MATCH Communications in Mathematical and in Computer Chemistry
MATCH Commun. Math. Comput. Chem. 54 (2005) 363-378
ISSN 0340 - 6253
On the extremal energies of trees with a given maximum degree* Wenshui Lin a, a
Xiaofeng Guo a, †,
Haohong Li b
School of Mathematical Sciences, Xiamen University, Xiamen Fujian 361005, P. R. China b
Department of Chemistry, Fuzhou University, Fuzhou Fujian 350002, P. R. China
(Received July 6, 2004)
Abstract The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Let T n denote the set of trees with n vertices. When n ≥ 6 , for a given integer ∆ ∈ [3, n − 2] , we characterize the tree in T n with
maximum degree ∆ and maximal energy. Furthermore, for ⎡ n3+1 ⎤ ≤ ∆(T ) ≤ n − 2 and n ≥ 7 , the tree in T n with maximum degree ∆ and minimal energy is also determined.
* †
The research is supported by National Natural Science Foundation of China. Corresponding author. E-mail address:
[email protected]
- 364 -
1. Introduction In the present paper we consider graphs without loops and multiple edges. Let G be such a graph with n vertices. A k -matching of G is a set of k independent edges in G , k = 1,2, L, ⎣n / 2⎦ . And m(G , k ) will denote the number of k -matchings of G . It is both consistent and convenient to define m(G , 0) = 1 , and m(G , k ) = 0 for k > ⎢⎣ n / 2⎥⎦ . Let G be a graph with vertex set {v1 , v2 ,L , vn } . Its adjacency matrix A(G ) = (aij ) is defined to be the n × n matrix ( aij ) , where aij = 1 if vi is
adjacent to v j , and aij = 0 otherwise. The characteristic polynomial of G is just φ (G ) = det( xI − A(G )) , where I denote the identity matrix of order n . The n roots of the equation φ (G ) = 0 , denoted by λ1 , λ2 ,L , λn , are called the
eigenvalues of graph G . In chemistry the (experimentally determined) heats of 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 [15] ) that of E (G ) = λ1 + λ2 + L + λn . n
If the characteristic polynomial of the graph G is φ (G ) = ∑ ai x n −i , then i =0
E (G ) can be expressed in terms of the Coulson integral [15] as
E (G ) =
1
π∫
+∞
−∞
⎣⎢ n / 2 ⎦⎥
⎣⎢ n / 2⎦⎥
k =0
j =0
x −2 ln[( ∑ (−1) j a2 j x 2 j ) 2 + ( ∑ (−1) j a2 j +1 x 2 j +1 ) 2 ]dx .
Furthermore it is well known [2] that if T is a tree with n vertices then φ (T ) =
⎢⎣ n / 2 ⎥⎦
∑ (−1) m(T , k ) x k
i =0
E (T ) =
2
π
∫
+∞
0
n−2 k
. Hence
⎣⎢ n / 2⎦⎥
x −2 ln[ ∑ m(T , k ) x 2 k ]dx . k =0
(*)
- 365 -
It is easily seen that E (T ) is a strictly monotonously increasing function of all matching numbers m(T , k ), k = 0,1,L, ⎣ n2 ⎦ . Based on this fact Gutman [4] introduced a quasi-ordering relation “ f ” (i.e. reflexive and transitive relation) on the set of all forests (acyclic graphs) with n vertices: if T1 and T2 are two forests with n vertices, then T1 f T2 ⇔ m(T1 , k ) ≥ m(T2 , k ) for all k = 0,1,L , ⎣⎢ n / 2⎦⎥ . If T1 f T2 and there exists j such that m(T1 , j ) > m(T2 , j ) , then we write T1 f T2 . If T1 f T2 ( T1 f T2 ), we also write T2 p T1 (resp. T2 p T1 ). Hence by (*) we have T1 f T2 ⇒ E (T1 ) ≥ E (T2 ) and T1 f T2 ⇒ E (T1 ) > E (T2 ) .
This quasi-ordering has been successfully applied in the study of the extremal values of energy over a significant class of graphs (see [3, 4, 5, 6-15]). In [4] Gutman determined the tree in T n with the maximal energy, namely, the path Pn . Furthermore, he obtained the following result. Lemma 1.1 [4].
Let T be a tree in T n \ { X n , Yn , Z n ,Wn } . If n ≥ 5 , then
E ( X n ) < E (Yn ) < E ( Z n ) < E (Wn ) ≤ E (T ) .
In Lemma 1.1, X n is the star K1,n −1 , Yn is the graph obtained by attaching a pendant vertex to a pendant vertex of K1,n − 2 , Z n by attaching two pendant vertex to a pendant vertex of K1,n −3 , Wn by attaching a P2 to a pendant vertex of K1,n −3 . Fig.1 shows the trees X 9 , Y9 , Z 9 and W9 .
X9
Y9
Z9
W9
Fig 1. The trees X 9 , Y9 , Z 9 and W9 . In this paper we use the quasi-ordering to determine the trees with a given
- 366 -
maximum degree and extremal energies. For a given integer ∆ ∈ [3, n − 2] and n ≥ 6 , the tree in T n with maximum degree ∆ and maximal energy is given.
Furthermore, for ⎡ n3+1 ⎤ ≤ ∆(T ) ≤ n − 2 and n ≥ 7 , the tree in T n with maximum degree ∆ and minimal energy is also determined. 2. Preliminaries First we need the following notations. The vertices of the path Pn will be labeled by v1 , v2 ,L , vn so that vi and vi +1 are adjacent.
Let G and H be two graphs whose vertex sets are disjoint. If v is a vertex of G and w a vertex of H , then G (v, w) H is the graph obtained by identifying the vertices v and w . In particular, the graph Pn (vr , v)G is obtained by identifying the vertex vr of Pn with the vertex v of G . Let u and v be two vertices of the graph G . Then G (u , v)(a, b) denotes the graph obtained from G by attaching a pendant vertices to the vertex u and by attaching b additional pendant vertices to the vertex v . Two vertices u and v of the graph G are called equivalent if the subgraphs G − u and G − v are isomorphic. With the above notations, Gutman and Zhang [1] have shown the following results. Lemma 2.1 [1]. If v is an arbitrary vertex of the graph G , then for n = 4k + i, i ∈ {−1, 0,1, 2}, k ≥ 1 , Pn (v1 , v)G f Pn (v3 , v)G f L f Pn (v2 k +1 , v)G
f Pn (v2 k , v)G f Pn (v2 k − 2 , v)G f L f Pn (v2 , v)G .
Lemma 2.2 [1]. If the vertices u and v of the graph G are equivalent, then
- 367 -
G (u, v)(0, n) p G (u, v)(1, n − 1) p L p G (u, v)( ⎣⎢ n / 2⎦⎥ , n − ⎣⎢ n / 2 ⎦⎥ ) .
Definition 2.1. We call the transformation from G1 = Pn (vr , v)G to Pn (v1 , v)G , where r ≥ 2 and n ≥ 3 , the α 1 -transformation of G1 . Definition 2.2. We call the transformation from G1 = Pn (vr , v)G to Pn (v3 , v)G , where r ≠ 1, 3 and n ≥ 6 , the α 3 -transformation of G1 . Definition
2.3. We call
the transformation from
G1 = G (u , v)(a, b)
to
G (u , v )( a − 1, b + 1) the β -transformation of G1 , and the transformation from
G1 = G (u , v)(a, b) to G (u , v )(0, n) the β ' -transformation of G1 , where 1 ≤ a ≤ b ,
and u, v are equivalent in G . By Lemmas 2.1 and 2.2, the following results are immediate. Corollary 2.1.
If G0 can be obtained from G by one step of α 1 - or
α 3 -transformation, then G0 f G .
Corollary 2.2.
If G0 can be obtained from G by one step of β - or
β ' -transformation, then G0 p G .
Definition 2.4. Let T be a tree in T n , and n ≥ 3 . Let e = uv be a nonpendant edge of T , and let T1 and T2 be the two components of T − e , u ∈ T1 , v ∈ T2 . T0 is the tree obtained from T in the following way.
(1) Contract the edge e = uv (i.e. identify u of T1 with v of T2 ). (2) Attach a pendant vertex to the vertex u ( = v ). The procedures (1) and (2) are called [16] the edge-growing transformation of T (on edge e = uv ), or e.g.t of T (on edge e = uv ) for short. Lemma 2.3.
Let T be a tree in T n with at least a nonpendant edge, and
n ≥ 3 . If T0 can be obtained from T by one step of e.g.t (on edge e = uv ), then T f T0 and E (T ) > E (T0 ) .
- 368 -
Proof.
On one hand, each k -matching of T0 corresponds a k -matching of
T , k = 0,1,L , ⎢⎣ n / 2⎥⎦ , thus m(T , k ) ≥ m(T0 , k ) and T f T0 . On the other hand, since
e = uv is a nonpendant edge of T , we can find a vertex adjacent to u (resp. v )
in T1 (resp. in T2 ), say u1 (resp. v1 ). Then {u1u, vv1} is a 2-matching of T , but not one of T0 , so m(T , 2) > m(T0 , 2) . Hence T f T0 , and E (T ) > E (T0 ) . Lemma 2.4 [1].
□
If e = uv is an edge of G , then for all k ≥ 1 ,
m(G , k ) = m(G − u − v, k − 1) + m(G − e, k ) .
The lemmas and corollaries above are often used to determine the quasi-order between two trees in the remainder of this paper. In order to formulate our results, we need to define three trees: S (n, m, r ) , where
n = 2m + r + 1 ≥ 5 , m ≥ 1 , r ≥ 0 , m + r ≥ 3 ;
Y ( n , m, r )
,
where
n = 2m + r + 1 , n ≥ 8 , m ≥ 2 , r ≥ 3 ; and D ( n, p, q ) ( n ≥ 4 , p ≥ q ≥ 1 , p + q = n − 2 ) as
following: S (n, m, r ) is obtained from the star K1,m + r by attaching one pendant vertex to each of m pendant vertices of K1,m + r . Y (n, m, r ) is obtained from the path Pr +1 by attaching m P2 to one end vertex of Pr +1 . D (n, p, q ) is obtained from the star K1, p +1 by attaching q pendant vertices to one pendant vertex of K1, p +1 . The three trees are shown in Fig.2.
r m
m S (n, m, r )
Pr +1
p
Y (n, m, r )
q
D(n, p, q )
Fig.2. The trees S (n, m, r ) , Y (n, m, r ) and D (n, p, q ) . For S (n, m, r ) , Hou has shown the following result [17]. Lemma 2.5 [17].
Let T be a tree in T n with m -matchings (i.e. m(T , m) > 0 ).
- 369 -
Then T f S (n, m − 1, n − 2m + 1) with the equality iff T ≅ S (n, m − 1, n − 2m + 1) . ∆
Let T n = {T ∈ T n | T consists of ∆ paths with a common end vertex}, ∆ = 3,4, L , n − 1 . Then S ( n, n − ∆ − 1, 2∆ − n + 1) and Y ( n, ∆ − 1, n − 2∆ + 1) are both in ∆
∆
∆
T n , and Pn ∉ T n . Obviously, if T ∈ T n , then T has exact one vertex of degree
more than 2. We call the vertex of degree ∆ (> 2) the root of T , and call each of the ∆ paths a pendant path of T (rooting at the root). We also let Tr2,s ,t and Tp3,q ,l ,m be the two trees shown in Fig. 3, where r , s ≥ 1 , t ≥ 0 and p, q ≥ 0 , l , m ≥ 1 .
q
p
r
s
m
l
t Tr2,s ,t
Tp3,q ,l ,m Fig 3. The trees Tr2,s ,t and Tp3,q ,l ,m .
3. Main results We first determine the tree with a given maximum degree and maximal energy. Theorem 3.1. Let T be a tree in T n and n ≥ 4 . Then T p T1* (n, ∆ ) and E (T ) ≤ E (T1* ( n, ∆ )) , with the equalities iff T ≅ T1* ( n, ∆ ) , where T1* (n, ∆ ) = S ( n, n − ∆ − 1, 2∆ − n + 1) if 3 ≤ ⎣ n2 ⎦ ≤ ∆(T ) ≤ n − 2 , T1* ( n, ∆ ) = Y ( n, ∆ − 1, n − 2∆ + 1)
if 3 ≤ ∆(T ) < ⎣ n2 ⎦ , and T1* (n, ∆) = Pn if ∆(T ) = 2 . Proof.
If ∆(T ) = 2 , then T ≅ Pn = T1* (n, ∆ ) , and the conclusion holds. So we ∆
suppose ∆ (T ) ≥ 3 hereafter. If T ∉ T n , then T can be transformed into a tree
- 370 -
∆
T '∈T n
by carrying out α 1 -transformation repeatedly. Thus T p T ' by ∆
Corollary 2.1. So it is sufficient to show that for any tree T ∈ T n and T ≠ T1* (n, ∆) , T p T1* (n, ∆) .
We distinguish the following two cases. Case 1.
∆(T ) = ∆(T ) ≥ ⎣n / 2⎦ ≥ 3 . Since T ≠ T1* ( n, ∆ ) = S ( n, n − ∆ − 1,2∆ − n + 1) , then
T has at least a pendant path with not less than 4 vertices. Moreover, there must
exist at least a pendant path P2 in T . (Otherwise the number of vertices of T
is n ≥ 2∆ + 2 ≥ 2 ⎢⎣ n / 2⎥⎦ + 2 > n , a contradiction.) So T can be transformed into T1* (n, ∆ ) by repeatedly carrying out α 3 -transformation, and T p T1* (n, ∆) by
Corollary 2.1. Case 2.
∆(T ) = ∆(T ) < ⎣n / 2⎦ . Since T ≠ T1* ( n, ∆ ) = Y ( n, ∆ − 1, n − 2∆ + 1) , then either
T has at least two different pendant paths with not less than 4 vertices, or T
has at most one pendant paths with not less than 4 vertices and at least a pendant path P2 . If T is the former case, then T can be transformed into a tree T '
with only one pendant path with not less than 4 vertices by carrying out α 3 -transformation
repeatedly.
Hence
T pT'
by
Corollary
2.1.
If
T ' ≅ T ( n, ∆ ) = Y ( n, ∆ − 1, n − 2∆ + 1) , then the result holds; otherwise, T ' is the latter * 1
case. Thus it remains to show that T ' p T1* (n, ∆ ) , where T ' has at most one
pendant path with not less than 4 vertices and at least a pendant path P2 . Then T ' must have at least one pendant path with not less than 5 vertices. Otherwise,
T ' has at most one pendant path with 4 vertices and at least one pendant path
P2 . Then, if n is odd, n ≤ 2∆ + 1 ≤ 2⎣n / 2⎦ < n , a contradiction; if n is even, n < 2∆ + 1 ≤ 2 ⎣n / 2⎦ = n , again a contradiction. Thus T ' can be transformed into a
tree T " which has a pendant path with at least 4 vertices by a step of α 3 -transformation. If T " still contains a pendant path P2 , by the same
- 371 -
reasoning, T " can be transformated into T1* (n, ∆ ) = Y (n, ∆ − 1, n − 2∆ + 1) by repeatedly carrying out α 3 -transformations. Hence T ' p T1* (n, ∆ ) by Corollary 2.1. □
The proof is thus completed.
Noting that, for each ∆ , 2 ≤ ∆ ≤ n − 2 , T1* (n, ∆ + 1) can be transformed into T1* (n, ∆ ) by exact one step of α 1 -transformation, we have T1* (n, ∆ + 1) p T1* ( n, ∆ ) by
Corollary 2.1, and the next result follows from Theorem 3.1 immediately. Corollary 3.1.
Let T be a tree in T n and n ≥ 5 . If ∆ (T ) ≥ l ≥ 3 , then
T p T (n, l ) , with the equality iff T ≅ T1* (n, l ) . * 1
Furthermore we can obtain the following interesting result. Corollary 3.2.
Let T1 and T2 be two trees in T n , and n ≥ 4 . If T1 has
m -matchings and ∆(T2 ) ≥ n − m , then T1 f T2 , with the equality iff T1 ≅ T2 ≅ S (n, m − 1, n − 2m + 1) .
Proof.
Immediate from Lemma 2.5 and Corollary 3.1.
□
Now we consider the tree with a given maximum degree and minimal energy. Theorem 3.2.
Let T be a tree in T n , and n ≥ 7 . If ⎡⎢ n3+1 ⎥⎤ ≤ ∆(T ) ≤ n − 2 , then
T f T2* ( n, ∆ ) and E (T ) ≥ E (T2* ( n, ∆ )) , with the equalities iff T ≅ T2* ( n, ∆ ) , where T2* (n, ∆ ) = D (n, ∆ − 1, n − ∆ − 1)
⎡⎢
n +1 3
if
n ⎢⎡ 2 ⎥⎤ ≤ ∆(T ) ≤ n − 2 ,
and
T2* (n, ∆) = T∆−2 1,∆−1,n − 2 ∆−1
if
⎤⎥ ≤ ∆(T ) ≤ ⎡⎢ ⎤⎥ − 1 . n 2
It is easy to see that, if 3 ≤ ⎢⎡ n2 ⎥⎤ ≤ ∆ ≤ n − 2 , T can be transformed into T2* (n, ∆ ) by carrying out e.g.t and β -transformation repeatedly, so T f T2* (n, ∆ ) with the equality iff T ≅ T2* (n, ∆ ) by Lemma 2.3 and Corollary 2.2. However, in order to prove the conclusion of Theorem 3.2 for ⎡⎢ n3+1 ⎥⎤ ≤ ∆(T ) ≤ ⎢⎡ n2 ⎥⎤ − 1 , we need more
- 372 -
preparations. Let T be a tree in T n with maximum degree ∆ , where
Lemma 3.1. ⎡⎢
n +1 3
⎤⎥ ≤ ∆(T ) ≤ ⎡⎢ ⎤⎥ − 1 , n 2
T f T (n, ∆) = T * 2
Proof.
2 ∆−1, ∆−1, n − 2 ∆−1
and
n≥7
.
If
T = Tr2,s ,t
or
T = Tp3,q ,l ,m
,
then
, with the equality iff T ≅ T (n, ∆ ) . * 2
We prove the result by the following three cases.
Case 1. T = Tr2,s ,t . Then either r = ∆ − 1 and s ≤ ∆ − 1 , t ≤ ∆ − 2 , or t = ∆ − 2 and s, r ≤ ∆ − 1 .
Subcase 1.1.
r = ∆ − 1 . Then s + t = n − ∆ − 2 . By repeatedly applying Lemma 2.4
∆ − 1 times, we have, for all k ≥ 1 ,
m(T , k ) = ( ∆ − 1) × m( D ( n − ∆, t , s ), k − 1) + m( D ( n − ∆ + 1, t + 1, s ), k ) , and
m(T2* ( n, ∆ ), k ) = ( ∆ − 1) × m( D ( n − ∆, ∆ − 1, n − 2∆ − 1), k − 1) + m( D ( n − ∆ + 1, ∆ − 1, n − 2∆ ), k ) .
Moreover by Lemma 2.2, D (n − ∆, t , s ) f D (n − ∆, ∆ − 1, n − 2∆ − 1) and D ( n − ∆ + 1, t + 1, s ) f D ( n − ∆ + 1, ∆ − 1, n − 2∆ ) , so m(T , k ) ≥ m(T2* ( n, ∆ ), k ) , for all k ≥ 1 .
Hence T f T2* (n, ∆ ) , with the equality iff T ≅ T2* (n, ∆ ) . Subcase 1.2. 2 ∆−1, n − 2 ∆ , ∆− 2
T
t = ∆ − 2 . Obviously, r + s ≥ ∆ . Thus T can be transformed into
by β -transformation, and by Corollary 2.2 and Subcase 1.1,
T f T∆−2 1,n −2 ∆ ,∆− 2 f T2* (n, ∆) , with the equality iff T ≅ T∆−2 1,n −2 ∆ ,∆− 2 ≅ T2* (n, ∆) .
Case 2. T = Tp3,q ,l ,m . Then either l = ∆ − 1 , p, q ≤ ∆ − 2 and m ≤ ∆ − 1 , or p = ∆ − 2 , q ≤ ∆ − 2 and l , m ≤ ∆ − 1 .
Subcase 2.1.
l = ∆ − 1 . By repeatedly applying Lemma 2.4 ∆ − 1 times, we
have, for all k ≥ 1 , m(T , k ) = (∆ − 1) × m(Tp2,m,q , k − 1) + m(Tp2+1,m,q , k ) and m(T2* ( n, ∆ ), k ) = ( ∆ − 1) × m( D ( n − ∆, ∆ − 1, n − 2∆ − 1), k − 1) + m( D ( n − ∆ + 1, ∆ − 1, n − 2∆ ), k ) .
If p + m ≥ ∆ − 1 , and without loss of generality assuming p ≥ m , then Tp2,m,q can be transformed into D (n − ∆, ∆ − 1, n − 2∆ −1) by exact (∆ − 1) − p = ∆ − p − 1 steps
- 373 -
of β -transformation, and followed one step of e.g.t if p + m ≥ ∆ . So by Corollary 2.2 and 2.3, Tp2,m,q f D(n − ∆, ∆ − 1, n − 2∆ − 1) . Otherwise p + m < ∆ − 1 , then Tp2,m,q can be transformed into D (n − ∆, p + m, q + 1) by one step of β ' -transformation,
so
Tp2,m,q f D(n − ∆, p + m, q + 1) f D(n − ∆, ∆ − 1, n − 2∆ − 1)
Corollary 2.2 and Lemma 2.2. Similarly, T
2 p +1, m , q
by
f D(n − ∆ + 1, ∆ − 1, n − 2∆) .
Therefore m(T , k ) ≥ m(T2* (n, ∆ ), k ) , and T f T2* (n, ∆ ) . Subcase 2.2.
3 3 p = ∆ − 2 . We will show that T = T∆− 2, q ,l , m f T ' = Tl −1, q , ∆−1, m , then the
case is deduced to Subcase 2.1. It is easy to see that for k ≥ 1 , m(T , k ) = m × m(Tl ,2q ,∆−2 , k − 1) + m(Tl ,2q +1,∆− 2 , k ) ,
and
m(T ', k ) = m × m(T∆−2 1,q ,l −1 , k − 1) + m(T∆−2 1,q +1,l −1 , k ) . Similarly, we have, for all k ≥ 2 , m(Tl ,2q ,∆−2 , k − 1) = q × m( D(l + ∆, l , ∆ − 2), k − 2) + m( D(l + ∆ + 1, l , ∆ − 1), k − 1) and m(T∆−2 1,q ,l −1 , k − 1) = q × m( D(l + ∆, l − 1, ∆ − 1), k − 2) + m( D(l + ∆ + 1, l , ∆ − 1), k − 1) . Noting
that l ≤ ∆ − 1 , for k ≥ 2 , we have m( D (l + ∆, l , ∆ − 2), k − 2) ≥ m( D (l + ∆ , l − 1, ∆ − 1), k − 2) , so m(Tl ,2q ,∆−2 , k ) ≥ m(Tl −21,∆−1,q , k ) .
Similarly m(T∆−2 2,l ,q +1 , k − 1) ≥ m(Tl −21,∆−1,q+1 , k − 1) for all k ≥ 2 . Therefore m(T , k ) ≥ m(T ', k ) for all k ≥ 2 , and T f T ' .
□
The proof is thus completed.
Let ε (G ) denote the number of edges of the graph G . Let K1,∆ be a star, and v1 , v2 ,L , v∆ its vertices of degree 1. Let H i be a tree with maximum degree at most ∆ , and ui a vertex of H i , i = 1, 2,L , ∆ , with degree at most ∆ − 1 . Then T (n, ∆; H1 , H 2 ,L , H ∆ ) will denote the tree K1, ∆ (v1 , u1 ) H1 (v2 , u2 ) H 2 L (v∆ , u∆ ) H ∆ . Let
ε i = ε ( H i ) , i = 1, 2,L , ∆ . Without loss of generality we assume that ε1 ≥ ε 2 ≥ L ≥ ε ∆ .
When ε i = 0 , H i is an isolated vertex ui , which will be denoted by K1 . If H i is a star K1,ε with the center ui , then we write H i as Cε . (A center of a star is i
i
the vertex of the star with maximum degree.) Obviously T (n, ∆; H1 , H 2 ,L , H ∆ ) has maximum degree ∆ , while every tree in
- 374 -
T n with maximum degree ∆ has the form T (n, ∆; H1 , H 2 ,L , H ∆ ) .
Let T = T (n, ∆; H1, H2 ,L, H∆ ) with ε 3 > 0 and ⎡ n3+1 ⎤ ≤ ∆(T ) ≤ ⎡ n−22 ⎤ .
Lemma 3.2.
Then there exists a tree T ' = T (n, ∆; H1 ', Ct , K1,L, K1 ) , where ε ( H1 ' ) ≥ ∆ − 1 , t = n − ∆ − ε ( H1 ') − 1 ≤ ∆ − 1 , such that T f T ' .
Proof.
If ε1 ≤ ∆ − 1 , then ε i ≤ ∆ − 1 , i = 2, 3,L , ∆ . Thus T can be transformed
into T ' = T (n, ∆; C∆−1 , Cn−2∆ , K1,L, K1 ) by a number of e.g.t and β -transformations, so the conclusion holds from Corollary 2.2 and Lemma 2.3. If ε1 ≥ ∆ , then t = ∑ i∆= 2 ε i ≤ ∆ − 1 . (Otherwise ε (T ) = n − 1 ≥ 3∆ ≥ 3⎡ n3+1 ⎤ > n , a contradiction.) Thus T can be transformed into T ' = T (n, ∆; H1 , Ct , K1 ,L, K1 ) by a number of e.g.t and
β -transformations, and so T f T ' .
□
Now we give the proof of Theorem 3.2. Proof of Theorem 3.2.
We have mentioned that the conclusion holds if
n−2 n +1 n ⎢⎡ 2 ⎥⎤ + 1 ≤ ∆ ≤ n − 2 . Now we only deal with the case when ⎡⎢ 3 ⎥⎤ ≤ ∆(T ) ≤ ⎢⎡ 2 ⎥⎤ − 1
here. By Lemma 3.2, it suffices to show the following statement: for each tree T = T (n, ∆; H1, Ct , K1 ,L, K1 ) T f T (n, ∆) = T * 2
2 ∆−1, ∆−1, n − 2 ∆−1
with
ε 1 = ε ( H1 ) ≥ ∆ − 1
and
t = n − ∆ − ε1 − 1 ≤ ∆ − 1 ,
, with the equality iff T ≅ T (n, ∆ ) . * 2
If ε1 = ∆ − 1 , then T can be transformed into T1 = T∆−1,t ,∆−2 by e.g.t, and the conclusion holds from Lemma 2.3 and 3.1. Hence assume ε1 ≥ ∆ . Then we can find an edge e = wu1 in H1 such that the degree of w is more than 1 in H1 . Let L1 and L2 be the two components of H1 − e such that w is in L1 ( L2 may be K1 ). We complete the proof by induction on τ (T ) , the number of nonpendant edges of T . When τ = 2 , T = Tr2,s ,t for some r , s and t with ∆ (T ) = ∆ , so the result holds from Lemma 3.1. Assume
that
the
statement
is
true
when
τ = l −1 .
Now
let
- 375 -
T = T (n, ∆; H1, Ct , K1 ,L, K1 ) be a tree with τ (T ) = l . We distinguish the following
two cases. Case 1. t > 0 , i.e. Ct ≠ K1 . Subcase 1.1. ε ( L1 ) ≤ ∆ − 1 and ε ( L2 ) ≤ ∆ − 2 . Thus by e.g.t T 3 ∆− 2,ε ( L2 ),t ,ε ( L1 )
transformed into T
can be
, so the statement is true by Lemma 3.1.
Subcase 1.2. ε ( L1 ) ≥ ∆ . Thus ε ( L2 ) + t ≤ ∆ − 2 . By e.g.t, T can be transformed into Tˆ so that L2 becomes K1,ε ( L ) with the center u1 . Then T f Tˆ by Lemma 2
2.3. Let T1 be the tree obtained from Tˆ by moving all the t pendant edges at vertex u2 to u1 . By induction hypothesis T1 f T2* (n, ∆) , with the equality iff T1 ≅ T2* ( n, ∆ ) , so it remains to show that Tˆ f T1 . Let m '( L1 , i ) denote the number
of i -matchings of L1 in which at least one edge is incident with the vertex w , and m ''( L1 , j ) the number of j -matchings of L1 which consist of edges not incident with the vertex w . Let T ' = Tε2( L ),t ,∆−2 and T '' = Tε2( L )+1,t ,∆−2 . Then for all 2
2
k ≥0, k
k
i =1
j =0
m(Tˆ , k ) = ∑ m '( L1 , i ) × m(T ', k − i ) + ∑ m ''( L1 , j ) × m(T '', k − j ) .
Similarly, D ' = D(∆ + ε ( L2 ) + t + 3, ∆ + 1, ε ( L2 ) + t ) and D '' = D(∆ + ε ( L2 ) + t + 4, ∆ + 1, ε ( L2 ) + t + 1) ). k
k
i =0
j =0
m(T1 , k ) = ∑ m' ( L1 , i ) × m( D' , k − i) + ∑ m" ( L1 , j ) × m( D", k − j ) .
Moreover by Lemma 2.2, T ' f D ' and T '' f D '' , so m(Tˆ , k ) ≥ m(T1 , k ) for all k ≥ 0 . Hence T f Tˆ f T1 f T2* (n, ∆) .
Subcase 1.3. ε ( L2 ) ≥ ∆ − 1 . Thus ε ( L1 ) + t ≤ ∆ − 1 . By e.g.t, T can be transformed into Tˆ so that L1 becomes K1,ε ( L ) with the center w . Let T1 be the tree 1
obtained from Tˆ by moving all the ε ( L1 ) at vertex w to u2 . Similar to Subcase 1.2, we have T f T1 , and the statement holds. Case 2. t = 0 , i.e. Ct = K1 .
- 376 -
Subcase 2.1. ε ( L1 ) ≤ ∆ − 1 and ε ( L2 ) ≤ ∆ − 2 . Thus by e.g.t T 2 ∆−1,ε ( L1 ),ε ( L2 )
transformed into T
can be
, and the statement holds by Lemma 3.1.
Subcase 2.2. ε ( L1 ) ≥ ∆ . Thus ε 2 ≤ ∆ − 2 . By e.g.t, T can be transformed into Tˆ so that L2 becomes K1,ε ( L ) with the center u1 . Let e ' = yw be a nonpendant 2
edge with an end vertex y in L1 . Let I1 and I 2 be the two components of L1 − e' such that y is in I1 ( I 2 may be K1 ).
Subsubcase 2.2.1. ε ( I1 ) ≤ ∆ − 1 and ε ( I 2 ) ≤ ∆ − 2 . Thus by e.g.t Tˆ can be transformed into Tε3( L ),ε ( I ),∆−1,ε ( I ) , and the statement holds by Lemma 3.1. 2
2
1
Subsubcase 2.2.2. ε ( I1 ) ≥ ∆ . Thus ε ( I 2 ) + ε ( L2 ) + 1 ≤ ∆ − 2 . Let T1 be the tree obtained from Tˆ by e.g.t on edge e = wu1 and nonpendant edges in L2 . Then T f T1 f T2* ( n, ∆ ) by Lemma 2.3 and induction hypothesis.
Subsubcase 2.2.3. ε ( I 2 ) ≥ ∆ − 1 . Thus ε ( I1 ) + ε ( L2 ) + 1 ≤ ∆ − 2 . Let T1 be the tree obtained from Tˆ by e.g.t on nonpendant edges in I1 and then moving all the ε ( I1 ) pendant edges at vertex y to u1 . Similar to Subcase 1.2, we have Tˆ f T1 ,
and the conclusion holds. Subcase 2.3. ε ( L2 ) ≥ ∆ − 1 . Then ε ( L1 ) ≤ ∆ − 2 . By e.g.t, T can be transformed into Tˆ so that L1 becomes K1,ε ( L ) with the center w . Let e '' = zu1 be a 1
nonpendant edge with an end vertex u1 in H1 . Let N1 and N 2 be the two components of L2 − e '' such that z is in N1 ( N 2 may be K1 ). Subsubcase 2.3.1. ε ( N1 ) ≤ ∆ − 1 and ε ( N 2 ) ≤ ∆ − 3 . By e.g.t, Tˆ
can be
transformed into T1 so that N1 becomes K1,ε ( N ) with the center z and N 2 1
becomes K1,ε ( N ) with the center u1 . If s = ε ( L1 ) + ε ( N1 ) ≤ ∆ − 1 , then by Lemma 2
2.2 and Lemma 3.1 we have T f Tˆ f T1 f T∆−1, s ,ε ( N )+1 f T2* (n, ∆) . Otherwise s ≥ ∆ . 2
Then T1 can be transformed into T (n, ∆) = T * 2
2 ∆−1, ∆−1, n − 2 ∆−1
by exact ∆ − 1 − ε (L1 )
- 377 -
steps of β -transformation and followed a step of e.g.t. on edge e '' = zu1 . Hence by Corollary 2.2, Lemma 2.3 and Lemma 3.1, T f T2* (n, ∆ ) . Subsubcase 2.3.2. ε ( N1 ) ≥ ∆ . Thus ε ( L1 ) + ε ( N 2 ) ≤ ∆ − 3 . Let T1 be the tree obtained from Tˆ by repeatedly carrying out e.g.t so that N 2 becomes K1,ε ( N ) 2
with the center u1 , and followed a step of e.g.t on edge e = wu1 . Hence T f Tˆ f T1 f T2* (n, ∆) by Lemma 2.3 and induction hypothesis.
Subsubcase 2.3.3. ε ( N 2 ) ≥ ∆ − 2 . Thus ε ( L1 ) + ε ( N1 ) ≤ ∆ − 1 . Let T1 be the tree obtained from Tˆ by repeatedly carrying e.g.t such that N1 becomes K1,ε ( N ) 1
with the center z , and then moving all the ε ( N1 ) pendant edges at vertex z to w (i.e. a step of β ' -transformation). Then T f Tˆ f T1 f T2* (n, ∆) by Corollary 2.2,
Lemma 2.3 and induction hypothesis. The proof is completed.
□
References [1] I. Gutman, Fuji Zhang, On the ordering of graphs with respect to their matching numbers, Discrete Appl. Math.,15(1986) 25-33. [2] D.M. Cvetkovic, M. Doob, H.Sachs, Spectra of Graphs, Academic Press, New York, 1980. [3] I. Gutman, Acyclic conjugated molecules, trees and their energies, J. Math. Chem., 1(1987), 123-143. [4] I. Gutman, Acyclic systems with extremal Huckel π -electronic energy, Theoret. Chim. Acta (Berlin), 45(1977), 79-87. [5] D. Cvetkovic, M. Doob, I. Gutman and A. Torgasev, Recent Results in the Theory of Graph Spectra (Elsevier, Amsterdam, 1986).
- 378 -
[6] I. Gutman, Y. Hou, Bipartite unicyclic graphs with greatest energy, MATCH Commun. Math. Comput. Chem., 43(2001), 17-28. [7] Y. Hou, Unicyclic graphs with minimal energy, J. Math. Chem., 29(2001). 163-168. [8] Y. Hou, Bicyclic graphs with minimal energy, Linear Multilinear Algebra, 49(2002), 347-354. [9] Y. Hou, I. Gutman, C.-W. Wou, Unicyclic graphs with maximal energy, Linear Algebra Appl., 356(2002), 27-36. [10] H. Li, On minimal energy ordering of acyclic conjugated molecules, J. Math. Chem.. 25(1999), 145-169. [11] J.Rada, A. Tineo, Polygonal chains with minimal energy, Linear Algebra Appl., 372(2003). 333-344. [12] F. Zhang, H. Li, On acyclic conjugated molecules with minimal energies, Discrete Appl. Math., 92(1999), 71-84. [13] F. Zhang, Z. Li, L. Wang, Hexagonal chains with minimal total π -energy, Chem. Phys. Lett., 337(2001), 125-130. [14] F. Zhang, Z. Li, L. Wang, Hexagonal chains with maximal total π -energy, Chem. Phys. Lett., 337(2001), 131-137. [15] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986, p.135-142. [16] G.H. Xu, On the spectral radius of trees with perfect matchings, in: Combinatorics and Graph Theory, World Scientific, Singapore, 1997. [17] Y. Hou, On trees with the least energy and a given size of matching, J. Sys. Sci. & Math. Scis., 23(4)(2003,10), 491-494 (in Chinese).
