Bulletin T.CXXXIV de l’Académie serbe des sciences et des arts − 2007 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 32

NORDHAUS-GADDUM-TYPE RELATIONS FOR THE ENERGY AND LAPLACIAN ENERGY OF GRAPHS

B. ZHOU, I. GUTMAN

(Presented at the 8th Meeting, held on November 24, 2006)

A b s t r a c t. Let \overline{G} denote the complement of the graph G. If I(G) is some invariant of G, then relations (identities, bounds, and similar) pertaining to I(G) + I(\overline{G}) are said to be of Nordhaus-Gaddum type. A number of lower and upper bounds of Nordhaus-Gaddum type are obtained for the energy and Laplacian energy of graphs. Also some new relations for the Laplacian graph energy are established.

AMS Mathematics Subject Classification (2000): 05C50

Key Words: spectrum (of graph), Laplacian spectrum (of graph), energy (of graph), Laplacian energy (of graph), Nordhaus-Gaddum-type relation

1. Introduction

In this paper we are concerned with simple graphs. Let G be such a graph, and let n and m denote, respectively, the number of its vertices and edges. Then G is said to be an (n, m)-graph.

The (ordinary) spectrum of G is the spectrum of its adjacency matrix [6], and consists of the numbers λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n}. The Laplacian spectrum of G is the spectrum of its Laplacian matrix [10, 11, 21, 22], and consists of the numbers μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{n} = 0 .

The energy of a graph G, denoted by E(G) , is defined as

E(G) = ∑_{i=1}^{n} |λ_{i}| .

This graph-spectrum-based invariant has its origin in theoretical chemistry (for details see [13, 14]), but has recently attracted the interest of mathematicians. The basic mathematical properties of graph energy can be found in the review [12], whereas some most recent mathematical studies in the papers [3, 4, 25–30, 32, 33, 35].

The Laplacian energy of a graph G, denoted by LE(G) , has been recently defined as [15]

LE(G) = ∑_{i=1}^{n} |μ_{i} − \frac{2m}{n}|

and was aimed at being the Laplacian-spectral analog of graph energy. Until now, only two papers [15, 37] are devoted to the study of Laplacian graph energy.

As usual, \overline{G} will symbolize the complement of the graph G. The number of vertices and edges of the complement of an (n, m)-graph will be denoted by \overline{n} and \overline{m}, respectively.

Nordhaus and Gaddum [23] reported bounds for the sum of the chromatic numbers of a graph and its complement. Eventually, Norhhaus-Gaddum-type relations were established for many other graph invariants [1, 2, 5, 8, 9, 16, 17, 20, 31, 34, 36]. In this paper we obtain bounds of this kind for the graph energy and Laplacian graph energy.

2. Nordhaus-Gaddum-Type Bounds for Graph Energy

Let \overline{λ_{1}} be the largest eigenvalue of \overline{G}. Nosal [24] demonstrated that for a graph G with n vertices,

n − 1 ≤ λ_{1} + \overline{λ_{1}} < \sqrt{2} n  (1)

which itself is a Nordhaus-Gaddum-type relation. In connection with the right-hand side inequality in (1), it was shown in [17] that

λ_{1} + \overline{λ_{1}} ≤ \sqrt{(2 − \frac{1}{ω} − \frac{1}{\overline{ω}} ) n(n − 1)} ,  (2)

where ω and \overline{ω} denote the clique numbers of G and \overline{G}, respectively.

Theorem 2.1. Let G be a graph with n vertices. Then

E(G) + E(\overline{G}) ≥ 2(n − 1)  (3)

with equality if and only if G is the complete graph K_{n} or its complement, the empty graph (the n-vertex graph without edges).

P r o o f. We first observe that E(G) ≥ 2 λ_{1} with equality if and only if G has at most one positive eigenvalue, i.e., if G is the empty graph or a complete multipartite graph [6]. Therefore,

E(G) + E(\overline{G}) ≥ 2(λ_{1} + \overline{λ_{1}}) ≥ 2(n − 1) .

If equality holds in (3), then both G and \overline{G} are empty or complete multipartite graphs, and so G must be the complete graph or the empty graph. Conversely, knowing the spectrum of K_{n} and \overline{K_{n}}, see [6], it is easily shown that (3) is an equality if G ≅ K_{n} or G ≅ \overline{K_{n}}. □

In [19] it was shown that for an (n, m)-graph G,

E(G) ≤ λ_{1} + \sqrt{(n − 1) (2m − λ_{1}^{2})} .  (4)

From this upper bound it could be deduced that [18]

E(G) ≤ \frac{n}{2} (\sqrt{n} + 1)

which immediately implies

E(G) + E(\overline{G}) ≤ n (\sqrt{n} + 1) .

In what follows we improve the latter upper bound.

Theorem 2.2. Let G be a graph with n vertices. Then

E(G) + E(\overline{G}) < \sqrt{2} n + (n − 1)\sqrt{n − 1} .  (5)

P r o o f. Let m and \overline{m} denote, respectively, the number of edges of G and \overline{G}. By (4) and (1), we have

E(G) + E(\overline{G}) ≤ λ_{1} + \overline{λ_{1}} + \sqrt{(n − 1) (2m − λ_{1}^{2})} + \sqrt{(n − 1) (2\overline{m} − \overline{λ_{1}}^{2})}

≤ λ_{1} + \overline{λ_{1}} + \sqrt{2(n − 1) [2m + 2\overline{m} − (λ_{1}^{2} + \overline{λ_{1}}^{2}) ]}

≤ λ_{1} + \overline{λ_{1}} + \sqrt{2(n − 1) [n(n − 1) − \frac{1}{2} (λ_{1} + \overline{λ_{1}})^{2} ]}

< \sqrt{2} n + \sqrt{2(n − 1) [n(n − 1) − \frac{1}{2} (n − 1)^{2}]}

= \sqrt{2} n + (n − 1) \sqrt{n − 1} .

This completes the proof. □

Remark 2.3. Let G be an n-vertex regular graph of degree r. Then (4) becomes E(G) ≤ r + \sqrt{(n − 1)r(n − r)} and we have

E(G) + E(\overline{G}) ≤ n − 1 + \sqrt{(n − 1)} [\sqrt{r(n − r)} + \sqrt{(r + 1)(n − r − 1)} ]
≤ (n − 1) (\sqrt{n + 1} + 1)

which for n ≥ 6 is better than (5).

Remark 2.4. A strongly regular graph G with parameters (n, r, ρ, σ) is an r-regular graph on n vertices, in which each pair of adjacent vertices has ρ common neighbors and each pair of non-adjacent vertices has σ common neighbors. If σ ≥ 1 and G is non-complete, then the eigenvalues of G are [6] r, s, and t, with multiplicities 1, m_{s}, and m_{t}, where s and t are the solutions of the equation x^{2} + (σ − ρ)x + (σ − r) = 0 , and m_{s} and m_{t} are determined by m_{s} + m_{t} = n − 1 and r + m_{s} s + m_{t} t = 0 . If G is a strongly regular graph with parameters (n, (n + \sqrt{n})/2, (n + 2\sqrt{n})/4, (n + 2\sqrt{n})/4) (for some conveniently chosen value of n), then

E(G)+E(\overline{G}) = \frac{n}{2} (\sqrt{n} + 1)+ \frac{n}{2} (\sqrt{n} + 1)−\sqrt{n}−2 = (n−1) (\sqrt{n} + 1)−1 .

If we consider a Paley graph H, which is a strongly regular graph with parameters (n, (n − 1)/2, (n − 5)/4, (n − 1)/4), then

E(H) + E(\overline{H}) = (n − 1)(\sqrt{n} + 1) .

The results stated in Remark 2.4 show that the bound (5) is asymptotically tight.

Remark 2.5. Using (2), from the proof of Theorem 2.2, we have

E(G) + E(\overline{G}) ≤ \sqrt{(2 − \frac{1}{ω} − \frac{1}{\overline{ω}} ) n(n − 1)} + (n − 1) \sqrt{n − 1} .

3. Some Properties of the Laplacian Graph Energy

Details of the theory of Laplacian graph spectra are found in the reviews [10, 11, 21, 22]. For the following consideration we need the properties: μ_{n} = 0 for all graphs, and μ_{n−1} > 0 if and only if G is connected.

Let G_{1} ∗ G_{2} denote the join of the graphs G_{1} and G_{2}, i.e., the graph obtained from the disjoint union of G_{1} and G_{2} by adding all possible edges between vertices of G_{1} and vertices of G_{2}.

Theorem 3.1. Let G_{1} and G_{2} be (n, m)-graphs. Then

LE(G_{1} ∗ G_{2}) = LE(G_{1}) + LE(G_{2}) + 2n − \frac{4m}{n} .

P r o o f. Let μ′_{1}, μ′_{2}, …, μ′_{n} be the Laplacian eigenvalues of G_{1} and μ′′_{1}, μ′′_{2}, …, μ′′_{n} the Laplacian eigenvalues of G_{2} . Then the Laplacian eigenvalues of G_{1} ∗ G_{2} are [22]

2n , n + μ′_{1} , n + μ′′_{1} , n + μ′_{2} , n + μ′′_{2} , …, n + μ′_{n−1} , n + μ′′_{n−1} , 0 .

Note that G_{1} ∗ G_{2} is a (2n, 2m + n^{2})-graph with average vertex degree (2m + n^{2})/n. Therefore,

LE(G_{1} ∗ G_{2}) = 2n + ∑_{i=1}^{n−1} |n + μ′_{i} − \frac{2m + n^{2}}{n} | + ∑_{i=1}^{n−1} |n + μ′′_{i} − \frac{2m + n^{2}}{n} |

= 2n + ∑_{i=1}^{n−1} |μ′_{i} − \frac{2m}{n}| + ∑_{i=1}^{n−1} |μ′′_{i} − \frac{2m}{n}|

= 2n + LE(G_{1}) − \frac{2m}{n} + LE(G_{2}) − \frac{2m}{n} .

The result follows. □

Remark 3.2. Let G_{1} and G_{2} be regular graphs of degrees r′ and r′′, respectively, with n′ and n′′ vertices, respectively. Then

E(G_{1} ∗ G_{2}) = E(G_{1}) + E(G_{2}) + \sqrt{(r′ − r′′)^{2} + 4 n′ n′′} − r′ − r′′ .

Let G_{1} × G_{2} denote the Cartesian product of graphs G_{1} and G_{2} . Then V (G_{1} × G_{2}) = V (G_{1}) × V (G_{2}) and (u_{1}, u_{2}) is adjacent to (v_{1}, v_{2}) if and only if u_{1} = v_{1} and (u_{2}, v_{2}) ∈ E(G_{2}), or u_{2} = v_{2} and (u_{1}, v_{1}) ∈ E(G_{1}) .

Theorem 3.3. Let G_{1} and G_{2} be, respectively, (n, m_{1})– and (n, m_{2})- graphs. Then

LE(G_{1} × G_{2}) ≤ n LE(G_{1}) + n LE(G_{2}) .

P r o o f. Let the notation be the same as in the proof of Theorem 3.1. Then the Laplacian eigenvalues of G_{1} × G_{2} are [22]

μ′_{i} + μ′′_{j} , i, j = 1, 2, …, n .

Note that G_{1} × G_{2} is an (n^{2}, n(m_{1} + m_{2}))-graph with average vertex degree (2m_{1} + 2m_{2})/n. Therefore,

LE(G_{1} × G_{2}) = ∑_{i=1}^{n} ∑_{j=1}^{n} |μ′_{i} + μ′′_{j} − \frac{2 m_{1} + 2 m_{2}}{n} |

≤ ∑_{i=1}^{n} ∑_{j=1}^{n} (|μ′_{i} − \frac{2 m_{1}}{n} | + |μ′′_{j} − \frac{2 m_{2}}{n} |)

= n LE(G_{1}) + n LE(G_{2}) .

The result follows. □

Let G be an (n, m)-graph. Note that μ_{1} ≥ 2m/n. Then

LE(G) = μ_{1} + ∑_{i=2}^{n−1} |μ_{i} − \frac{2m}{n}| .

If G is not a complete graph, then μ_{n−1} ≤ 2m/n [7], and therefore

LE(G) = μ_{1} − μ_{n−1} + \frac{2m}{n} + ∑_{i=2}^{n−2} |μ_{i} − \frac{2m}{n}| .

Theorem 3.4. Let G be an (n, m)-graph with n ≥ 2 and m ≥ 1 . Then LE(G) ≥ μ_{1} , with equality if and only if G ≅ K_{n/2,n/2} , in which case, of course, n must be even.

P r o o f. It is easy to see that LE(G) ≥ μ_{1} , with equality if and only if n = 2 or for n ≥ 3 , if μ_{2} = ⋯ = μ_{n−1} = \frac{2m}{n} . Suppose that n ≥ 3 and LE(G) = μ_{1} . Then by a result from [37], G is a regular complete k-partite graph with 1 < k ≤ n. Then

n − \frac{n}{k} + (k − 1) \frac{n}{k} = n ,

implying k = 2 . Thus, G ≅ K_{n/2,n/2} . Conversely, if G ≅ K_{n/2,n/2} , then it is easy to verify that LE(G) = μ_{1} . □

In a similar manner we arrive at

Theorem 3.5. Let G be an (n, m)-graph, such that n ≥ 3 and m ≥ 1 . Then

LE(G) ≥ μ_{1} − μ_{n−1} + \frac{2m}{n}

with equality if and only if n = 3 or for n ≥ 4 , if μ_{2} = ⋯ = μ_{n−2} = 2m/n.

4. Nordhaus-Gaddum-Type Bounds for Laplacian Graph Energy

Lemma 4.1. If G is not the complete graph, and has at least one edge, then μ_{1} − μ_{n−1} > 1 .

P r o o f. Since G has at least one edge, μ_{1} ≥ Δ + 1 , where Δ is the maximum vertex degree of G [10, 21]. If G is connected, then equality holds if and only if Δ = n − 1 .

Suppose that G is connected. Then μ_{1} − μ_{n−1} ≥ Δ − 2m/n + 1 ≥ 1 . If μ_{1} − μ_{n−1} = 1 , then 2m/n = Δ = n − 1 and then it would be G ≅ K_{n}, a contradiction.

If G is not connected, then μ_{1} − μ_{n−1} = μ_{1} ≥ Δ + 1 > 1 . □

Theorem 4.2. Let G be a graph with n vertices. Then

LE(G) + LE(\overline{G}) ≥ 2n − 2

with equality if and only if G is isomorphic to K_{n} or \overline{K_{n}}.

P r o o f. If G is isomorphic to K_{n} or \overline{K_{n}}, then it is easy to show that LE(G) + LE(\overline{G}) = 2n − 2 . Suppose that n ≥ 3 and that G is different from K_{n} and \overline{K_{n}}. Then

LE(G) + LE(\overline{G}) = μ_{1} − μ_{n−1} + \frac{2m}{n} + ∑_{i=2}^{n−2} |μ_{i} − \frac{2m}{n}|

+ μ_{1} − μ_{n−1} + \frac{2 \overline{m}}{n} + ∑_{i=2}^{n−2} |n − μ_{i} − \frac{2 \overline{m}}{n}|

≥ 2 (μ_{1} − μ_{n−1}) + n − 1 + ∑_{i=2}^{n−2} 1 = 2 (μ_{1} − μ_{n−1}) + 2n − 4 .

By Lemma 4.1, LE(G) + LE(\overline{G}) > 2n − 2 . □

Theorem 4.3. Let G be a graph with n vertices. Then

LE(G) + LE(\overline{G}) < n \sqrt{n^{2} − 1} .

P r o o f. Denote by d_{1}, d_{2}, …, d_{n} the vertex degrees of G. Assume that n ≥ 2 . Let the auxiliary quantity M be defined as [15]

M = M(G) = m + \frac{1}{2} ∑_{i=1}^{n} (d_{i} − \frac{2m}{n})^{2} .

Then

M(\overline{G}) = \overline{m} + \frac{1}{2} ∑_{i=1}^{n} (d_{i} − \frac{2m}{n})^{2} .

Using the fact

∑_{i=1}^{n}(d_{i})^{2} ≤ 2(n − 1)m

with equality if and only if G is the empty graph or the complete graph, we have

M(G) + M(\overline{G}) = \frac{1}{2} n(n − 1) + ∑_{i=1}^{n} (d_{i} − \frac{2m}{n})^{2}

= \frac{1}{2} n(n − 1) + ∑_{i=1}^{n}(d_{i})^{2} − \frac{4m^{2}}{n}

≤ \frac{1}{2} n(n − 1) + 2(n − 1)m − \frac{4m^{2}}{n}
≤ \frac{1}{2} n(n − 1) + \frac{1}{4} n(n − 1)^{2} = \frac{1}{4} (n − 1)n(n + 1) .

Now, because for n ≥ 2 the number of edges of K_{n} and \overline{K_{n}} differs from n(n − 1)/4 , we have

M(G) + M(\overline{G}) < \frac{1}{4} (n − 1)n(n + 1) .  (6)

In [15] it has been shown that LE(G) ≤ \sqrt{2nM} , which combined with (6) implies

LE(G) + LE(\overline{G}) ≤ \sqrt{4n [M(G) + M(\overline{G})]} < n \sqrt{n^{2} − 1} . □

Example 4.4. Let G ≅ K_{n/2} ∪ \overline{K_{n/2}} . Then the Laplacian eigenvalues of G are

\frac{n}{2} (\frac{n}{2} − 1 times ) and 0 (\frac{n}{2} + 1 times )

and therefore

LE(G) = (\frac{n}{2} − 1) \frac{n + 2}{4} + (\frac{n}{2} + 1) \frac{n − 2}{4} = \frac{1}{4} (n^{2} − 4) .

The Laplacian eigenvalues of \overline{G} are

n (\frac{n}{2} times ) , \frac{n}{2} (\frac{n}{2} − 1 times ) and 0 (1 time )

and therefore

LE(\overline{G}) = \frac{n}{2} \frac{n + 2}{4} + (\frac{n}{2} − 1) \frac{n − 2}{4} + \frac{3n − 2}{4} = \frac{1}{4} (n^{2} + 2n) .

This implies

LE(G) + LE(\overline{G}) = \frac{1}{2} (n^{2} + n − 2) .

Acknowledgement. This work was supported by the National Natural Science Foundation of China through Grant no. 10671076, and by the Serbian Ministry of Science and Environmental Protection, through Grant no. 144015G.

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Department of Mathematics South China Normal University Guangzhou 510631 P. R. China Faculty of Science University of Kragujevac P. O. Box 60 34000 Kragujevac Serbia