Graphs with the second and third maximum Wiener index over the 2-vertex connected graphs
Stéphane Bessy, François Dross, Martin Knor, Riste Škrekovski
aa r X i v : . [ c s . D M ] M a y Graphs with the second and thirdmaximum Wiener index over the 2-vertexconnected graphs
St´ephane Bessy , Fran¸cois Dross , Martin Knor , Riste ˇSkrekovski Laboratoire d’Informatique, de Robotique et de Micro´electronique de Montpellier(LIRMM), CNRS, Universit´e de Montpellier, France, [email protected] Universit´e Cˆote d’Azur, I3S, CNRS, Inria, France, [email protected] Slovak Technical University in Bratislava,Faculty of Civil Engineering, Department of Mathematics, Bratislava, Slovakia, [email protected] Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia, [email protected]
Abstract
Wiener index, defined as the sum of distances between all unordered pairsof vertices, is one of the most popular molecular descriptors. It is well knownthat among 2-vertex connected graphs on n ≥ C n attainsthe maximum value of Wiener index. We show that the second maximumgraph is obtained from C n by introducing a new edge that connects two ver-tices at distance two on the cycle if n = 6. If n ≥
11, the third maximumgraph is obtained from a 4-cycle by connecting opposite vertices by a path oflength n −
3. We completely describe also the situation for n ≤ Keywords: Wiener index, 2-vertex connected graphs, gross status, distance,transmission
The sum of distances between all pairs of vertices in a connected graph was firstintroduced by Wiener [16] in 1947. He observed a correlation between boiling pointsof paraffins and this invariant, which has later become known as the Wiener indexof a graph. Today, Wiener index is one of the most used descriptors in chemicalgraph theory.Wiener index was used by chemists decades before it attracted attention of math-ematicians. In fact, it was studied long before the branch of discrete mathematics,which is now known as Graph Theory, was developed. Many years after its intro-duction, the same quantity has been studied and referred to by mathematicians as1he gross status [9], the distance of graphs [5] and the transmission [15]. A greatdeal of knowledge on the Wiener index is accumulated in several survey papers, seee.g. [3, 10, 12, 17].In what follows, we formally define this index. Let d G ( u, v ) denote the distancebetween vertices u and v in G . The transmission of a vertex v is the sum of distancesfrom v to other vertices of G , i.e., w G ( v ) = P u ∈ V ( G ) d G ( u, v ). Then the Wiener indexof G equals W ( G ) = 12 X u ∈ G w G ( u ) = X u,v ∈ V ( G ) d G ( u, v ) . Due to big importance and popularity, there are many results about graphswith extremal (either maximum or minimum) values of Wiener index in particularclasses, see the surveys mentioned above. However, only few papers are devoted tothe second, third, etc extremal graphs, although it is important to understand theordering of graphs by Wiener index. One of the reasons is that results of this typeare much more complicated, often including the extremal graph as a trivial case. Ofcourse, the situation is known for trees. In [4] there are described the first 15 treeswith the smallest value of Wiener index. Analogously, in [2, 13] there are the first15 trees with the greatest value of Wiener index. Graphs with the second minimumand second maximum value of Wiener index over the class of unicyclic graphs arefound in [6]. In this paper we describe graphs with the second and third maximumvalue of Wiener index over the class of 2-vertex connected graphs.We use the following notation. As usual, C n is the cycle on n vertices. Let H n,p,q be a graph on n vertices comprised of three internally disjoint paths with the sameend-vertices, where the first one has length p , the second one has length q , and thelast one has length n − p − q + 1. Notice that H n,p,q has n vertices. Also observe that H n, , is the cycle on n vertices plus an edge linking two vertices at distance two onthe cycle. When using the notation H n,p,q we assume that 1 ≤ p ≤ q ≤ n − p − q + 1and q >
1. Our main result is the following theorem.
Theorem 1.
Let n ≥ and let G be a 2-vertex connected graph on n verticesdifferent from C n , H n, , and H n, , . Then W ( G ) < W ( H n, , ) < W ( H n, , ) < W ( C n ) . We start with some definitions. For two vectors ω and ω ′ of the same finite dimen-sion, we write ω (cid:22) ω ′ if for every coordinate i we have ω i ≤ ω ′ i . Moreover we define h ω i as the value P i iω i . It is clear that ω (cid:22) ω ′ implies h ω i ≤ h ω ′ i .Let G be a connected graph on n vertices and let v be a vertex of G . The distancevector of v is the ( n − ω G ( v ) given by ω G ( v ) i = |{ x ∈ G : d G ( v, x ) = i }| . Observe that h ω G ( v ) i = w G ( v ).If n is even, the vector n has dimension n/ n is odd, n has dimension ( n − / = (2 , ,
1) and = (2 , , G be a 2-vertex connected graph and let v be a vertex of G . Since G has nocut-vertices, every coordinate of ω G ( v ) has value at least 2, except for the last one2hich can be 1. In other words, for every vertex v of a 2-vertex connected graph G we have h ω G ( v ) i ≤ h n i = ⌊ n ⌋ . This implies the following classical result. Theorem 2.
For every n ≥ , the cycle C n is the unique graph which has themaximum Wiener index over the class of 2-vertex connected graphs on n vertices.Moreover, W ( C n ) = n h n i . Now we describe the structure of graphs with the second and third maximumWiener index over the class of 2-vertex connected graphs on n vertices. First weneed some definitions and lemmas.We denote by k ( v ) the first coordinate i of ω G ( v ) such that ω G ( v ) i >
2. If such acoordinate does not exist, we set k ( v ) = ⌊ n ⌋ . Notice that if ω G ( v ) = n , then k ( v ) < ⌊ n ⌋ . For a graph G on n vertices we denote by k ( G ) = ( k i ( G )) ≤ i ≤ n the sequenceformed by the values k ( v ) of all v ∈ V ( G ) given in non-decreasing order. Forinstance, the sequence k ( H n, , ) is given by k i ( H n, , ) = ⌊ i +12 ⌋ for every i = 1 , . . . , n − k n ( H n, , ) = ⌊ n ⌋ . In other words we have k ( H n, , ) = (1 , , , , , , . . . , ⌊ n ⌋ )with twice the value ⌊ n ⌋ at the end if n is even and three times if n is odd. Similarlyfor n ≥
5, the sequence k ( H n, , ) is given by k ( H n, , ) = k ( H n, , ) = 1, k ( H n, , ) = k ( H n, , ) = 2 and k i ( H n, , ) = ⌊ i − ⌋ for every i = 5 , . . . , n . In other words we have k ( H n, , ) = (1 , , , , , , , , , , . . . , ⌊ n − ⌋ ) with once the value ⌊ n − ⌋ if n is oddand twice if it is even.As previously precised, we write k ( G ) (cid:22) k ( G ′ ) if for every i ∈ { , . . . , n } we have k i ( G ) ≤ k i ( G ′ ). Moreover, if k j ( G ) < k j ( G ′ ) for some j ∈ { , . . . , n } , then we write k ( G ) ≺ k ( G ′ ) .The next two lemmas give necessary conditions to bound the Wiener index of agraph by the ones of H n, , and H n, , . Lemma 3.
Let G be a 2-vertex connected graph on n ≥ vertices. If k ( G ) ≺ k ( H n, , ) , then W ( G ) < W ( H n, , ) . Similarly if k ( G ) ≺ k ( H n, , ) , then W ( G ) Let H be either H n, , or H n, , . Let G be a 2-vertex-connected graphon n ≥ vertices such that P x ∈ V ( G ) k ( x ) < b ( G ) + P x ∈ V ( H ) k ( x ) . Then W ( G ) 6, let H n be the class of graphs comprised of H n, ,q for q = 3 , . . . , ⌊ n ⌋ . We have the following claim. Proposition 5. Let n = 9 or n ≥ , and let G be a graph of H n . Then W ( G ) < W ( H n, , ) < W ( H n, , ) . For n ≤ and n = 10 , the values of W ( G ) for G ∈ H n ∪ { H n, , , H n, , } aresummarised in the table below. n H n H , , H , , H , , H , , H , , H , , H , , H , , H , , W n H n H , , H , , H , , H , , H , , H , , H , , H , , H , , W 58 58 55 56 115 113 107 109 112 Table 1: Values of W ( H n,p,q ) for n ≤ n = 10. Proof. Assume that n ≥ 9. As noticed below Lemma 3, we have W ( H n, , ) 2) = n − 5; and if n is even, then P x ∈ V ( H n, ,q ) k ( x ) − P x ∈ V ( H n, , ) k ( x ) ≤ n − 2) + 2 = n − q ≥ n − q ≥ u and v are bad vertices with k ′ = 2.For instance, u has three distinct vertices at distance two, which are x , y and y n − q − . Moreover, since n ≥ y and y n − q − are bad vertices with k ′ = 3. Forinstance, if q = 3 then n − q − ≥ x , y , and y n − q − are distinct vertices atdistance 3 from y , and if q ≥ 4, then x , x q − , and y are distinct vertices at distance3 from y . Moreover, if n ≥ 11 we can find another bad vertex. Indeed, if q ≥ 5, then n − q ≥ q ≥ x , y and y n − q − are distinct vertices at distance 3 from x , whichis bad then. If q = 4, then n − q ≥ x , x q − and y are at distance 4 from y ,which is bad. And if q = 3, then n − q ≥ x , y and y n − q − are at distance 4 from y which is bad. Therefore b ( H n, ,q ) ≥ ·⌊ n − ⌋− (2 · · n ≥ 11. If n = 9 then b ( H n, ,q ) ≥ P x ∈ V ( H n, ,q ) k ( x ) − P x ∈ V ( H n, , ) k ( x ) ≤ 4. If n = 11 then b ( H n, ,q ) ≥ 10 and P x ∈ V ( H n, ,q ) k ( x ) − P x ∈ V ( H n, , ) k ( x ) ≤ 6. If n ≥ b ( H n, ,q ) > n − ≥ n − P x ∈ V ( H n, ,q ) k ( x ) − P x ∈ V ( H n, , ) k ( x ) ≤ n − W ( H n, ,q ) < W ( H n, , ), by Lemma 4.For n ≥ 4, let I n be the class of graphs built from C n by adding two distinctedges, each linking two vertices at distance precisely 2 along C n . That is, a graph G belongs to I n if V ( G ) = { x , . . . , x n } and E ( G ) = { x i x i +1 : i = 1 , . . . , n − } ∪{ x n x , x x , x i x i +2 } , where 1 < i ≤ n − 2. Further, by G we denote a graph from I when i = 4. So G consists of two disjoint triangles connected by two independentedges. We have the following claim. Proposition 6. Let n ≥ . Every graph G of I n satisfies the inequality W ( G ) 5, with a uniqueexception when n = 6 and W ( H , , ) − W ( G ) = 0. In this case we must have d ( u , u ) = n , and consequently G = G .Consider H n, , with vertices { x , . . . , x n } and edges { x i x i +1 : i = 1 , . . . , n − } ∪{ x n x , x x } . Let G be obtained from H , , by adding the edge x x , and let G be obtained from H n, , by adding the edge x x . Denote G = { G , G , G } .Moreover, let G be obtained from H , , by adding the edge x x . Observe that W ( G ) = W ( H , , ) when G ∈ G , and W ( G ) = W ( H , , ). The following theoremimplies and precises Theorem 1. Theorem 7. For n = 4 , there are three -vertex connected graphs and they satisfy W ( K ) < W ( H , , ) < W ( C ) . For every n ≥ , let G be a 2-vertex connected graphon n vertices different from C n , H n, , , H n, , and H n, , . Moreover, assume G / ∈ G if n = 6 and G = G if n = 8 . We have : • W ( G ) < W ( H n, , ) = W ( H n, , ) for n = 5 , • W ( G ) < W ( H n, , ) < W ( H n, , ) < W ( H n, , ) for n = 6 , • W ( G ) < W ( H n, , ) = W ( H n, , ) < W ( H n, , ) for n = 7 , • W ( G ) < W ( H n, , ) < W ( H n, , ) = W ( H n, , ) for n = 8 , • W ( G ) < W ( H n, , ) < W ( H n, , ) < W ( H n, , ) for n = 10 , and • W ( G ) < W ( H n, , ) < W ( H n, , ) and W ( H n, , ) < W ( H n, , ) for n = 9 and n ≥ .Proof. For n ≥ 5, let C be the class of 2-vertex connected graphs on n verticesdifferent from C n , H n, , , H n, , and H n, , . Let G be a graph with the maximumWiener index over C . We want to prove W ( G ) < W ( H n, , ) except when G ∈ G or G = G . Notice that no proper subgraph of G is in C , since otherwise this propersubgraph would have a bigger Wiener index than G . First suppose that G has aHamiltonian cycle C . We distinguish three cases. Case 1: G contains an edge xy where x and y are at distance at least 4 along C . Then C + xy itself is a graph from C . Thus, by the choice of G we have G = C + xy and G = H n, ,q for some q ≥ 4. By Proposition 5, W ( G ) < W ( H n, , ). Case 2: G contains an edge xy where x and y are at distance 3 along C . Sincethe Hamiltonian cycle C with xy is H n, , , G must contain one more edge, say st .Since H n, , with st is in C , there are no other edges in G . By Case 1 we may assumethat s and t are at distance 2 or 3 along C . Since G is a supergraph of H n, , ,we have W ( G ) < W ( H n, , ) if n / ∈ { , , } , by Proposition 5. The cases when n ∈ { , , } were checked by a computer and it was found that W ( G ) < W ( H n, , )with two exceptions if n = 6, namely when G = G and G = G , and with oneexception if n = 8, namely when G = G .6 ase 3: The edges of G not belonging to C link vertices at distance 2 along C . Let us denote by e , . . . , e ℓ these edges. Since C + e + e itself is a graph from C ,we have G = C + e + e . Now Proposition 6 concludes the proof.So assume that G has no Hamiltonian cycle. Let v be a vertex of G with themaximum value of k , that is k ( v ) = k n ( G ). We denote this value by p . If p = 1,it is clear that k ( G ) ≺ k ( H n, , ) and Lemma 3 implies the result. So assume that p ≥ 2. We know that there are exactly two vertices at distance i from v for every i ∈ { , . . . , p − } . Denote these vertices by u i and v i . Notice that for i = 1 , . . . , p − u i and v i are contained in { u i − , v i − , u i , v i , u i +1 , v i +1 } (with u = v = v ). Moreover, since G is 2-vertex connected, there exists a match-ing of size 2 between { u i , v i } and { u i +1 , v i +1 } for i = 1 , . . . , p − 2. So we as-sume that u i u i +1 and v i v i +1 are edges of G for i = 1 , . . . , p − P = u p − , . . . , u , v, v , . . . , v p − is a path of G . Finally, denote by X the set( N G ( u p − ) ∪ N G ( v p − )) \ { u p − , v p − , u p − , v p − } . Let G ′ be the subgraph of G ob-tained by removing the edges of G [ V ( P )] which do not belong to P . Notice firstthat G ′ is a 2-vertex connected graph. Indeed, since G ′ \ { u p − , u p − , . . . , v p − , v p − } is connected (otherwise u p − or v p − would be a cut-vertex of G ), no vertex of P is a cut-vertex of G ′ . Moreover, no vertex of G ′ \ P is a cut-vertex of G ′ otherwiseit would be a cut-vertex of G . Furthermore, G ′ is not a cycle or H n, , or H n, , ,otherwise G would have a Hamiltonian cycle. We may also assume that G ′ is not H n, , , since otherwise G is a supergraph of H n, , and W ( G ) < W ( H n, , ). Hence, G ′ belongs to C , and by the choice of G we have G ′ = G . We consider two cases. Case 1: p = ⌊ n ⌋ . In this case | X | = 1 or | X | = 2. If | X | = 1, denote by x theunique vertex of X . Since G is 2-vertex connected, u p − x and v p − x are edges of G and G has a Hamiltonian cycle, contradicting a previous assumption. If | X | = 2,denote by u p and v p the vertices of X . Analogously as above, since G is 2-vertexconnected, we can assume that u p − u p and v p − v p are edges of G . Since G has noHamiltonian cycle, u p v p is not an edge of G . But G is 2-vertex connected, and so u p − v p and v p − u p are edges of G . Hence G = H n, , . Case 2: p < ⌊ n ⌋ . Below we will show that k ( G ) admits a non-decreasingsubsequence κ = ( k , . . . , k p +1 ) with κ ≺ (1 , , , , , , , , , , . . . , p − , p − , p ),and the existence of a coordonate i for which k i ( G ) < k i ( H n, , ). Then we willconclude that W ( G ) < W ( H n, , ). Indeed, for every value k i ( G ) > i < n therewill exist at least 2 k i ( G ) elements before it in k ( G ), which means that i ≥ k i ( G )+1,and hence k i ( G ) ≤ ⌊ i − ⌋ = k i ( H n, , ). If k i ( G ) = 1, then we have k i ( G ) ≤ k i ( H n, , ).Further, G has at least two vertices of degree at least 3, for otherwise G would not be2-vertex connected. Hence, if k i ( G ) = 2, then we have i ≥ k i ( G ) ≤ k i ( H n, , ).Moreover, as k i ( G ) < k i ( H n, , ) we have k ( G ) ≺ k ( H n, , ). By Lemma 3, weconclude that W ( G ) < W ( H n, , ).Thus, all that remains to show is the existence of subsequence κ of k ( G ) with κ ≺ (1 , , , , , , , , , , . . . , p − , p − , p ), and the existence of a coordonate i for which k i ( G ) < k i ( H n, , ). Since p < ⌊ n ⌋ we have | X | ≥ 3, and we mayassume that u p − has at least two neighbours in X . We find a special path Q in G . There are two cases to consider. First, if v p − has at least two neighbours in X ,then k ( u i ) = k ( v i ) = p − i for every i = 1 , . . . , p − 1. In this case we set x = u p − , y = v p − and Q = P . So Q is an induced path, the only neighbours of Q in G \ Q are those of x and y and κ Q = (1 , , , , . . . , p − , p − , p ) is a subsequence of k ( G )7chieved only by vertices of Q . If v p − has only one neighbour in X , then v p − hasdegree 2, and we denote by v p its neighbour different from v p − . The degree of v p isat least 3, for otherwise we would have k ( v ) ≥ p + 1, which contradicts the fact that v has the maximum value in k ( G ). So we have k ( v ) = p and k ( u i ) = k ( v i +1 ) = p − i for i = 1 , . . . , p − 1. In this case we set x = u p − , y = v p and Q = P ∪ { v p } . If Q isnot an induced path in G , then u p − v p = xy is an edge of G . But since v p is not acut-vertex of G , G − xy is 2-vertex connected. Since G has no Hamiltonian cycle, G − xy is different from C n , H n, , and H n, , . And analogously as before Case 1we may assume that G − xy is different from H n, , . So G − xy ∈ C which is notpossible. Thus here again, Q is an induced path in G , the only neighbours of Q in G \ Q are those of x and y and κ Q = (1 , , , , . . . , p − , p − , p, p ) is a subsequenceof k ( G ) achieved only by vertices of Q . To conclude the proof, we analyse threedifferent cases.First assume that G contains a vertex z with degree at least 4. If z ∈ { x, y } then z has at least three neighbours in G \ Q . On the other hand, if z / ∈ { x, y } then z and at least two its neighbours are in G \ Q . In any case, at least threevertices of { z } ∪ N ( z ) are in G \ Q . We show that all these vertices have k atmost 2. Obviously k ( z ) = 1. So let z be a neighbour of z outside Q . Suppose that k ( z ) > 2. If the degree of z is at least 3, then k ( z ) = 1, a contradiction. Therefore z has exactly two neighbours. Observe that d ( z , s ) ≤ s is z or one ofits neighbours. Since { z } ∪ N ( z ) has at least five vertices, the other neighbour of z (different from z ) must be in N ( z ). Denote this neighbour by z . Since z is not acut-vertex, z has a neighbour, say q , which is different from z and z . If q ∈ N ( z )then z, z , z , q form a 4-cycle with a chord in G , a contradiction. On the other handif q / ∈ N ( z ) then k ( z ) = 2 which contradicts our assumption that k ( z ) > 2. Hence k ( z ) ≤ 2, and the same holds for all neighbours of z outside Q . It means that thethree vertices of { z } ∪ N ( z ) outside Q together with κ Q yield a sequence, first 2 p + 1members of which form the desired sequence κ . Moreover, we know that k ( G ) isat most the seventh value of κ , and so k ( G ) ≤ < k ( H n, , ) = 3. Consequently, W ( G ) < W ( H n, , ). Therefore, in the next we assume that every vertex of G hasdegree at most 3.Now suppose that G contains at least four vertices z , z , z and z of de-gree 3. Two of these vertices at least, say z and z , do not belong to Q , and so { k ( z ) , k ( z ) }∪ κ Q contains the desired sequence κ = (1 , , , , , , , , . . . , p − , p − , p ). Since k ( G ) = 1 < k ( H n, , ) = 2, we have W ( G ) < W ( H n, , ).Finally, suppose that G has exactly two vertices of degree 3, while all the othervertices have degree 2. So G is H n,a,b , and x, y are connected by paths of length a , b and n − a − b + 1, where the last one is Q . Since G has no Hamiltonian cycle, a ≥ 2. And since G is different from H n, , , we have b ≥ 3. Then G \ Q has atleast three vertices, say z , z and z , which are adjacent to x or y . Since k i ( z i ) = 2,where 1 ≤ i ≤ κ Q ∪ { k ( z ) , k ( z ) , k ( z ) } contains the desired sequence κ . Since k ( G ) = 2 < k ( H n, , ) = 3, we conclude that W ( G ) < W ( H n, , ).For the sake of completeness, in Table 2 we present Wiener indices of C n , H n, , , H n, , and H n, , . These indices are calculated using the fact that all the consideredgraphs have one large isometric cycle of length t ≤ n plus some extra vertices. Noticethat t = t − if t is odd and t = t if t is even.8dd n even nW ( C n ) ( n − n ) ( n ) W ( H n, , ) ( n − n + 3 n − ( n − n + 2 n ) W ( H n, , ) ( n − n − n + 17) ( n − n − n + 16) W ( H n, , ) ( n − n + 11 n − ( n − n + 12 n − C n , H n, , , H n, , and H n, , .Let H + n, , be the graph obtained from H n, , by adding an edge between twovertices that are at distance 1 from the vertices of degree 3. As a remark, we notethat H + n, , has Wiener index exactly W ( H n, , ) − 1, so it is the (possibly not unique)fourth 2-connected graph by decreasing Wiener index for n = 9 and n ≥ 11. Weconjecture that for n large enough, it is the unique fourth 2-connected graph bydecreasing Wiener index, and that the unique fifth such graph is H n, , . Acknowledgements. The third author acknowledges partial support by Slovak re-search grants APVV-15-0220, APVV-17-0428, VEGA 1/0142/17 and VEGA 1/0238/19.The research was partially supported by Slovenian research agency ARRS, programno. P1-0383. 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