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Discrete Applied Mathematics 157 (2009) 869–873

Contents lists available at ScienceDirect

Discrete Applied Mathematics

journal homepage: www.elsevier.com/locate/dam

Note

On bipartite graphs with minimal energy$

Xueliang Lia,∗, Jianbin Zhanga, Lusheng Wangb

aCenter for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, PR China

bDepartment of Computer Science, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

a r t i c l e i n f o

Article history:

Received 11 December 2007

Received in revised form 25 July 2008

Accepted 30 July 2008

Available online 29 August 2008

Keywords:

Minimal energy

Characteristic polynomial

Eigenvalue

Bipartite graph

a b s t r a c t

The energy of a graph is the sum of the absolute values of the eigenvalues of the graph.

In a paper [G. Caporossi, D. Cvetkovi, I. Gutman, P. Hansen, Variable neighborhood search

for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci.

39 (1999) 984–996] Caporossi et al. conjectured that among all connected graphs G with

n ≥ 6 vertices and n − 1 ≤ m ≤ 2(n − 2) edges, the graphs with minimum energy

are the star Snwith m − n + 1 additional edges all connected to the same vertices for

m ≤ n + ?(n − 7)/2?, and the bipartite graph with two vertices on one side, one of which

is connected to all vertices on the other side, otherwise. The conjecture is proved to be

true for m = n − 1,2(n − 2) in the same paper by Caporossi et al. themselves, and for

m = n by Hou in [Y. Hou, Unicyclic graphs with minimal energy, J. Math. Chem. 29 (2001)

163–168]. In this paper, we give a complete solution for the second part of the conjecture

on bipartite graphs. Moreover, we determine the graph with the second-minimal energy

in all connected bipartite graphs with n vertices and m(n ≤ m ≤ 2n − 5) edges.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Let G = (V,E) be a graph without loops or multiple edges with vertex set V = {v1,v2,...,vn} and edge set E. Denote

the degree of vertexviby d(vi). The adjacency matrix A(G) = [aij] of G is an n×n symmetric matrix of 0’s and l’s with aij= 1

if and only if viand vjare joined by an edge.

Wedenotebyφ(G,x)thecharacteristicpolynomialdet(xI−A(G))ofGandcalltherootsofdet(xI−A(G))theeigenvalues

of G. It is well known [3] that if G is a bipartite graph, then

?n

?

where b2i(G) = (−1)ia2iand b2i(G) ≥ 0 for all i = 1,...,?n

of G.

In chemistry, the experimental heats from the formation of conjugated hydrocarbons are closely related to the total

π-electron energy. And the calculation of the total energy of all π-electrons in conjugated hydrocarbons can be reduced to

(within the framework of HMO approximation) [6] that of

φ(G,x) =

2?

i=0

a2iλn−2i=

?n

?

2?

i=0

(−1)ib2iλn−2i,

2?. Clearly, b0(G) = 1 and b2(G) equals the number of edges

E = E(G) =

n

?

i=0

|λi|,

(1)

$Supported by CityU Project No. 7001996, PCSIRT, NSFC and the ‘‘973’’ program.

∗Corresponding author.

E-mail addresses: lxl@nankai.edu.cn (X. Li), zhangjb.cx@gmail.com (J. Zhang), cswangl@cityu.edu.hk (L. Wang).

0166-218X/$ – see front matter © 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.dam.2008.07.008

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X. Li et al. / Discrete Applied Mathematics 157 (2009) 869–873

Fig. 1. Graphs Bn,mand B?

n,m.

whereλiare the eigenvalues of the corresponding graph G. The right-hand side of Eq. (1) is defined for all graphs (no matter

whether they represent the carbon-atom skeleton of a conjugated electron system or not). In view of this, if G is any graph,

then by means of Eq. (1) one defines E(G) and calls it the energy of the graph G. Recently, it has been intensively studied by

some researchers (see [1,4,7–18]). For a survey of the mathematical properties and results on E(G), see the recent review

paper [5].

It is known [6] that for bipartite graph G, E(G) can be also expressed as the Coulson integral formula

?

i=0

If for two bipartite graphs G1and G2, b2i(G1) ≤ b2i(G2) holds for all i = 1,2,...,?n/2?, we say that G1is smaller than G2,

and write G1? G2or G2? G1. Moreover, if b2i(G1) < b2i(G2) holds for some i, we write G1≺ G2or G2? G1. From Eq. (2)

we know that for two bipartite graphs G1and G2

E(G) =

2

π

?+∞

0

1

x2ln1 +

?n/2?

?

b2ix2i

?

dx.

(2)

G1? G2⇒ E(G1) ≤ E(G2)

G1≺ G2⇒ E(G1) < E(G2).

If a graph G has n vertices, then we say that G is an n-graph; if G has n vertices and m edges, then G is called an (n,m)-

graph. Many results on the minimal energy have been obtained for various classes of graphs. In [1], Caporossi et al. gave the

following conjecture.

Conjecture 1. Among all connected graphs G with n ≥ 6 vertices and n − 1 ≤ m ≤ 2(n − 2) edges, the graphs with minimum

energy are the star Snwith m − n + 1 additional edges all connected to the same vertices for m ≤ n + ?(n − 7)/2?, and the

bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise.

This conjecture is proved to be true for m = n − 1,2(n − 2) [1, Theorem 1], and m = n [9].

The main purpose of this paper is to consider connected bipartite graphs. Clearly, the extremal bipartite graph of

Conjecture 1 is a connected (n,m)-graph such that n ≤ m ≤ 2(n − 2).

Let Bn,mbe the bipartite(n,m)-graph with two vertices on one side, one of which is connected to all vertices on the other

side, otherwise. Let B?

degree in Bn−1,m−1(see Fig. 1).

In this paper, we will show that Bn,mis the unique graph with minimal energy in all bipartite connected (n,m)-graphs

for n ≤ m ≤ 2(n − 2), giving a complete solution to the above Conjecture 1 on bipartite graphs. Moreover, we prove that

B?

n,mbe the graph obtained from Bn−1,m−1by adding a pendant edge to the vertex of second-maximal

n,mis the unique graph with second-minimal energy in all bipartite connected (n,m)-graphs for n ≤ m ≤ 2n − 5.

2. Main result

Let G be a graph with characteristic polynomial φ(G,λ) =?n

ai=

S∈Li

where Lidenotes the set of Sachs subgraphs (see [3]) of G with i vertices, that is, the graphs S in which every component is

either a K2or a cycle, p(S) is the number of components of S and c(S) is the number of cycles contained in S. In addition,

a0= 1.

By Sachs’ Theorem [3] we have

i=0aiλn−i. Then for i ≥ 1

?

(−1)p(S)2c(S),

Lemma 1 ([18]). b4(G) = m(G,2) − 2q(G), where q(G) denotes the number of quadrangles in G.

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X. Li et al. / Discrete Applied Mathematics 157 (2009) 869–873

871

Lemma 2. Let uv be a cut edge in G. Then

b4(G) = b4(G − uv) + e(G − u − v).

In particular, let uv be a pendant edge of G with the pendant vertex v; then

b4(G) = b4(G − v) + e(G − u − v).

Proof. Since uv is a cut edge, we have q(G) = q(G − uv). By Lemma 1 we have

b4(G − uv) = m(G − uv,2) − 2q(G − uv),

b4(G) = m(G,2) − 2q(G)

= m(G − uv,2) + m(G − u − v,1) − 2q(G)

= b4(G − uv) + e(G − u − v).

We thus obtain the result.

?

Theorem 1. Bn,m(n ≤ m ≤ 2(n − 2)) is the unique graph with minimal energy in all bipartite connected (n,m)-graphs.

Proof. Let G be a bipartite connected (n,m)-graph. Then ∆(G) ≤ n − 2.

Since bi(Bn,m) = 0 for i ?= 0,2,4, we have that b0(G) = 1 and b2(G) = m. It suffices to prove that b4(G) ≥ b4(Bn,m). We

apply induction on n to prove it. From the table of [2] the result is true for n = 7. So we suppose that n ≥ 8 and the result

is true for smaller n.

Case 1. There is a pendant edge uv in G with pendant vertex v. Then, by Lemma 2 we have

b4(G) = b4(G − v) + e(G − u − v).

Since ∆(G) ≤ n − 2, we have e(G − u − v) ≥ m − ∆(G) ≥ m − n + 2 = e(Sm−n+3). By the induction hypothesis,

b4(G − v) ≥ b4(Bn−1,m−1). Since b4(Bn,m) = b4(Bn−1,m−1) + e(Sm−n+3), we get the result b4(G) ≥ b4(Bn,m).

Case 2. There are no pendant vertices in G.

Claim 1. Let G be a connected bipartite (n,m)-graph for n ≤ m ≤ 2(n − 2). Then q(G) ≤

number of quadrangles in G.

Proof. We apply induction on m. The result is obvious for m = n. So we suppose n ≤ m ≤ 2(n − 2) and the result is true

for smaller m.

Let e be an edge of a cycle in G. Then G contains at most m − n + 1 quadrangles containing the edge e. Otherwise, we

suppose that there are m − n + a(a ≥ 2) quadrangles containing e = uv. Let U be a set of neighbor vertices of u except v,

and let V be a set of neighbor vertices of v except u. Then there are just m− n+ a edges between U and V. Let X be a subset

of U such that each vertex in X is incident to some of the above m−n+a edges and Y be a subset of V defined similarly to X.

Assume|X| = x,|Y| = y. Let G0be a subgraph of G induced by V(G0) = u∪v∪X ∪Y. There are at least m−n+a+x+y+1

edges and exactly x + y + 2 vertices in G0. In order for the remaining vertices to connect to G0, the number of remaining

edges must be not less than that of the remaining vertices:

?

m−n+2

2

?

, where q(G) denotes the

m − (m − n + a + x + y + 1) ≥ n − (x + y + 2),

that is,

(3)

n − a − 1 ≥ n − 2 (a ≥ 2).

This is a contradiction. Note that inequality (3) still holds when there are no remaining vertices.

Let qG(e) denote the number of quadrangles in G. Thus we have

q(G) = qG(e) + q(G − e)

≤ m − n + 1 +

2

?

A nonincreasing sequence (d)G = (d1,d2,...,dn) of positive integers is said to be graphic if there a simple graph G

having degree sequence (d)G.

?m − 1 − n + 2

m−1−n+2

2

?

=

?m − n + 2

2

?

,

where q(G − e) ≤

?

, obtained by the induction hypothesis.

?

Claim 2. Let G be a bipartite connected (n,m)-graph. If G has no pendant vertices, then

?

v∈V(Bn,m)

?d(v)

2

?

≥

?

v∈V(G)

?d(v)

2

?

.

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X. Li et al. / Discrete Applied Mathematics 157 (2009) 869–873

Proof. Let

(d)G= (d1,d2,...di−1,di,...,dj,dj+1,...,dn),

and

(d)?= (d1,d2,...di,di+ 1,...,dj− 1,dj+1,...,dn)

where di≥ dj. By writing

(d)?= (d?

we can obtain that?n

?

Notice that d1≥ d2≥ ··· ≥ dn≥ 2 and d1+d2+···+dn= 2m. Repeating this procedure, we can obtain the sequence

2n−m−4

?

where d??

the above procedure repeatedly we finally obtain the degree sequence (d)Bn,m:

1,d?

2,...d?

i−1,d?

?

?dt

i,...,d?

>?n

=

j,d?

?

j+1,...,d?

dt

2

?

n),

t=1

d?

2

t

?

t=1

?

+

, since

n

t=1

?d?

t

2

?

−

n

?

t=1

2

??di+ 1

2

?dj− 1

2

?

−

??di

2

?

+

?dj

2

??

= di− dj+ 2 > 0.

(d??) = (d??

1≤ n − 2, d??

1,d??

2,d3,...,dm−n+4,

2≥ d2if d??

???

1,1,...,1)

1= n − 2, and d??

2= d2if d??

1< n − 2. Since d??

1≥ d??

2≥ d3≥ ··· ≥ dm−n+4≥ 2, by applying

(d)Bn,m= (n − 2,m − n + 2,

which has the maximum value of?

For a simple graph G, we have m(G,2) =?m

b4(G) =

2

v∈V(G)

By Claims 1 and 2, we can easily obtain the result.

Combining all the above cases we thus complete the proof.

m−n+2

??

v∈V(G)

??

2

2,2,...,2,

2n−m−4

??

. The proof of the claim is thus complete.

?

??

1,1,...,1),

?

?−?

?

d(v)

?

2

v∈V(G)

d(v)

2

?

. By Lemma 1 we know that

?m

?

−

?

?d(v)

2

?

− 2q(G).

?

Since the characteristic polynomial of Bn,mis

φ(Bn,m,x) = xn−4[x4− mx2+ (m − n + 2)(2n − m − 4)],

by simple computation we have

Corollary 1. Let G be a bipartite connected (n,m)-graph with n ≤ m ≤ 2n − 4. Then

?

with equality if and only if G∼= Bn,m.

Theorem 2. B?

E(G) ≥ 2

m + 2

?

(m − n + 2)(2n − m − 4),

n,m(n ≤ m ≤ 2n − 5) is the unique graph with second-minimal energy in all bipartite connected (n,m)-graphs.

Proof. Since b0(B?

integers i, like in the proof of Theorem 1 we can obtain the result.

n,m) = 1,b2(B?

n,m) = m,b4(B?

n,m) = (m − n + 3)(2n − m − 4) − 1 and bi(B?

?

n,m) = 0 for other positive

Analogously, we have

φ(B?

n,m,x) = xn−4{x4− mx2+ [(m − n + 3)(2n − m − 4) − 1]},

and therefore

Corollary 2. Let G be a bipartite connected (n,m)-graph with n ≤ m ≤ 2n − 5. If G ?∼= Bn,m, then

?

with equality if and only if G∼= B?

E(G) ≥ 2

m + 2

?

(m − n + 3)(2n − m − 4) − 1,

n,m.

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