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On some degree-and-distance-based graph invariants of trees
Applied Mathematics and Computation 289 (2016) 1–6
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On some degree-and-distance-based graph invariants of trees Ivan Gutman a,b,∗, Boris Furtula a, Kinkar Ch. Das c a b c
Faculty of Science, University of Kragujevac, Kragujevac, Serbia State University of Novi Pazar, Novi Pazar, Serbia Department of Mathematics, Sungkyunkwan University, Suwon, Republic of Korea
a r t i c l e
i n f o
a b s t r a c t Let G be a connected graph with vertex set V(G). For u, v ∈ V (G ), d (v ) and d (u, v ) denote the degree of the vertex v and the distance between the vertices u and v. A much studied degree–and–distance–based graph invariant is the degree distance, defined as DD = {u,v}⊆V (G ) [d (u ) + d (v )] d (u, v ). A related such invariant (usually called “Gutman in dex”) is ZZ = {u,v}⊆V (G ) [d (u ) · d (v )] d (u, v ). If G is a tree, then both DD and ZZ are linearly related with the Wiener index W = {u,v}⊆V (G ) d (u, v ). We examine the difference DD − Z Z for trees and establish a number of regularities.
Keywords: Distance (in graph) Degree distance Wiener index Gutman index
© 2016 Elsevier Inc. All rights reserved.
1. Introduction Let G be a connected graph of order n with vertex set V (G ) = {v1 , v2 , . . . , vn } and edge set E(G). The degree of the vertex v ∈ V (G ), denoted by dG (v ) = d (v ), is the number of first neighbors of v in the graph G. The distance of the vertices u, v ∈ V (G ), denoted by dG (u, v ) = d (u, v ) is the length of (= number of edges in) a shortest path in G, connecting u and v. In this paper we are concerned with two degree–and–distance–based graph invariants, namely with the degree distance defined as
DD = DD(G ) =
d (u ) + d (v ) d (u, v )
(1)
{u,v}⊆V (G )
and another closely related invariant, defined as
Z Z = Z Z (G ) =
d (u ) · d (v ) d (u, v ).
(2)
{u,v}⊆V (G )
In addition, we recall the definition of the Wiener index:
W = W (G ) =
d (u, v ).
(3)
{u,v}⊆V (G )
The Wiener index was introduced in 1947 by Harold Wiener [15] and since then became one of the most extensively studied distance–based graph invariants; for details see the surveys [3,17] and the recent papers [7,9–11,14]. The degree distance, as defined by Eq. (1), was put forward by Dobrynin and Kochetova in 1994 [4]. In the meantime, this degree–and–distance–based graph invariant became a popular topic for mathematical studies; for details see the recent papers [1,2,5,12,16,18] and the references cited therein. ∗
Corresponding author. Tel.: +381-34-331876. E-mail addresses: [email protected] (I. Gutman), [email protected] (B. Furtula), [email protected] (K.Ch. Das).
http://dx.doi.org/10.1016/j.amc.2016.04.040 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
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I. Gutman et al. / Applied Mathematics and Computation 289 (2016) 1–6
The fact that in the case of trees there is a simple linear relation between DD and the Wiener index was first noticed in [13], and then mathematically proven by Klein [8]. An independent proof was offered by one of the present authors [6]. The respective result can be stated as: Theorem 1 [6,8]. Let T be a tree of order n. Then its degree distance and Wiener index are related as
DD(T ) = 4 W (T ) − n(n − 1 ).
(4)
In [6], it was noticed that an identity analogous to Eq. (4) holds if the sum d (u ) + d (v ) is replaced by d (u ) · d (v ): Theorem 2 [6]. Let T be a tree of order n. Then
Z Z (T ) = 4 W (T ) − (2n − 1 )(n − 1 ).
(5)
Theorem 2 was the sole reason for considering the invariant ZZ, namely the multiplicative analogue of degree distance. Eventually and unfortunately, this invariant is nowadays referred to as the Gutman index. Combining Eqs. (4) and (5), one immediately realizes that DD(T ) − Z Z (T ) = (n − 1 )2 . Curiously, whereas both Eqs. (4) and (5) were discovered in the 1990s, the fact that their difference depends solely on the number of vertices of the underlying tree seems to have so far eluded attention. We state the same observation in a slightly different manner: Corollary 3. Let T be a tree of order n. Then the expression
dT (u ) + dT (v ) − dT (u ) dT (v ) dT (u, v )
{u,v}⊆V (T )
is independent of the structure of T, and is equal to (n − 1 )2 . 2. Elaborating Corollary 3 In the case of trees, it is purposeful to divide the Wiener index, Eq. (3), into two parts as W = Wex + Win , where
Wex = Wex (T ) =
dT (u, v )
(6)
{u,v}⊆V (T )
min{d (u ),d (v )}=1
Win = Win (T ) =
dT (u, v ).
{u,v}⊆V (T )
min{d (u ),d (v )}≥2
We refer to these as to the “external” and “internal” Wiener index of the tree T. Evidently, the external Wiener index pertains to the distances between pairs of vertices of T, of which at least one is pendent. Note now that the term d (u ) + d (v ) − d (u ) d (v ) is equal to unity if either d (u ) = 1 or d (v ) = 1, equal to zero if d (u ) = d (v ) = 2, and negative–valued in all other cases. Define an auxiliary quantity as
= (G ) =
dG (u ) dG (v ) − dG (u ) − dG (v ) dG (u, v )
{u,v}⊆V (G )
min{d (u ),d (v )}≥2
and note that its value is non-negative for all graphs G. Bearing in mind the above, Corollary 3 can be re-stated as: Corollary 4. Let T be a tree of order n. Then
DD(T ) − Z Z (T ) = Wex (T ) − (T ) = (n − 1 )2 i.e.,
Wex (T ) = (n − 1 )2 + (T ) . Let, as usual, Sn and Pn denote the star and path of order n. Theorem 5. Let T be a tree of order n. Then
Wex (T ) ≥ (n − 1 )2 . Equality holds if and only if T∼ =Sn or T∼ =Pn for all n ≥ 2. Proof. The inequality in Theorem 5 is a direct consequence of (T) ≥ 0. Equality (T ) = 0 happens in two cases: (a) if the tree T does not have a pair of non-pendent vertices, and (b) if all non-pendent vertices of the tree T are of degree two.
I. Gutman et al. / Applied Mathematics and Computation 289 (2016) 1–6
3
Fig. 1. Trees considered in Theorem 7.
From (a) and (b) follows T∼ =Sn and T∼ =Pn , respectively.
Remark 6. In graph theory, numerous invariants are known, which for trees with a fixed number of vertices are minimal for the star and maximal for the path (or vice versa). The external Wiener index is one of the rare examples of a graph invariant for which the star and the path both assume minimal values. Let Ti , i = 1, 2, . . . , 10 be the trees depicted in Fig. 1. Theorem 7. If T is a tree of order n and Wex is its external Wiener index, Eq. (6), then the following holds: (a) (b) (c) (d)
Wex (T ) = Wex (T ) = Wex (T ) = Wex (T ) =
( n − 1 )2 + 1 ( n − 1 )2 + 2 ( n − 1 )2 + 3 ( n − 1 )2 + 4
if if if if
and and and and
Proof. We first note that
d (u ) d (v ) −d (u ) −d (v ) =
only only only only
if if if if
⎧ 0 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎨ 2 3
4 ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎩>4 >4
n=5 n=6 n=6 n=7
if if if if if if if if
and T∼ =T1 . and T ∈ {T2 , T3 }. or n = 7 and T ∈ {T4 , T5 , T6 , T7 }. or n = 8 and T ∈ {T8 , T9 , T10 }.
d (u ) = d (v ) = 2 {d ( u ), d ( v )} = {2, 3} {d ( u ), d ( v )} = {2, 4} {d ( u ), d ( v )} = {2, 5} {d ( u ), d ( v )} = {2, 6} d (u ) = d (v ) = 3 {d ( u ), d ( v )} = {2, k} , k ≥ 7 {d (u ), d (v )} = {, k} , ≥ 3, k ≥ 4 .
(7)
(a) Bearing in mind Eq. (7), we see that Wex (T ) = (n − 1 )2 + 1 i.e., (T ) = 1 will hold only if T has a single pair of nonterminal vertices, of degrees 2 and 3, at distance 1. This implies T∼ =T1 . (b) Wex (T ) = (n − 1 )2 + 2 i.e., (T ) = 2 will hold in two cases: First: T has a single pair of non-terminal vertices of degrees 2 and 4, at distance 1, implying T∼ =T2 . Second: T has two pairs of non-terminal vertices of degrees 2 and 3, at distance 1, implying T∼ =T3 . (c) Wex (T ) = (n − 1 )2 + 3 i.e., (T ) = 3 will hold in four cases: First: T has a single pair of non-terminal vertices, both of degree 3, at distance 1, implying T∼ =T4 . Second: T has a single pair of non-terminal vertices of degrees 2 and 5, at distance 1, implying T∼ =T5 . Third: T has three pairs of non-terminal vertices of degrees 2 and 3, at distance 1, implying T∼ =T6 . Fourth: T has a pair of non-terminal vertices of degrees 2 and 3 at distance 1, and another such pair at distance 2 (plus a third pair both having degree 2), which implies T∼ =T7 .
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I. Gutman et al. / Applied Mathematics and Computation 289 (2016) 1–6
The proof of statement (d) of Theorem 7 is analogous.
Theorem 8. Let k be a positive integer. The number of trees satisfying the relation Wex (T ) = (n − 1 )2 + k is finite. Proof. In view of Corollary 4, equivalent to Theorem 8 is the claim that there is a finite number of trees for which (T ) = k holds. First note, that (T ) = k > 0 cannot hold for trees with less than five vertices, since such trees are stars or paths. We therefore assume that n ≥ 5. Let T0 be a tree on n vertices, possessing a vertex x of degree greater than 2. Thus, T0 ∼ = Pn . Let y be any other vertex of T0 . Construct the tree T1 from T0 by attaching a pendent vertex to its vertex y. Thus, T1 ∼ = Sn+1 We have to separately examine the cases dT0 (y ) = 1 and dT0 (y ) ≥ 2 Bearing in mind the definition of , and recalling that dT1 (x, y ) = dT0 (x, y ) ≥ 1, if dT0 (x ) ≥ 3 and dT0 (y ) = 1, then
(T1 ) − (T0 ) ≥ dT1 (x ) dT1 (y ) − dT1 (x ) − dT1 (y ) d (x, y ) = dT0 (x ) · 2 − dT0 (x ) − 2 d (x, y ) = dT0 (x ) − 2 d (x, y ) ≥ 1 .
If dT0 (x ) ≥ 3 and dT0 (y ) ≥ 2, then,
(T1 ) − (T0 ) ≥ dT1 (x ) dT1 (y ) − dT1 (x ) − dT1 (y ) d (x, y ) − dT0 (x ) dT0 (y ) −dT0 (x ) −dT0 (y ) d (x, y ) = dT0 (x )[dT0 (y ) + 1] − dT0 (x ) − [dT0 (y ) + 1] d (x, y ) − dT0 (x ) dT0 (y ) − dT0 (x ) − dT0 (y ) d (x, y ) = dT0 (x ) − 1 d (x, y ) ≥ 2 .
Thus, the transformation T0 → T1 increases the -value by at least one. Therefore, if T ∼ = Sn , Pn , then (T ) ≥ n − 4. This implies that if (T ) = k, then the tree T must have at most k + 4 vertices. The number of such trees is evidently finite. Corollary 9. Let k be a positive integer. Any tree satisfying the relation Wex (T ) = (n − 1 )2 + k has k + 4 or fewer vertices. For any positive integer k, there exists a tree with n = k + 4 vertices, for which Wex (T ) = (n − 1 )2 + k holds. Such a tree is the double broom DB(n − 2, 2, 0 ), defined below. Remark 10. Theorem 7 may serve as an illustration of Theorem 8 and Corollary 9 for k = 1, 2, 3, 4. The double broom DB = DBa,b,c is the tree obtained from the path Pc+2 , by attaching a − 1 pendent vertices to its one end, and b − 1 pendent vertices to its the other end. Thus, DBa, b, c has a vertex of degree a, a vertex of degree b and c vertices of degree 2 in between. Its order is n = a + b + c. If |a − b| ≤ 1 then DBa, b, c is said to be a balanced double broom. In view of Theorem 5, it may be of some interest to determine the tree(s) with maximal external Wiener index. By means of a computer–aided search (done until n = 24), we arrived at: Conjecture 11. Among trees of a fixed order, the trees with greatest external Wiener index i.e., with greatest -value are double brooms. By direct calculation, it can be shown that
(DBa,b,c ) = (ab − a − b)(c + 1 ) + (a + b − 4 )
c (c + 1 ) . 2
(8)
We now intend to determine the maximal value of Wex (DB) for a given value of n. Assume that n is fixed. Consider the case when also the parameter c has some fixed value. Then a + b also has a fixed value. Then from the form of Eq. (8) we immediately see that (DB) will be maximal if the product ab is maximal. This happens if the parameters a and b (which are integers!) differ as little as possible, i.e., if |b − a| ≤ 1. Thus, the double broom with maximal Wex must be a balanced double broom. We have separately examine two cases: b = a and b = a + 1. Case 1: b = a Then,
= (a2 − 2a )(c + 1 ) + (2a − 4 )
c (c + 1 ) 2
and c = n − 2a. This yields
= (a − 2 )(n − a )(n + 1 − 2a ) = 2a3 − (3n + 5 )a2 + (n2 + 7n + 2 )a − 2n(n + 1 )
I. Gutman et al. / Applied Mathematics and Computation 289 (2016) 1–6
5
Table 1 The parameters a, b, c maximizing Wex (DBa, b, c ) and (DBa, b, c ). In most cases, for a given value of n, the triplet (a, b, c) is unique. For n ≤ 100 double solutions were found for n = 6, 9, 28, 58, 99. For n = 59, (a, b, c ) = (13, 14, 31 )&(14, 14, 30 ). For n = 99, (a, b, c ) = (22, 22, 55 )&(22, 23, 54 ). n
a
b
c
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 7 7 7 7 7 8
3 3 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8
0 1 1 2 3 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 11 12 13 14 14 14
a
b
c
3
3
0
3
4
2
7
8
13
a cubic polynomial in variable a. It is easy to
see that the zeros of this polynomial are 2, (n + 1 )/2 and n. Consequently it has a single maximum, lying in the interval 2, (n + 1 )/2 . By standard calculus we get that the maximum is at
a=
1 3n + 5 − 3n2 − 12n + 13 . 6
Bearing in mind that a, b, c are integers, we conclude that in the Case 1, the double broom with maximal external Wiener index is either DBa1 ,b1 ,c1 or DBa2 ,b2 ,c2 where
a1 = and
a2 =
1 6
1 6
3n + 5 −
3n + 5 −
3n2 − 12n + 13
,
3n2 − 12n + 13
,
b1 = a1 ,
c1 = n − 2 a1
b2 = a2 ,
c2 = n − 2 a2 .
Case 2: b = a + 1 An analogous consideration show that the double broom with maximal Wex -value is either DBa3 ,b3 ,c3 or DBa4 ,b4 ,c4 where
a3 = and
a4 =
1 6
1 6
3n + 2 −
3n + 2 −
3n2 − 12n + 10
,
3n2 − 12n + 10
,
b3 = a3 + 1 ,
c3 = n − 2 a3 − 1
b4 = a4 + 1 ,
c4 = n − 2 a4 − 1 .
Summarizing the above, we get: Theorem 12. The double broom of order n, with maximal external Wiener index is one or more among DBa1 ,b1 ,c1 , DBa2 ,b2 ,c2 , DBa3 ,b3 ,c3 , DBa4 ,b4 ,c4 . Conjecture 13. The tree of order n, with maximal external Wiener index is one or more among DBa1 ,b1 ,c1 , DBa2 ,b2 ,c2 , DBa3 ,b3 ,c3 , DBa4 ,b4 ,c4 . For the first few values of n, the parameters a, b, c that maximize Wex (DB) and (DB) are given in Table 1. It is not easy to envisage any regularity for their n-dependence.
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I. Gutman et al. / Applied Mathematics and Computation 289 (2016) 1–6
Acknowledgment The second author was partially supported by the Serbian Ministry of Science and Education, through the Grant No. 174033. The third author was supported by the National Research Foundation of the Korean government by Grant No. 2013R1A1A2009341. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
P. Ali, S. Mukwembi, S. Munyira, Degree distance and vertex–connectivity, Discret. Appl. Math. 161 (2013) 2802–2811. P. Ali, S. Mukwembi, S. Munyira, Degree distance and edge–connectivity, Australas. J. Combin. 60 (2014) 50–68. A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001) 211–249. A.A. Dobrynin, A.A. Kochetova, Degree distance of a graph: a degree analogue of the wiener index, J. Chem. Inf. Comput. Sci. 34 (1994) 1082–1086. L. Feng, W. Liu, A. Ilic´ , G. Yu, The degree distance of unicyclic graphs with given matching number, Graphs Combin. 29 (2013) 449–462. I. Gutman, Selected properties of the schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34 (1994) 1087–1089. A. Kelenc, S. Klavžar, N. Tratnik, The edge–wiener index of benzenoid systems in linear time, MATCH Commun. Math. Comput. Chem. 74 (2015) 521–532. D.J. Klein, Z. Mihalic´ , D. Plavšic´ , N. Trinajstic´ , Molecular topological index: a relation with the wiener index, J. Chem. Inf. Comput. Sci. 32 (1992) 304–305. H. Lin, On the wiener index of trees with given number of branching vertices, MATCH Commun. Math. Comput. Chem. 72 (2014) 301–310. H. Lin, Extremal wiener index of trees with given number of vertices of even degree, MATCH Commun. Math. Comput. Chem. 72 (2014) 311–320. M. Knor, B. Luzar, R. Škrekovski, I. Gutman, On wiener index of common neighborhood graphs, MATCH Commun. Math. Comput. Chem. 72 (2014) 321–332. S. Mukwembi, S. Munyira, Degree distance and minimum degree, Bull. Austral. Math. Soc. 87 (2013) 255–271. D. Plavšic´ , S. Nikolic´ , N. Trinajstic´ , D.J. Klein, Relation between the wiener index and the schultz index for several classes of chemical graphs, Croat. Chem. Acta 66 (1993) 345–353. R. Škrekovski, I. Gutman, Vertex version of the wiener theorem, MATCH Commun. Math. Comput. Chem. 72 (2014) 295–300. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20. K. Xu, S. Klavžar, K.C. Das, J. Wang, Extremal (n, m)-graphs with respect to distance–degree–based topological indices, MATCH Commun. Math. Comput. Chem. 72 (2014) 865–880. K. Xu, M. Liu, K.C. Das, I. Gutman, B. Furtula, A survey on graphs extremal with respect to distance–based topological indices, MATCH Commun. Math. Comput. Chem. 71 (2014) 461–508. K. Xu, J. Wang, K.C. Das, S. Klavžar, Weighted harary indices of apex trees and k-apex trees, Discrete Appl. Math. 189 (2015) 30–40.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On some degree-and-distance-based graph invariants of trees Ivan Gutman a,b,∗, Boris Furtula a, Kinkar Ch. Das c a b c
Faculty of Science, University of Kragujevac, Kragujevac, Serbia State University of Novi Pazar, Novi Pazar, Serbia Department of Mathematics, Sungkyunkwan University, Suwon, Republic of Korea
a r t i c l e
i n f o
a b s t r a c t Let G be a connected graph with vertex set V(G). For u, v ∈ V (G ), d (v ) and d (u, v ) denote the degree of the vertex v and the distance between the vertices u and v. A much studied degree–and–distance–based graph invariant is the degree distance, defined as DD = {u,v}⊆V (G ) [d (u ) + d (v )] d (u, v ). A related such invariant (usually called “Gutman in dex”) is ZZ = {u,v}⊆V (G ) [d (u ) · d (v )] d (u, v ). If G is a tree, then both DD and ZZ are linearly related with the Wiener index W = {u,v}⊆V (G ) d (u, v ). We examine the difference DD − Z Z for trees and establish a number of regularities.
Keywords: Distance (in graph) Degree distance Wiener index Gutman index
© 2016 Elsevier Inc. All rights reserved.
1. Introduction Let G be a connected graph of order n with vertex set V (G ) = {v1 , v2 , . . . , vn } and edge set E(G). The degree of the vertex v ∈ V (G ), denoted by dG (v ) = d (v ), is the number of first neighbors of v in the graph G. The distance of the vertices u, v ∈ V (G ), denoted by dG (u, v ) = d (u, v ) is the length of (= number of edges in) a shortest path in G, connecting u and v. In this paper we are concerned with two degree–and–distance–based graph invariants, namely with the degree distance defined as
DD = DD(G ) =
d (u ) + d (v ) d (u, v )
(1)
{u,v}⊆V (G )
and another closely related invariant, defined as
Z Z = Z Z (G ) =
d (u ) · d (v ) d (u, v ).
(2)
{u,v}⊆V (G )
In addition, we recall the definition of the Wiener index:
W = W (G ) =
d (u, v ).
(3)
{u,v}⊆V (G )
The Wiener index was introduced in 1947 by Harold Wiener [15] and since then became one of the most extensively studied distance–based graph invariants; for details see the surveys [3,17] and the recent papers [7,9–11,14]. The degree distance, as defined by Eq. (1), was put forward by Dobrynin and Kochetova in 1994 [4]. In the meantime, this degree–and–distance–based graph invariant became a popular topic for mathematical studies; for details see the recent papers [1,2,5,12,16,18] and the references cited therein. ∗
Corresponding author. Tel.: +381-34-331876. E-mail addresses: [email protected] (I. Gutman), [email protected] (B. Furtula), [email protected] (K.Ch. Das).
http://dx.doi.org/10.1016/j.amc.2016.04.040 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
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I. Gutman et al. / Applied Mathematics and Computation 289 (2016) 1–6
The fact that in the case of trees there is a simple linear relation between DD and the Wiener index was first noticed in [13], and then mathematically proven by Klein [8]. An independent proof was offered by one of the present authors [6]. The respective result can be stated as: Theorem 1 [6,8]. Let T be a tree of order n. Then its degree distance and Wiener index are related as
DD(T ) = 4 W (T ) − n(n − 1 ).
(4)
In [6], it was noticed that an identity analogous to Eq. (4) holds if the sum d (u ) + d (v ) is replaced by d (u ) · d (v ): Theorem 2 [6]. Let T be a tree of order n. Then
Z Z (T ) = 4 W (T ) − (2n − 1 )(n − 1 ).
(5)
Theorem 2 was the sole reason for considering the invariant ZZ, namely the multiplicative analogue of degree distance. Eventually and unfortunately, this invariant is nowadays referred to as the Gutman index. Combining Eqs. (4) and (5), one immediately realizes that DD(T ) − Z Z (T ) = (n − 1 )2 . Curiously, whereas both Eqs. (4) and (5) were discovered in the 1990s, the fact that their difference depends solely on the number of vertices of the underlying tree seems to have so far eluded attention. We state the same observation in a slightly different manner: Corollary 3. Let T be a tree of order n. Then the expression
dT (u ) + dT (v ) − dT (u ) dT (v ) dT (u, v )
{u,v}⊆V (T )
is independent of the structure of T, and is equal to (n − 1 )2 . 2. Elaborating Corollary 3 In the case of trees, it is purposeful to divide the Wiener index, Eq. (3), into two parts as W = Wex + Win , where
Wex = Wex (T ) =
dT (u, v )
(6)
{u,v}⊆V (T )
min{d (u ),d (v )}=1
Win = Win (T ) =
dT (u, v ).
{u,v}⊆V (T )
min{d (u ),d (v )}≥2
We refer to these as to the “external” and “internal” Wiener index of the tree T. Evidently, the external Wiener index pertains to the distances between pairs of vertices of T, of which at least one is pendent. Note now that the term d (u ) + d (v ) − d (u ) d (v ) is equal to unity if either d (u ) = 1 or d (v ) = 1, equal to zero if d (u ) = d (v ) = 2, and negative–valued in all other cases. Define an auxiliary quantity as
= (G ) =
dG (u ) dG (v ) − dG (u ) − dG (v ) dG (u, v )
{u,v}⊆V (G )
min{d (u ),d (v )}≥2
and note that its value is non-negative for all graphs G. Bearing in mind the above, Corollary 3 can be re-stated as: Corollary 4. Let T be a tree of order n. Then
DD(T ) − Z Z (T ) = Wex (T ) − (T ) = (n − 1 )2 i.e.,
Wex (T ) = (n − 1 )2 + (T ) . Let, as usual, Sn and Pn denote the star and path of order n. Theorem 5. Let T be a tree of order n. Then
Wex (T ) ≥ (n − 1 )2 . Equality holds if and only if T∼ =Sn or T∼ =Pn for all n ≥ 2. Proof. The inequality in Theorem 5 is a direct consequence of (T) ≥ 0. Equality (T ) = 0 happens in two cases: (a) if the tree T does not have a pair of non-pendent vertices, and (b) if all non-pendent vertices of the tree T are of degree two.
I. Gutman et al. / Applied Mathematics and Computation 289 (2016) 1–6
3
Fig. 1. Trees considered in Theorem 7.
From (a) and (b) follows T∼ =Sn and T∼ =Pn , respectively.
Remark 6. In graph theory, numerous invariants are known, which for trees with a fixed number of vertices are minimal for the star and maximal for the path (or vice versa). The external Wiener index is one of the rare examples of a graph invariant for which the star and the path both assume minimal values. Let Ti , i = 1, 2, . . . , 10 be the trees depicted in Fig. 1. Theorem 7. If T is a tree of order n and Wex is its external Wiener index, Eq. (6), then the following holds: (a) (b) (c) (d)
Wex (T ) = Wex (T ) = Wex (T ) = Wex (T ) =
( n − 1 )2 + 1 ( n − 1 )2 + 2 ( n − 1 )2 + 3 ( n − 1 )2 + 4
if if if if
and and and and
Proof. We first note that
d (u ) d (v ) −d (u ) −d (v ) =
only only only only
if if if if
⎧ 0 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎨ 2 3
4 ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎩>4 >4
n=5 n=6 n=6 n=7
if if if if if if if if
and T∼ =T1 . and T ∈ {T2 , T3 }. or n = 7 and T ∈ {T4 , T5 , T6 , T7 }. or n = 8 and T ∈ {T8 , T9 , T10 }.
d (u ) = d (v ) = 2 {d ( u ), d ( v )} = {2, 3} {d ( u ), d ( v )} = {2, 4} {d ( u ), d ( v )} = {2, 5} {d ( u ), d ( v )} = {2, 6} d (u ) = d (v ) = 3 {d ( u ), d ( v )} = {2, k} , k ≥ 7 {d (u ), d (v )} = {, k} , ≥ 3, k ≥ 4 .
(7)
(a) Bearing in mind Eq. (7), we see that Wex (T ) = (n − 1 )2 + 1 i.e., (T ) = 1 will hold only if T has a single pair of nonterminal vertices, of degrees 2 and 3, at distance 1. This implies T∼ =T1 . (b) Wex (T ) = (n − 1 )2 + 2 i.e., (T ) = 2 will hold in two cases: First: T has a single pair of non-terminal vertices of degrees 2 and 4, at distance 1, implying T∼ =T2 . Second: T has two pairs of non-terminal vertices of degrees 2 and 3, at distance 1, implying T∼ =T3 . (c) Wex (T ) = (n − 1 )2 + 3 i.e., (T ) = 3 will hold in four cases: First: T has a single pair of non-terminal vertices, both of degree 3, at distance 1, implying T∼ =T4 . Second: T has a single pair of non-terminal vertices of degrees 2 and 5, at distance 1, implying T∼ =T5 . Third: T has three pairs of non-terminal vertices of degrees 2 and 3, at distance 1, implying T∼ =T6 . Fourth: T has a pair of non-terminal vertices of degrees 2 and 3 at distance 1, and another such pair at distance 2 (plus a third pair both having degree 2), which implies T∼ =T7 .
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The proof of statement (d) of Theorem 7 is analogous.
Theorem 8. Let k be a positive integer. The number of trees satisfying the relation Wex (T ) = (n − 1 )2 + k is finite. Proof. In view of Corollary 4, equivalent to Theorem 8 is the claim that there is a finite number of trees for which (T ) = k holds. First note, that (T ) = k > 0 cannot hold for trees with less than five vertices, since such trees are stars or paths. We therefore assume that n ≥ 5. Let T0 be a tree on n vertices, possessing a vertex x of degree greater than 2. Thus, T0 ∼ = Pn . Let y be any other vertex of T0 . Construct the tree T1 from T0 by attaching a pendent vertex to its vertex y. Thus, T1 ∼ = Sn+1 We have to separately examine the cases dT0 (y ) = 1 and dT0 (y ) ≥ 2 Bearing in mind the definition of , and recalling that dT1 (x, y ) = dT0 (x, y ) ≥ 1, if dT0 (x ) ≥ 3 and dT0 (y ) = 1, then
(T1 ) − (T0 ) ≥ dT1 (x ) dT1 (y ) − dT1 (x ) − dT1 (y ) d (x, y ) = dT0 (x ) · 2 − dT0 (x ) − 2 d (x, y ) = dT0 (x ) − 2 d (x, y ) ≥ 1 .
If dT0 (x ) ≥ 3 and dT0 (y ) ≥ 2, then,
(T1 ) − (T0 ) ≥ dT1 (x ) dT1 (y ) − dT1 (x ) − dT1 (y ) d (x, y ) − dT0 (x ) dT0 (y ) −dT0 (x ) −dT0 (y ) d (x, y ) = dT0 (x )[dT0 (y ) + 1] − dT0 (x ) − [dT0 (y ) + 1] d (x, y ) − dT0 (x ) dT0 (y ) − dT0 (x ) − dT0 (y ) d (x, y ) = dT0 (x ) − 1 d (x, y ) ≥ 2 .
Thus, the transformation T0 → T1 increases the -value by at least one. Therefore, if T ∼ = Sn , Pn , then (T ) ≥ n − 4. This implies that if (T ) = k, then the tree T must have at most k + 4 vertices. The number of such trees is evidently finite. Corollary 9. Let k be a positive integer. Any tree satisfying the relation Wex (T ) = (n − 1 )2 + k has k + 4 or fewer vertices. For any positive integer k, there exists a tree with n = k + 4 vertices, for which Wex (T ) = (n − 1 )2 + k holds. Such a tree is the double broom DB(n − 2, 2, 0 ), defined below. Remark 10. Theorem 7 may serve as an illustration of Theorem 8 and Corollary 9 for k = 1, 2, 3, 4. The double broom DB = DBa,b,c is the tree obtained from the path Pc+2 , by attaching a − 1 pendent vertices to its one end, and b − 1 pendent vertices to its the other end. Thus, DBa, b, c has a vertex of degree a, a vertex of degree b and c vertices of degree 2 in between. Its order is n = a + b + c. If |a − b| ≤ 1 then DBa, b, c is said to be a balanced double broom. In view of Theorem 5, it may be of some interest to determine the tree(s) with maximal external Wiener index. By means of a computer–aided search (done until n = 24), we arrived at: Conjecture 11. Among trees of a fixed order, the trees with greatest external Wiener index i.e., with greatest -value are double brooms. By direct calculation, it can be shown that
(DBa,b,c ) = (ab − a − b)(c + 1 ) + (a + b − 4 )
c (c + 1 ) . 2
(8)
We now intend to determine the maximal value of Wex (DB) for a given value of n. Assume that n is fixed. Consider the case when also the parameter c has some fixed value. Then a + b also has a fixed value. Then from the form of Eq. (8) we immediately see that (DB) will be maximal if the product ab is maximal. This happens if the parameters a and b (which are integers!) differ as little as possible, i.e., if |b − a| ≤ 1. Thus, the double broom with maximal Wex must be a balanced double broom. We have separately examine two cases: b = a and b = a + 1. Case 1: b = a Then,
= (a2 − 2a )(c + 1 ) + (2a − 4 )
c (c + 1 ) 2
and c = n − 2a. This yields
= (a − 2 )(n − a )(n + 1 − 2a ) = 2a3 − (3n + 5 )a2 + (n2 + 7n + 2 )a − 2n(n + 1 )
I. Gutman et al. / Applied Mathematics and Computation 289 (2016) 1–6
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Table 1 The parameters a, b, c maximizing Wex (DBa, b, c ) and (DBa, b, c ). In most cases, for a given value of n, the triplet (a, b, c) is unique. For n ≤ 100 double solutions were found for n = 6, 9, 28, 58, 99. For n = 59, (a, b, c ) = (13, 14, 31 )&(14, 14, 30 ). For n = 99, (a, b, c ) = (22, 22, 55 )&(22, 23, 54 ). n
a
b
c
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 7 7 7 7 7 8
3 3 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8
0 1 1 2 3 3 3 4 5 5 6 6 7 8 8 9 9 10 11 11 11 12 13 14 14 14
a
b
c
3
3
0
3
4
2
7
8
13
a cubic polynomial in variable a. It is easy to
see that the zeros of this polynomial are 2, (n + 1 )/2 and n. Consequently it has a single maximum, lying in the interval 2, (n + 1 )/2 . By standard calculus we get that the maximum is at
a=
1 3n + 5 − 3n2 − 12n + 13 . 6
Bearing in mind that a, b, c are integers, we conclude that in the Case 1, the double broom with maximal external Wiener index is either DBa1 ,b1 ,c1 or DBa2 ,b2 ,c2 where
a1 = and
a2 =
1 6
1 6
3n + 5 −
3n + 5 −
3n2 − 12n + 13
,
3n2 − 12n + 13
,
b1 = a1 ,
c1 = n − 2 a1
b2 = a2 ,
c2 = n − 2 a2 .
Case 2: b = a + 1 An analogous consideration show that the double broom with maximal Wex -value is either DBa3 ,b3 ,c3 or DBa4 ,b4 ,c4 where
a3 = and
a4 =
1 6
1 6
3n + 2 −
3n + 2 −
3n2 − 12n + 10
,
3n2 − 12n + 10
,
b3 = a3 + 1 ,
c3 = n − 2 a3 − 1
b4 = a4 + 1 ,
c4 = n − 2 a4 − 1 .
Summarizing the above, we get: Theorem 12. The double broom of order n, with maximal external Wiener index is one or more among DBa1 ,b1 ,c1 , DBa2 ,b2 ,c2 , DBa3 ,b3 ,c3 , DBa4 ,b4 ,c4 . Conjecture 13. The tree of order n, with maximal external Wiener index is one or more among DBa1 ,b1 ,c1 , DBa2 ,b2 ,c2 , DBa3 ,b3 ,c3 , DBa4 ,b4 ,c4 . For the first few values of n, the parameters a, b, c that maximize Wex (DB) and (DB) are given in Table 1. It is not easy to envisage any regularity for their n-dependence.
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Acknowledgment The second author was partially supported by the Serbian Ministry of Science and Education, through the Grant No. 174033. The third author was supported by the National Research Foundation of the Korean government by Grant No. 2013R1A1A2009341. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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