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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

Wiener index of Eulerian graphs Ivan Gutman a,b,∗ , Roberto Cruz c , Juan Rada c a

Faculty of Science, University of Kragujevac, P. O. Box 60, Kragujevac, Serbia

b

Chemistry Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

c

Instituto de Matemáticas, Universidad de Antioquia, Medellín, Colombia

article

info

Article history: Received 5 April 2013 Received in revised form 17 August 2013 Accepted 19 August 2013 Available online xxxx

abstract The Wiener index of a connected graph G is the sum of distances between all pairs of vertices of G. We characterize Eulerian graphs (with a fixed number of vertices) with smallest and greatest Wiener indices. © 2013 Elsevier B.V. All rights reserved.

Keywords: Distance (in graph) Wiener index Eulerian graph Extremal graph

1. Introduction An Eulerian cycle is a walk in a graph that visits every edge exactly once, and that starts and ends on the same vertex. A graph possessing an Eulerian cycle is said to be Eulerian. According to the classical result by Euler [1], a graph is Eulerian if and only if it is connected and all its vertices have even degrees. Denote by En the set of all Eulerian graphs of order n. Because the smallest number of vertices of an Eulerian graph is 3, throughout this paper it will be always assumed that n ≥ 3. Let G be a connected graph and u and v its two vertices. The distance between u and v is the length ( = number of edges) of a shortest path connecting u and v . The Wiener index of the graph G, denoted by W = W (G), is the sum of distances between all pairs of vertices of G. The Wiener index of graphs has been studied in much detail (see the reviews [6,3,4], the recent papers [5,8,11,12,2,9,10,7], and the references cited therein). Yet, Wiener indices of Eulerian graphs seem to have evaded the attention of scholars. The aim of the present paper is to contribute towards filling this gap. In the present paper, we characterize the elements of En having the first few smallest and the first few greatest Wiener indices. When searching for graphs with extremal values of W , the following result is frequently used: Lemma 1. Let G be a connected graph, and e its edge. Let G′ be obtained from G by deleting the edge e. If G′ is connected, then W (G′ ) > W (G). For our purposes, Lemma 1 is useless because if G ∈ En , then G′ ̸∈ En . However, the following variant of Lemma 1 would be purposeful:

∗

Corresponding author at: Faculty of Science, University of Kragujevac, P. O. Box 60, Kragujevac, Serbia. Tel.: +381 34 331876; fax: +381 34335040. E-mail addresses: [email protected] (I. Gutman), [email protected] (R. Cruz), [email protected] (J. Rada).

0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.08.024

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Lemma 2. Let G be a connected graph, possessing (as subgraph) a cycle Z of size t. Let e1 , e2 , . . . , et be the edges of G belonging to Z . Let G′ be obtained from G by deleting the edges e1 , e2 , . . . , et . If G′ is connected, then W (G′ ) > W (G). The advantage of Lemma 2 is that if G ∈ En and G′ is connected, then also G′ ∈ En . Now, in order to apply Lemma 2, we first have to determine the elements of En whose Wiener index is minimal. 2. Eulerian graphs with smallest Wiener indices Denote by Kn the complete graph of order n. If n is odd, then Kn ∈ En . If n is even, denote by CPn the cocktail-party graph of order n (obtained by deleting n/2 independent edges from Kn ) and note that CPn ∈ En . Theorem 3. (a) If n is odd, then the unique Eulerian graph of order n with minimal Wiener index is Kn . (b) If n is even, then the unique Eulerian graph of order n with minimal Wiener index is CPn . Proof. Case (a) is evident, since Kn has minimal W -value for all (connected) graphs of order n. If n is even, then the maximum vertex degree of an Eulerian graph is n − 2. The degree of all vertices of the cocktail-party graph is n − 2, and thus this Eulerian graph has the smallest possible number (= n/2) of pairs of non-adjacent vertices. Any pair vertices of CPn is at distance 2. Consequently, the Wiener index of CPn has the smallest possible value 1of 2non-adjacent = 2n . By means of Lemma 2 we can now characterize the first few elements of En with small Wiener indices. Theorem 4. (a) If n is odd, n ≥ 5, then the graphs obtained from Kn by deleting the edges of a triangle, of a quadrangle, and of a pentagon have, respectively, second-minimal, third-minimal, and fourth-minimal Wiener index in En . (b) If n is even, n ≥ 6, then the graphs obtained from CPn by deleting the edges of a triangle, of a quadrangle, and of a pentagon have, respectively, second-minimal, third-minimal, and fourth-minimal Wiener index in En . 3. Eulerian graphs with greatest Wiener indices The edge connecting the vertices u and v will be denoted by (u, v). Denote by Cn the cycle of order n and label its vertices by 1, 2, . . . , n, so that its edges are (1, n) and (i, i + 1) for i = 1, . . . , n − 1. Evidently, Cn ∈ En . Theorem 5. If G ∈ En , then W (G) ≤ W (Cn ) with equality if and only if G ∼ = Cn . Proof. We have to separately consider the cases of odd and even n. If n is odd, then by Theorem 3, the Eulerian graph with smallest Wiener index is Kn . The edges of Kn can be partitioned into edge-disjoint cycles. Label the vertices of Kn by 1, 2, . . . , n. For instance, these cycles of K5 are [1, 2, 3, 4, 5, 1] and [1, 3, 5, 2, 4, 1] whereas of K9 are [1, 2, 3, 4, 5, 6, 7, 8, 9, 1], [1, 3, 5, 7, 9, 2, 4, 6, 8, 1], [1, 4, 7, 1], [2, 5, 8, 2], [3, 6, 9, 3], and [1, 5, 9, 4, 8, 3, 7, 2, 6, 1]. Deleting from Kn the edges of any of these cycles, we obtain a connected subgraph, which by Lemma 2 has greater Wiener index than Kn . Deleting from Kn the edges of all these cycles, except of [1, 2, . . . , n, 1], we arrive at Cn , which thus has the greatest possible W -value. If n is even, then we construct the cocktail-party graph CPn by deleting from Kn the edges (1, n/2 + 1), (2, n/2 + 2), . . . , (n/2, n). The edges of CPn can also be partitioned into edge-disjoint cycles. For instance, if n = 6, then these cycles are [1, 2, 3, 4, 5, 6, 1], [1, 3, 5, 1] and [2, 4, 6, 2]. If n = 8, then these cycles are [1, 2, 3, 4, 5, 6, 7, 8, 1], [1, 3, 5, 7, 1], [2, 4, 6, 8, 2], and [1, 4, 7, 2, 5, 8, 3, 6, 1]. Deleting from CPn the edges of any of these cycles, we obtain a connected subgraph, which by Lemma 2 has greater Wiener index than CPn . Deleting from CPn the edges of all these cycles, except of [1, 2, . . . , n, 1], we arrive at Cn , which thus has the greatest possible W -value. We see that the Eulerian graph with greatest Wiener index is the (connected) graph whose all vertices have the smallest possible degree (equal to 2). Therefore, it is plausible to expect that the element of En with second-greatest Wiener index be a graph with n − 1 vertices of degree 2 and a single vertex of degree 4. Graphs with this property will be denoted by Cn,a , a = 3, 4, . . . , ⌊(n + 1)/2⌋, and are defined as follows. Definition 6. Let Ca and Cb be cycles with disjoint vertex sets, possessing, respectively, a and b vertices, a ≤ b. The bicyclic graph Cn,a of order n = a + b − 1 is obtained by coalescing one vertex of Ca with one vertex of Cb . Denote by d(Cp ) the sum of distances between a vertex of the cycle Cp and all its other vertices. It is easy to show that d(Cp ) =

2 p 2

and

W (Cp ) =

p

2 p

2

2

.

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Using the fact that for b = n − a + 1, W (Cn,a ) = W (Ca ) + W (Cb ) + (a − 1) d(Cb ) + (b − 1) d(Ca ) we get W (Cn,a ) =

=

a

a2

2 a

+

4

a

2

2

4

+

n−a+1

(n − a + 1)2

2

b

b

2

2

4

4

+ (b − 1)

a

2

4

+ ( n − a)

+ (a − 1)

a2

2

4

b

+ (a − 1)

(n − a + 1)2

4 (1)

4

where 3 ≤ a ≤ ⌈n/2⌉. Proposition 7. Let n be an even number. Then, W (Cn,3 ) > W (Cn,4 ) > · · · > W (Cn, n −1 ) > W (Cn, n ). 2

2

Proof. Case 1. Let a be odd. Then b = n − a + 1 is even. Using Eq. (1), we have W (Cn,a ) − W (Cn,a+1 ) =

(a + b − 5)(b − a − 1) 8

=

(n − 4)(n − 2a)

=

(n − 2)(n − 2a)

8

>0

since a + 1 ≤ n/2 and n ≥ 5. Case 2. Let a be even. Then b is odd and by Eq. (1), W (Cn,a ) − W (Cn,a+1 ) =

(a + b − 3)(b − a − 1) 8

since a + 1 ≤ n/2 and n ≥ 5.

8

>0

Proposition 8. Let n be an odd number. Then for n = 7 and n = 9, W (Cn,4 ) > W (Cn,3 ) and

W (Cn,4 ) > W (Cn,3 ) > W (Cn,5 )

respectively. For n = 4k + 3, k ≥ 2, W (Cn,3 ) > W (Cn,4 ) > · · · > W (Cn,2k ) > W (Cn,2k+2 ) > W (Cn,2k+1 ) whereas for n = 4k + 1, k ≥ 3, W (Cn,3 ) > W (Cn,4 ) > · · · > W (Cn,2k−2 ) > W (Cn,2k ) > W (Cn,2k−1 ) > W (Cn,2k+1 ). Proof. The cases n = 7 and n = 9 can be easily checked by direct calculation. Case 1. Let a be odd. Then b = n − a + 1 is odd and by Eq. (1) we get W (Cn,a ) − W (Cn,a+1 ) =

(a + b − 4)(b − a − 4) − 8 8

=

(n − 3)(n − 2a − 3) − 8 8

.

For n = 4k + 3, W (Cn,a ) > W (Cn,a+1 ) for all odd values of a such that a < 2k. In addition, if a = 2k + 1, then W (Cn,2k+1 ) < W (Cn,2k+2 ). For n = 4k + 1, W (Cn,a ) > W (Cn,a+1 ) for all odd values of a such that a < 2k − 1. In addition, if a = 2k − 1, then W (Cn,2k−1 ) < W (Cn,2k ). Case 2. Let a be even. Then b = n − a + 1 is even. From Eq. (1) follows: W (Cn,a ) − W (Cn,a+1 ) =

(a + b − 4)(b − a + 2) 8

=

(n − 3)(n − 2a + 3) + 8 8

.

For n = 4k + 3, W (Cn,a ) > W (Cn,a+1 ) for all even values of a such that a < 2k + 3. This implies W (Cn,2k ) > W (Cn,2k+1 ). For n = 4k + 1, W (Cn,a ) > W (Cn,a+1 ) for all even values of a such that a < 2k + 2. This implies W (Cn,2k ) > W (Cn,2k+1 ). It now only remains to prove the inequality W (Cn,2k ) > W (Cn,2k+2 ) for n = 4k + 3 and the inequalities W (Cn,2k−2 ) > W (Cn,2k ) and W (Cn,2k−1 ) > W (Cn,2k+1 ) for n = 4k + 1. This we do by direct calculation: W (C4k+3,2k ) − W (C4k+3,2k+2 ) = 6k3 + 16k2 + 16k + 4 − 6k3 + 16k2 + 14k + 4 = 2k > 0

W (C4k+1,2k−2 ) − W (C4k+1,2k ) = 6k3 + 7k2 + 7k − 2 − 6k3 + 7k2 + 3k = 4k − 2 > 0

W (C4k+1,2k−1 ) − W (C4k+1,2k+1 ) = 6k3 + 7k2 + 3k − 1 − 6k3 + 7k2 + k = 2k − 1 > 0.

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Fig. 1. Eulerian graphs of order n with second-maximal Wiener index. For all other values of n, n ≥ 5, there is a unique graph with second-maximal Wiener index, isomorphic to Cn,3 .

Based on the above results, one may expect that the element of En with second-maximal Wiener index is Cn,3 for all n ≥ 5, except for n = 7 and n = 9, when this would be Cn,4 . Numerical testing revealed that this is not exactly the case, but that some violations exist for smaller values of n. We present the results of our numerical testing in the form of the following: Conjecture 9. (a) If n = 5, 6, 12, and n ≥ 14, then Cn,3 has second-maximal Wiener index in En . (b) If n = 7, 8, 9, 10, 11, and 13, then the Eulerian graphs of order n, with second-maximal Wiener index are those displayed in Fig. 1. For n = 8 and n = 13 there are two such graphs, of which one is of the form Cn,3 . The correctness of Conjecture 9 was checked by testing all Eulerian graphs for n ≤ 17, and — based on these results — by testing the Eulerian graphs with large W -values for 18 ≤ n ≤ 25. Finding a proof of the conjecture remains a task for the future. Acknowledgments The authors thank the anonymous referee for pointing out the exceptional case of n = 8 (depicted in Fig. 1), and Boris Furtula for assisting in the numerical work needed for the formulation of Conjecture 9. References [1] N.L. Biggs, E.K. Lloyd, R.J. Wilson, Graph Theory, Clarendon Press, Oxford, 1976, pp. 1736–1936. [2] G. Caporossi, M. Paiva, D. Vukičević, M. Segatto, Centrality and betweenness: vertex and edge decomposition of the Wiener index, MATCH Commun. Math. Comput. Chem. 68 (2012) 293–302. [3] A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001) 211–249. [4] A.A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002) 247–294. [5] M. Eliasi, G. Raeisi, B. Taeri, Wiener index of some graph operations, Discrete Appl. Math. 160 (2012) 1333–1344. [6] R.C. Entringer, Distance in graphs: trees, J. Combin. Math. Combin. Comput. 24 (1997) 65–84. [7] B. Furtula, I. Gutman, H. Lin, More trees with all degrees odd having extremal Wiener index, MATCH Commun. Math. Comput. Chem. 70 (2013) 293–296. [8] M. Knor, P. Potočnik, R. Škrekovski, The Wiener index in iterated line graphs, Discrete Appl. Math. 160 (2012) 2234–2245. [9] H. Lin, Extremal Wiener index of trees with all degrees odd, MATCH Commun. Math. Comput. Chem. 70 (2013) 287–292. [10] H. Lin, A congruence relation for the Wiener index of trees with path factors, MATCH Commun. Math. Comput. Chem. 70 (2013) 575–582. [11] N.S. Schmuck, S.G. Wagner, H. Wang, Greedy trees, caterpillars, and Wiener-type graph invariants, MATCH Commun. Math. Comput. Chem. 68 (2012) 273–292. [12] A.V. Sills, H. Wang, On the maximal Wiener index and related questions, Discrete Appl. Math. 160 (2012) 1615–1623.

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