returns to (0, T ) along a semicircle with radius T For n being a constant, if 2ni (11) r r Because of '(7),(8) and (9) the integrand on the right-hand side of (11) has the ~ro~ertv - A " lim [f(r)- n] = 0 Izl-+m In view of it, if r += oa (see 1) then the integrand f(z) - n vanishes everywhere on r, except on the y-axis. This change of r will, however, not affect the value of the contour integral itself. Thus for r += oa, Corollary 4.3. If G is a graph on n vertices, then Tdx E(G) = - n c~ log Izn $(G, i/x)l. - m Corollary 4.4 (1131). Let ( be any real number greater than XI. Then n where Mp = Jgl(xj)P is the p-th spectral moment of the graph G. The infinite summation on the right-hand side of (13) is, of course, convergent. Numerous authors attempted to express the energy in terms of spectral moments!14]-[19]. Corollary 4.4 may be understood as the final (negative) solution of this problem. If B is a bipartite graph on n vertices, then its characteristic polynomial' is of the form [I] Formula (6) is a direct consequence of Eqs. (10)-(12) and Eq. (5). The Coulson integral formula and its various modifications (see below) have important chemical applications. Namely, a theorem by Sachs [1,9] establishes the explicit dependence of the coefficients of the characteristic polynomial of a graph on the structure of this graph. The Coulson formula establishes the explicit dependence of the energy of a graph on the characteristic polynomial of this graph. By combining the Coulson integral formula with the Sachs theorem we gain insight into the dependence of the energy of a graph on the structure of this graph, hence a complete information on the with b(b, k) > 0 for k = 1,2,..., n+ and b(b, k) = 0 for k > n+, where n+ is the number of positive eigenvalues of B. Recall that n+ is also the number of negative eigenvalues of B, and therefore B has n - 2 n+ zero eigenvalues. If F is an acyclic bipartite graph (= a forest), then b(f, k) = m(f, k), where m(f, k) is the number of k-matchings of F, that is the number of k-element independent edge sets of F [I].

5 202 Ivan Gutman Corollary 4.5 ([20] ). If B and F are a bzpartzte graph and a forest, respectivzly, then 5 Bounds for the Energy The Energy of a Graph: Old and New Results 203 Theorem 5.1 ( (McClelland, 1971 [27])). If G is a graph with n vertices, m edges and adjacency matrix A, then Proof. We recall a well known relation for graph eigenvalues: Corollary 4.5 implies that the energy of a bipartite graph B is a monotonically increasing function of each of the parameters b(b, 1), b(b, Z),..., b(b,n+), and that the energy of a forest F is a monotonically increasing function of each of the parameters m(f,l),m(f,2),...,m(f,n+).. Corollary 4.6 ([20] ). (a) If for two bipartite graphs B1 and B2 (not necessarily with equal number of vertices), the relation and start with (19) where AM{ Ixj[ (xki } indicates the arithmetic mean of the (n2-n)/2 distinct terms Jxjl lxki, j < k. The geometric mean of the same terms is is satisfied for all k 2 1, then E(Bl) 5 E(B2). If, in addition, b(b1, k) < b(b2, k) for at least one value of k, then E(B1) < E(B2). (b) If for two forests Fl and Fz (not necessarily with equal number of vertices), the relation m(f1, k) I m(f2, k) (16) is satisfied for all k > 1, then E(F1) 5 E(F2). If, in addition, b(fl, k) < b(f2, k) for at least one value of k, then E(Fl) < E(F2). Relations (15) and (16) have been established for numerous pairs of graphs [21]-[26],- inferring inequalities between their energies. Of them we mention. only the following [20,21]: Let, as usual, z, Sn and Pn denote the graph without edges, the star and the path, respectively, each on n vertices. If Fn is an n-vertex forest, different from and Pn, then where we have taken into account that I1 xj = det A j=1 The lower bound is now a consequence'of the fact that the geometric mean of non-negative numbers cannot exceed their arithmetic mean. The variance of the numbers JxjJ ; j = 1,2,..., n, is equal to: If Tn is an n-vertex tree, different from Sn and Pn, then and the upper bound follows from the fact that the variance is a non-negative quantity. Corollary 5.2. If det A # 0, then E(G) >_ d2 m + n(n - 1) 2 n.

6 204 Ivan Gutman A large number of other bounds for the energy has been reported in the literature (see the subsequent section). Curiously, however, the following simple estimates have evaded attention until quite recently [28]. In addition to (18) we now need also the relation Then, in analogy to Eq. (19), The Energy of a Graph: Old and New Results 205 The estimate (22) is better than the lower bound in (21) if n2/4 < m 5 n(n - 1)/2. Besides, equality in (22) occurs if and only if G is a complete multipartite graph [28]. At this point it is natural to ask about bounds for E, depending solely on n. Then, of course, the consideration has to be restricted to graphs without isolated vertices. In addition to Corollary 5.2 we have: Theorem 5.4. If G is an n-vertex graph without isolated vertices, then with equality if and only if G is the n-vertex star. implying E If G has isolated vertices (i. e., vertices to which no other vertex is adjacent), then each isolated vertex results in an eigenvalue equal to zero. Adding isolated vertices to G will thus change neither m nor E. In view of this consider, for a moment, graphs having m edges and no isolated vertices. The maximum number of vertices of such graphs is 2m, which happens if G = m Kz, i. e., if the graph G consists of m isolated edges. For all.other graphs, n < 2 m. Bearing this in mind, we have 5 = 2 m which combined with the upper bound (17) yields E 5 2 m. As explained above, the latter inequality holds also if G possesses isolated vertices. By this we deduced: Proof. Case (a): G is connected. If G is a connected graph, then m 2 n- 1 and (23) follows from (20). Equality is attained if G is the complete bipartite graph with m = n - 1, which is the star. Case (b): G is disconnected. Let G be composed of p > 1 components, G1, G2,..., G,, having nl, n2,..., n, vertices, respectively, nl + nz n, = n. Note that since G has no isolated vertices, for all values of j it must be nj 2 2 and therefore Then E(G) = E(G1) + E(G2) E(G,) and (what just has been proven), E(Gj) 2 2 JT for j = 1,2,...,p. Consequently, Corollary 5.3. If G is a graph containing m edges, then A more detailed analysis [28] reveals that the bounds (21) are sharp: E(G) = 2 J5;i 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 consists of m isolated edges and of arbitrarily many isolated vertices. Thus, among lower and upper bounds for E, depending solely on m, those given by Eq. (21) are the best possible. In some cases the lower bound (21) can be improved. It is known (see [I]) that the greatest eigenvalue of a graph cannot be less than the average vertex degree 2m/n. Therefore, resulting in because there are p(p - 1)/2 summands of the form JTJx, each being greater than or equal to unity.. We thus arrived at a stronger result than needed: For a graph G with p components, p 2 1, and without isolated vertices, from which Theorem 5.4 follows immediately. Another way to formulate Theorem 5.4 is: Corollary 5.5. Among n-vertex graphs without isolated vertices, the star has minimal energy.

7 206 Ivan Gutman Clearly, among n-vertex graphs, the edgeless graph has minimal energy, equal to zero. What is missing from Theorem 5.4 is a sharp upper bound for the energy of a graph, depending solely on the number n of vertices. Finding of such a bound is related to the problem of identifying the n-vertex graph(s) with maximal energy. 6 More Bounds for the Energy Let U = 2 m - n I det A['/". Then the McClelland bounds (17) can be rewritten as 0 < 2mn-~'<(n-1)~. These bounds were improved [29]. For all graphs: whereas for bipartite graphs Let n+ and n- be the number of positive and negative eigenvalues of the graph G. Then [30] If t and q are, respectively, the number of triangles and quadrangles in the graph G, and if D is the sum of squares of its vertex degrees, then [30] Let B be a bipartite graph with n vertices and m edges. Let b(b, 2) and b(b, 3) be quantities defined via Eq. (14), and recall that b(b, 1) = m. Introduce the auxiliary quantity ET : The Energy of a Graph: Old and New Results 207 Additional bounds for E, applicable to all graphs are reported in [33,34]. The bounds valid for various special classes of graphs are too numerous to be considered here; for some of them see [35]. 7 Searching for the Graph(s) with Maximal Energy: Hyperenergetic Graphs Various empirical and statistical studies (usually performed on graphs of chemical interest, which are connected and possess relatively few edges), point towards a simple regularity: The energy of a graph increases with the increase of the number of vertices and edges. A famous approximation, quantifying the above regularity, is the McClelland formula [27] which was found to be chemically quite satisfactory (for details see [35]). A naive extension of this rule to all graphs resulted in the conjecture [7] that among n vertex graphs, the complete graph Kn has maximal energy. It was soon shown (first by Chris Godsil in the early 1980s) that there exist graphs whose energy exceeds E (K,). Because the spectrum of Kn consists of the numbers n - 1 and -1 (n - 1 times) [I], it follows that E(Kn) = 2n - 2. Definition 7.1. A graph G, such that E(G) > 2n - 2, is said to be hyperenergetic. The fact that hyperenergetic graphs exist has been known for a long time (C. D. Godsil, unpublished, [36]), but a systematic construction of such graphs was achieved only quite recently. Proposition 7.2 ([37]). The line graph of the complete graph on n vertices is hyperenergetic if n 2 5. Proof. The complete graph Kn is regular of.degree n-1. The line graph of Kn has n(n - 1)/2 vertices. It is known 11,381 that the characteristic polynomial of a regular graph R and of its line graph L(R) are related as Then [31] and furthermore [32] E(B) 5 ET 5 G where n and r are the number of vertices and the degree of R. Bearing in mind that $(Kn, x) = (x- n + l)(x + I)"-' we calculate which for n 2 5 is greater than

8 208 Ivan Gutman Proposition 7.2 provides hyperenergetic graphs with n(n - 1)/2 vertices for n > 5, that is hyperenergetic graphs with 10,15,21,28,36,... vertices. In what follows we construct hyperenergetic graphs for all n = 9,10,11,.... Definition 7.3. Let Kn be the complete graph - on n vertices, n > 3. Let v be a vertex of Kn and let ei, i = 1,...,r, 1 <_ r 5 n - 1, be edges of Kn, all incident to u. The graph obtained by deleting ei, i = 1,..., r, from Kn will be denoted by KL. Besides, for r = 0, K;. By performing a number of pertinent transformations on the determinant of XI - A(KL) (whose details are given elsewhere [39,40]), we arrive at Proposition 7.4. For 0 5 r 5 n - 1, From Proposition 7.4 one straightforwardly deduces the spectrum of KL, and from it the respective energy: Kn The Energy of a Graph: Old and New Results 209 There are hyperenergetic graphs also with 8 vertices. For instance, let the edges f 1, f2, f3, f4 lie on a quadrangle of Kg. Then E(K8 - f - f2 - f3 - f4) > E(K8). By designing hyperenergetic graphs we are still far from characterizing graphs with maximal energy. Computer-aided searches of such graphs were performed (28,361 and the maximum-energy graphs up to n = 12 reported. Their inspection, however, gives no clue about the structure of such graphs in the general case. Another approach to the same problem was done in [41], where edges were added one-by-one, in a random manner, starting with K, (for which m = 0) and ending with Kn (for which m = n(n - 1)/2). This construction produces labeled (n, m)-graphs uniformly at random. By repeating the construction many times, a statistical regularity emerges: if n > 9 then by varying m between 0 and n(n - 1)/2 the expectation value of the energy of a random, (n,m)-graph first increases, attains a maximum Em,, at some m = m, and then decreases. The following approximate behavior was found for 9 5 n 5 30 [41]: Proposition 7.5. For n > 4 and 2 5 r 5 n - 2, where xi, i = 1,2,3, are the three solutions of the cubic equation of which xi, 22 are positive and 23 is negative. The inequality E(KL) > E(Kn) will be satisfied if and only if x1+x2-x3>n+1. (24) Determining the conditions under which (24) is obeyed is an elementary (yet somewhat tedious) exercise from algebra. Skipping its details we state our main result: Theorem 7.6. The energy of the graph KL exceeds the energy of Kn for: r=2andn>10 ; r =3andn>9 ; r =4andn>9 ; r =5and n>_10; r>6andn>r+4. Corollary 7.7. Each choice of the parameters n and r, specified in Theorem 7.6, provides a hyperenergetic graph. Corollary 7.8. There exist hyperenergetic graphs on n vertices, for every n>9. These results give a hint where one should look for graphs with maximum energy, but the structure of these graphs remains almost completely obscure. In the author's opinion, finding and characterizing the maximum-energy graphs is the most challenging open problem in the, now more than half a century old, theory of graph energy. At this stage the solving of the problem deserves to be attacked both by means of computer-aided combinatorial search and optimization algorithms, and by the proof techniques of algebraic graph theory. It is easy to prove that for u being any vertex of the graph G, E(G - u) 5 E(G). Another hard-to-crack problem would be to characterize the graphs G and their edges e for which E(G - e) 5 E(G). References 1. CvetkoviC, D., Doob, M., Sachs, H.: Spectra of Graphs - Theory and Application. Academic Press, New York, 1980; 2nd revised ed.: Barth, Heidelberg, Streitwieser, A.: Molecular Orbital Theory for Organic Chemists. Wiley, New York, Coulson, C. A., O'Leary, B., Mallion, R. B.: Hiickel Theory for Organic Chemists Academic Press, London, Giinthard, H. H., Primas, H.: Zusammenhang von Graphentheorie und MO- Theorie von Molekeln mit Systemen konjugierter Bindungen. Helv. Chim. Acta

210 Ivan Gutman 5. Gutman, I., Polansky, 0. E.: Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin, 1986 6. Dias, J. R.: Molecular Orbital Calculations Using Chemical Graph Theory.

9 210 Ivan Gutman 5. Gutman, I., Polansky, 0. E.: Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin, Dias, J. R.: Molecular Orbital Calculations Using Chemical Graph Theory. Springer-Verlag, Berlin, Gutman, I.: The energy of a graph. Ber. Math.-Statist. Sekt. Forschungszentrum Graz 103 (1978) Coulson, C. A.: On the calculation of the energy in unsaturated hydrocarbon molecules. Proc. Cambridge Phil. Soc. 36 (1940) Sachs, H.: Beziehungen zwischen den in einem Graphen entaltenen Kreisen und seinem charakteristischen Polynom. Publ. Math. (Debrecen) 11 (1964) Gutman, I., TrinajstiC, N.: Graph theory and molecular orbitals. XV. The Huckel rule. J. Chem. Phys. 64 (1976) Gutman, I.: A class of approximate topological formulas for total n-electron energy. J. Chem. Phys. 66 (1977) Gutman, I.: Proof of the Huckel rule. Chem. Phys. Lett. 46 (1977) Gutman, I.: Remark on the moment expansion of total A-electron energy. Theor. Chim. Acta 83 (1992) Hall, G. G.: The bond orders of alternant hydrocarbon molecules. Proc. Roy. Soc. London A 229 (1955) Stepanov, N. F., Tatevskii, V. M.: Decomposing the energy of n-electrons in terms of bonds in the simplest variant of the molecular orbital method (in Russian). Zh. Strukt. Khim. 2 (1961) Gutman, I., TrinajstiC, N.: Graph theory and molecular orbitals. Total n- electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17 (1972) Gutman, I., TrinajstiC, N.: Graph theory and molecular orbitals. The loop rule. Chem. Phys. Lett. 20 (1973) Y. Jiang, Y., Tang, A., Hoffmann, R.: Evaluation of moments and their application to Huckel molecular orbital theory. Theor. Chim. Acta 65 (1984) Schmalz, T. G., ~ivkovit, T., Klein, D. J.: Cluster expansion of the Huckel molecular orbital energy of acyclics: Application to pi resonance theory. Stud. Phys. Theor. Chem. 54 (1988) Gutman, I.: Acyclic systems with extremal Huckel n-electron energy. Theor. Chim. Acta 45 (1977) Gutman, I.: Partial ordering of forests according to their characteristic polynomials. in: Hajnal, A., S6s, V. T. (Eds.), Combinatorics, North-Holland, Amsterdam, 1978, pp Zhang, F.: Two theorems of comparison of bipartite graphs by their energy. Kexue Tongbao 28 (1983) Zhang, F., Lai, Z.: Three theorems of comparison of trees by their energy. Science Exploration 3 (1983) Gutman, I., Zhang, F.: On a quasiordering of bipartite graphs. Publ. Inst. Math. (Beograd) 40 (1986) Gutman, I., Zhang, F.: On the ordering of graphs with respect to their matching numbers. Discr. Appl. Math. 15 (1986) Zhang, Y., Zhang, F., Gutman, I.: On the ordering of bipartite graphs with respect to their characteristic polynomials. Coll. Sci. Papers Fac. Sci. Kragujevac 9 (1988) McCelland, B. J.: Properties of the latent roots of a matrix: The estimation of n-electron energies. J. Chem. Phys. 54 (1971) The Energy of a Graph: Old and New Results Caporossi, G., CvetkoviC, D., Gutman, I., Hansen, P.: Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy. J. Chem. Inf. Comput. Sci. 39 (1999) Gutman, I. Bounds for total n-electron energy. Chem. Phys. Lett. 24 (1974) Gutman, I.: Bounds for total n-electron energy of conjugated hydrocarbons. Z. Phys. Chem. (Leipzig) 266 (1985) Turker, L.: An upper bound for total n-electron energy of alternant hydrocarbons. Commun. Math. Chem. (MATCH) 16 (1984) Gutman, I., Tiirker, L., Dias J. R.: Another upper bound for total A-electron energy of alternant hydrocarbons. Commun. Math. Chem. (MATCH) 19 (1986) Gutman, I.: Bounds for Hiickel total n-electron energy. Croat. Chem. Acta 51 (1978) D. A. Bochvar, D. A., Stankevich, I. V.: Approximate formulas for some characteristics of the electronic structure of molecules. 1. Total electron energy (in Russian). Zh. Strukt. Khim. 21 (1980) Gutman, I.: Total n-electron energy of benzenoid hydrocarbons. Topics Curr. Chem. 162 (1992) CvetkoviC, D., Gutman, I.: The computer system GRAPH: A useful tool in chemical graph theory. J. Comput. Chem. 7 (1986) Walikar, H. B., Ramane, H. S., Hampiholi, P. R.: On the energy of a graph. in: Balakrishnan, R., Mulder, H. M., Vijayakurnar, A. (Eds.), Graph Connections, Allied Publishers, New Delhi, 1999, pp Sachs, H.: ~ ber Teiler, Faktoren und charakteristische Polynome von Graphen, Teil 11. Wiss. Z. TH Ilrnenau 13 (1967) Gutman, I.: Hyperenergetic molecular graphs. J. Serb. Chem. Soc. 64 (1999) Gutman, I., PavloviC, L.: The energy of some graphs with large number of edges. Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.) 118 (1999) Gutman, I., SoldatoviC, T., VidoviC, D.: The energy of a graph and its size dependence. A Monte Carlo approach. Chem. Phys. Lett. 297 (1998)

EQUIENERGETIC GRAPHS

5 Kragujevac J. Math. 26 (2004) 5 13. EQUIENERGETIC GRAPHS Harishchandra S. Ramane, 1 Hanumappa B. Walikar, 2 Siddani Bhaskara Rao, 3 B. Devadas Acharya, 4 Prabhakar R. Hampiholi, 1 Sudhir R. Jog, 1 Ivan