Graphs and Combinatorics (2003) 19:131–135 Digital Object Identifier (DOI) 10.1007/s00373-002-0487-7
Graphs and Combinatorics Ó Springer-Verlag 2003
Maximal Energy Bipartite Graphs Jack H. Koolen1 and Vincent Moulton2 1 FSPM-Strukturbildungsprozesse, University of Bielefeld, D-33501 Bielefeld, Germany. e-mail:
[email protected] 2 Physics and Mathematics Department (FMI), Mid Sweden University, Sundsvall, S 851-70, Sweden. e-mail:
[email protected]
Abstract. Given a graph G, its energy EðGÞ is defined to be the sum of the absolute values of the eigenvalues of G. This quantity is used in chemistry to approximate the total p-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. Here we show that if G is a bipartite graph with n vertices, then n pffiffiffi pffiffiffi EðGÞ pffiffiffi ð 2 þ nÞ 8 must hold, characterize those bipartite graphs for which this bound is sharp, and provide an infinite family of maximal energy bipartite graphs.
Given a graph G, define the energy of G, denoted EðGÞ, by X EðGÞ :¼ jkj; k an eigenvalue of G
where the eigenvalues of G are simply those of the adjacency matrix of G. In chemistry, the energy of a graph is intensively studied since it can be used to approximate, the total p-electron energy of a molecule (see, for example, [3, 6, 8]). In [12], we considered maximal energy graphs (see also [9, 10, 13, 14, 17] for related results). In particular, for any graph G with n vertices, we derived an improvement of the well-known McClelland bound [15] for the energy of a graph, showing that pffiffiffi n EðGÞ ð1 þ nÞ 2 must hold. We also characterized those graphs for which this bound is sharp, i.e. the maximal energy graphs, and provided an infinite family of such graphs.
The author thanks the Swedish Natural Science Research Council (NFR) – grant M12342-300– for its support.
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J.H. Koolen and V. Moulton
Here we restrict our attention to bipartite graphs, which arise in chemistry from alternant hydrocarbons (see, for example, [11, 13]). In particular, given a bipartite graph G with n vertices, we show n pffiffiffi pffiffiffi EðGÞ pffiffiffi ð 2 þ nÞ 8 must hold, characterize those graphs for which this bound is sharp, and provide an infinite family of maximal energy bipartite graphs. This result is expected to have applications within chemistry, in particular within the study of hyperenergetic graphs [8]. We begin by recalling that a 2-(v, k, k)-design is a collection of k-subsets or blocks of a set of v points, such that each 2-set of points lies in exactly k blocks. The design is called symmetric (or square) in case the number of blocks b equals v. The incidence matrix B of a 2-(v, k, k)-design is the v b matrix defined by setting, for each x a point and S a block, Bx;s :¼ 0 if x 2 = S and Bx;S :¼ 1 otherwise. The incidence graph of a design is defined to be the graph with adjacency matrix
0 Bt
B : 0
Note that the incidence graph of a symmetric 2-(v, k, k)-design pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiwith v > k > k > 0 has eigenvalues k, k k (with multiplicity v 1), k k (with multiplicity v 1), and k (for more details on 2-(v, k, k)-designs see, for example, [5, 10.3] and [16]). We now give an ðn; mÞ-bound for the energy of a bipartite graph with n vertices and m edges (see also [13]) and characterize those graphs for which this bound is sharp. Theorem 1. If n 2m and G is a bipartite graph with n > 2 vertices and m edges, then the inequality vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u 2 # u 2m 2m t EðGÞ 2 þ ðn 2Þ 2m 2 n n
ð1Þ
holds. Moreover, equality holds in (1) if and only if at least one of the following statements holds: (i) n ¼ 2m and G ¼ mK2 ; (ii) n ¼ 2t, mp¼ffiffiffiffit2 , and G ¼ Kt;t ; (iii) n ¼ 2v, 2 m < n < 2m, and G is the incidence graph of a symmetric 2-(v, k, k)kðk 1Þ design with k ¼ 2m n and k ¼ v 1 . Proof. Suppose that g1 g2 gn are the eigenvalues of G (which are real since the adjacency matrix of G is real and symmetric). Then, as is well-known, we have
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g1
2m n
(see [4] for example). Moreover, since n X
g2i ¼ 2m
i¼1
holds (for example see [1]) and g1 ¼ gn as G is bipartite (by the pairing theorem – see e.g. [3]), we have n 1 X
g2i ¼ 2m 2g2i :
i¼2
Using this together with the Cauchy-Schwartz inequality, applied to the vectors ðjg2 j; . . . ; jgn 1 jÞ and ð1; 1; . . . ; 1Þ with n 2 entries, we obtain the inequality n 1 X
jgi j
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 2Þð2m 2g2i Þ:
i¼2
Thus, we must have EðGÞ 2g1 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 2Þð2m 2g21 Þ:
ð2Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 decreases on the Now,psince ffiffiffiffiffiffiffiffiffiffiffithe function pffiffiffiffi F ðxÞ :¼ 2x þ ðn 2Þð2m 2xpÞffiffiffiffiffiffiffiffiffiffiffi interval 2m=n < x m, in view of 2m n we see that 2m=n 2m=n g1 must hold, and hence F ðg1 Þ F ð2m=nÞ must hold as well. From this fact, and Inequality (2), it immediately follows that Inequality (1) holds. It is straight-forward to check that if G is one of the graphs in (i)–(iii), then equality must hold in (1). Conversely, if equality holds in (1), then by the previous discussion on the function F ðxÞ we see that g1 ¼ 2m n must hold. It follows that G is regular with valence 2m [4]. Now, since equality must also hold in theffi Cauchy-Schwartz inequality given q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 above, we have jgi j ¼ ð2m 2ð2m n Þ Þ=ðn 2Þ, for 2 i n 1. Hence, by the pairing theorem, we are reduced to the following possibilities; either G has two eigenvalues with equal absolute values, in which case G must equal mK2 so that (i) holds,pG eigenvalues, i.e. jgi j ¼ 0 for 2 i n 1, and hence ffiffiffiffi has three distinct ffiffiffi pffiffiffi so that (ii) holds, or G has four distinct eigenvalues in n ¼ 2 m and G ¼ Kpm ; m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 which case G is connected, 2m=n > ð2m 2ð2m n Þ Þ=ðn 2Þ holds, and G is the incidence graph of a symmetric 2-ðv; 2m n ; kÞ-design [4, p. 166] so that (iii) holds. h Remark 1. In case 2m n, [12, Theorem 2] states that if G is a graph on n vertices with m edges, then EðGÞ 2m with equality holding if and only if G is the disjoint union of edges and isolated vertices.
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Infinite families of symmetric 2-ðv; k; kÞ-designs are known to exist. For example, the symmetric designs with k ¼ 1 are exactly the projective planes. However, it is not known for fixed k > 1 whether or not there are infinitely many symmetric 2-ðv; k; kÞ-designs. We now give a bound for the energy of a bipartite graph G in terms of the number of its vertices, and characterize those graphs for which this bound is sharp. Theorem 2. Let G be a bipartite graph on n > 2 vertices. Then n pffiffiffi pffiffiffi EðGÞ pffiffiffi ð n þ 2Þ 8
ð3Þ
holds, with equality holding if and only if n ¼ 2v and G is the incidence graph of a pffi pffiffi 2- v; vþ2 v ; vþ24 v -design. Proof. Suppose that G is a graph with n vertices and m edges. If 2m > n, then using routine calculus, it is seen that the left hand side of Inequality (1) – considered as a function of m – is maximized when pffiffiffiffiffi n2 þ n 2n m¼ 8 holds. Inequality (3) now follows by substituting this value of m into (1). pffiffiffiffi 2 2 Now note that m ¼ n þn8 2n < n4 holds if n > 2. Thus it follows from Theorem 1 that holds in (3) if and only if n ¼ 2v and G is the incidence graph of a equality pffi pffi 2- v; vþ2 v ; vþ24 v -design. n pffiffiffi pffiffiffi n pffiffiffi ð n þ 2Þ 8 for n 2, the theorem follows.
h
Using Theorem 2, we now provide an infinite family of maximal energy bipartite graphs. First, recall that a k-regular graph G on n vertices is called strongly regular with parameters ðn; k; k; lÞ if each pair of adjacent vertices has the same number k 0 of common neighbors, and each pair of non-adjacent vertices has the same number l 0 of common neighbors (for more details see [1] or [2]). Now, given a strongly regular graph G with k ¼ l, we can construct a 2-ðn; k; kÞdesign from its adjacency matrix by taking the point set to be the vertex set of G, and defining a block for each row of the adjacency matrix of G by simply taking those vertices which give rise to a 1 in that row. This is a symmetric 2-ðn; k; kÞdesign with incidence matrix equal to the adjacency matrix of G (see [16]). In [12] we show that any maximal energy graph must be strongly regular with parameters
pffiffiffi pffiffiffi pffiffiffi pþ p pþ2 p pþ2 p p; ; ; 2 4 4
ð4Þ
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and, in addition, we provide an infinite family of such graphs. Using these facts, it is now a simple matter to construct an infinite family of maximal energy bipartite graphs. Namely, if A is the adjacency matrix of a maximal energy graph, then it is pffiffi pffiffi pþ p pþ2 p the incidence matrix of a 2- p; 2 ; 4 -design, and therefore by Theorem 2 we immediately see that 0 A A 0 is the adjacency matrix of a maximal energy bipartite graph. Acknowledgments. The authors would like to thank Ivan Gutman for encouraging them to write this paper, and for helpful discussions on this topic. They also would like to thank Edwin van Dam for his reference concerning connected bipartite regular graphs with four eigenvalues.
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