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Nordhaus-Gaddum type results for the Harary index of graphs. (English) Zbl 1406.92790

Summary: The Harary index \(H(G)\) of a connected graph \(G\) is defined as \(H(G)=\sum_{u,v\in V(G)}\frac{1}{d_G(u,v)}\) where \(d_G(u,v)\) is the distance between vertices \(u\) and \(v\) of \(G\). The Steiner distance in a graph, introduced by G. Chartrand et al. [Čas. Pěstování Mat. 114, No. 4, 399–410 (1989; Zbl 0688.05040)], is a natural generalization of the concept of classical graph distance. For a connected graph \(G\) of order at least \(2\) and \(S\subseteq V(G)\), the Steiner distance \(d_G(S)\) of the vertices of \(S\) is the minimum size of a connected subgraph whose vertex set contains \(S\). In [Iran. J. Math. Chem. 7, No. 1, 61–68 (2016; Zbl 1406.92758)], B. Furtula et al. introduced the concept of the Steiner Harary index and gave its chemical applications. The \(k\)-center Steiner Harary index \(SH_k(G)\) of \(G\) is defined by \(SH_k(G)=\sum_{S\subseteq V(G),| S| =k}\frac{1}{d_G(S)}\). In this paper, we get the sharp upper and lower bounds for \(SH_k(G)+SH_k(\overline{G})\) and \(SH_k(G)\cdot SH_k(\overline{G})\), valid for any connected graph \(G\) whose complement \(\overline {G}\) is also connected.

MSC:

92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
05C90 Applications of graph theory
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[1] MAIN RESULTSLet ( ) be a graph invariant and a positive integer, ≥ 2. The Nordhaus-Gaddum Problem is to determine sharp bounds for ( ) + ( ) and ( ) · ( ), as ranges over the class of all graphs of order , and to characterize the extremal graphs, i.e., graphs that achieve the bounds. Nordhaus-Gaddum type relations have received wide attention; see the recent survey [2] by Aouchiche and Hansen. Denote by ( ) the class of connected graphs of order whose complements are also connected. In the studies of Nordhaus-Gaddum-type relations it must be assumed that ( ) and ( ) exist. Therefore, such relations are examined in the case of Wiener and Steiner Wiener indices, one must restrict the consideration to the class ( ), ≥ 2. Mao et al. [28] studied the Nordhaus-Gaddum type results for the Wiener index. In this paper, we investigate the analogous problem for the Steiner Harary index. Our basic idea is from [28]. 2.1 RESULTS PERTAINING TO GENERAL For general , we obtain the following result: Theorem 2.1 Let ∈ ( ) and let be an integer such that 3 ≤ ≤ . Then: (1) { (),() }≤( ) +≤()(). (2) { (),() }≤( ) ∙≤(). Moreover, the lower bounds are sharp. Proof. Proof of part (1): For any ⊆ ( ) and | | = , from the definition of Steiner diameter, we have ( ) +( ) ≤ max{ + − 2,2 − 2} = + − 2. Then 186WANG,MAO,WANG AND WANG( ) +=1( )+1( )⊆⊆ ( )=( )+ ( )( ) ( )⊆ ( )≤( + − 2)( − 1). By the same reason, Lemma 1.4 implies ( ) += ∑( )( )( )( )⊆ ( )≥{ (),() }. Proof of part (2): For any ⊆ ( ), | | = and any ⊆ ( ̅), | | = , from the definition of Steiner diameter and Lemma 1.4, we have ( ) ∙̅ ( ) ≤ max{ ( − 1), (2 − 1) }. Then ( ) ⋅=1( )⊆ ( )∙1( )⊆=1( )⊆ ( ),⊆∙1( )≥1max{ ( − 1), (2 − 1) }. For any ⊆ ( ), | | = and any ⊆ ( ), | | = , from the definition of Steiner diameter and Lemma 1.4, we have ( ) ∙( ) ≥ ( − 1) . Then ( ) ⋅=1( )⊆ ( )∙1( )⊆=1( )⊆ ( ),⊆∙1( )≤1( − 1), as desired.
[2] FOR SOME For = , − 1, 3, we can improve the results in Theorem 2.1. 3.1 THE CASE = , - For = , the following result is immediate. Observation 3.1 Let ∈ ( ). Then Computing Szeged Index of Graphs on Triples 187 (1) ( ) +=; (2) ( ) ⋅=(). Akiyama and Harary [1] characterized the graphs for which both and are connected. Lemma 3.1 [1] Let be graph with vertices and maximal vertex degree ∆( ). Then ( ) == 1 if and only if satisfies the following conditions. i. ( ) = 1 and ∆( ) = − 2; ii. ( ) = 1,∆( ) ≤ − 3, and has a cut vertex with pendent edge , such that − contains a spanning complete bipartite subgraph. For = − 1, we have the following result: Proposition 3.1 Let be a graph of order ( ≥ 5).
[3] If and are both 2-connected, then ( ) += and ( ) ⋅=().
[4] If ( ) = 1 and is 2-connected, then ( ) +=+and ( ) ⋅=()()+()() , where is the number of cut vertices in .
[5] If ( )== 1, ∆( ) ≤ − 3, and has a cut vertex with pendent edge such that − contains a spanning complete bipartite subgraph, and ∆≤ − 3 and has a cut vertex with pendent edge such that − contains a spanning complete bipartite subgraph, then ( ) +=()() and ( ) ⋅=()() ().
[6] If ( ) == 1, ∆= − 2, ∆( ) ≤ − 3 and has a cut vertex with pendent edge such that − contains a spanning complete bipartite subgraph, then ( ) +=()() or ( ) +188WANG,MAO,WANG AND WANG=()()and ( ) ⋅=()() () or ( ) ⋅=()()() ().
[7] If ( ) == 1, ∆( ) = ∆= − 2, then ()≤( ) +≤()() and ()()≤( ) ⋅≤()() (). Proof. (1): From Lemma 1.5, if and are both connected, then ( ) = − 2 and ( ) = − 2 for any ⊆ ( ) and | | = − 1. Therefore, ( ) += and ( ) ⋅=(). (2): Since is 2-connected, it follows that ( ) = − 2 for any ⊆ ( ) and | | =− 1, and hence =. Note that ( ) = 1 and there are exactly cut vertices in . For any ⊆ ( ) and | | = − 1, if the unique vertex in ( ) is a cut vertex, then ( ) = − 1. If the unique vertex in ( )is not a cut vertex, then ( ) = − 2. Therefore, we have ( ) =+, and hence ( ) +=+ and ( ) ⋅=()()+()(), where is the number of cut vertices in . (3), (4), (5): We have ( ) = κ= 1. By condition ( ) of Lemma 3.1, since ∆( ) =− 2, there is a vertex of degree − 2, say . Let the set of first neighbors of be ( ) = { , , ⋯ ,}. Let \(( ){ } ∪( )) = { }. Since ∉ ( ), there must exist a vertex in ( ), say , such that ∈ ( ), since is connected. Since , may be the cut vertices in , it follows that there are one or two cut vertices in . So ( ) =+=()() or ( ) =+=. By condition ( ) of Lemma 3.1, since ∆( ) ≤ − 3 and has a cut vertex with pendent edge such that − contains a spanning complete bipartite subgraph, it follows that is the unique cut vertex. So ( ) =+=()(). From this argument, (3), (4), (5) are true. 3.2 THE CASE = The following lemmas and corollaries will be used later. Computing Szeged Index of Graphs on Triples 189 Lemma 3.2 [28] Let be a tree of order , and let be an integer such that 3 ≤ ≤ . Then there exist at least ( − + 1) subsets of ( ) for which the Steiner k-distance is equal to − 1. Corollary 3.1 [28] Let be a connected graph of order , and let be an integer such that 3 ≤ ≤ . Then there exist at least ( − + 1) subsets of ( ) whose Steiner k-distance is − 1. Lemma 3.3 [28] Let be a tree of order , and let be an integer such that 3 ≤ ≤ −
[8] Then there exist at least ( − ) subsets of ( ) whose Steiner k-distance is . In this section, we focus our attention on the case = 3. For = 3 and ≥ 10, from Theorem 2.1, we have ()≤( ) +≤() and ()≤( ) ⋅≤. We improve these bounds and prove the following result. Theorem 3.1 Let ∈ ( ) with ≥ 4. Then
[9] ≥( ) +≥+− = 6,7 ( ) = 5 = 6,7 ( ̅) = 5− ∑+ ℎ.
[10] ≥( ) ⋅≥+()()()−()()(). Moreover, the bounds are sharp. We first need the following lemma. Lemma 3.4 [28] Let be a connected graph. If ( ) = 5, then ≤ 4. Lemma 3.5 Let ∈ ( ). Then ( ) +≤ (3.1) ( ) ⋅≤ (3.2) 190WANG,MAO,WANG AND WANGand ( ) ⋅≥+()()()−()()() . (3.3) Moreover, the bounds are sharp. Proof. (1) For any ⊆ ( ) and | | = 3, ≅ or ≅ or ≅∪ or ≅ 3 . If ≅ or ≅, then ( ) = 2. If ≅∪ or ≅3 , then ( ) ≥ 3. Let , , ⋯ , be all the 3-subsets of ( ). Without loss of generality, let , , ⋯ , be all the 3-subsets of ( ) such that ≅ or ≅, where 1 ≤ ≤ . Therefore, ( ) = 2 and ( ) ≥ 3 for each (1 ≤ ≤ ). Furthermore, for any ( + 1 ≤ ≤), we have ( ) ≤ +=+ ≤ +=− S ( ) ≥ +=+()(). and S≥+=−()(). implying inequality (3.1). By Corollary 3.1, there exist at least − 2 subsets of ( ) whose Steiner 3-distances are equal to 2. The same is true for . Therefore, − 2 ≤ ≤− + 2, and hence S ( ) ∙ S≤13 3+612 3−6 =16 3+36 3−36 ≤13663+14 3 =25144 3i.e., inequality (3.2) holds. Computing Szeged Index of Graphs on Triples 191 S ( ) ∙ S≥1− 1 3+( − 3)2( − 1)12 3−( − 3)2( − 1)=12( − 1) 3+( − 3)4( − 1) 3−( − 3)4( − 1)≥1− 1 3+( − 3)( − 2)2( − 1)12 3−( − 3)( − 2)2( − 1)i.e., inequality (3.3) holds. The sharpness of the above bounds is illustrated by the following example. Example 3.2 Let ≅. Then ≅. By Lemma 1.7, ( ) == , and hence S ( ) + S==and S ( ) ∙ S===+()()()−()()(), which confirms that the lower and upper bounds are sharp. Let ∗ be a tree obtained from a star of order − 2 and a path of length 2 by identifying the center of the star and a vertex of degree one in the path. Then ∗ is a graph obtained from a clique of order − 1 by deleting an edge and then adding an pendent edge at . Observation 3.2 (1)(∗) =++− 3; (2) ∗=++− . Proof. From the structure of ∗and ∗, we conclude S (∗) = 14− 32+12− 32+ ( − 3) + 1 +13− 32+− 33+ 2( − 3) =1312− 32+13− 33+76− 3 and 192WANG,MAO,WANG AND WANG S∗=122− 32+ 2( − 3) +− 33+13− 32+ ( − 2)=43− 32+12− 33+43−113. In order to show the sharpness of the above bounds, we consider the following example. Example 3.3 Let ∗ be the same tree as before. From Observation 3.2, we have S (∗) + S∗=2912− 32+56− 33+156−203and S (∗) ∙ S∗=5236− 32+16− 33+7172− 32− 33 +279−28736− 32+3736−4918− 33 +43−11376− 3 . The following lemmas are preparations for deducing an upper bound on ( ) +. Lemma 3.6 Let be a connected graph of order , and let be a spanning tree of . If = 3, then S ( ) +≤ S ( ) +. Proof. Note that is a spanning subgraph of . It suffices to prove that S−≤ S ( ) −( ). Since sdia= 3, it follows that ( ) = 2 or ( ) = 3 for any ⊆ ( ) and | | = 3. Since is a spanning subgraph of and = 3, it follows that ≤ 3, and hence ( ) = 2 or ( ) = 3 for any ⊆ ( ) and | | = 3. Then 0 ≤( )−( )≤ . We claim that ( )−( )≤( )−( ) for ⊆ ( ) and Computing Szeged Index of Graphs on Triples 193 | | = 3. Because is a spanning subgraph of , ( )≤( ) for any ⊆ ( ) and | | = 3. Similarly, since is a spanning subgraph of , ( )≤( ) for any ⊆ ( ) and | | = 3. If ( )−( )= 0, then ( )−( )= 0 ≤( )−( ), as desired. If ( )−( )= , then ( ) = 3 and ( ) = 2, and hence ( ) = 2 and ( ) ≥ 3. Therefore,( )−( )≥ =( )−( ), as desired. The result follows from the arbitrariness of and the definition of Steiner Wiener index. Lemma 3.7 Let be a tree of order , different from the star . Let ∗ be the tree same as in Observation 3.2. If = 3, then S ( ) +∗≤ S ( ) +. Proof. Note first that the complements of all trees, except of the star, are connected. Therefore, in Lemma 3.7 is always well defined. By Lemma 1.6 and 1.7, S ( ) ≤( ). It suffices to prove ∗≤. Since≤ 3, it follows that ≤ 3. For any ⊆ ( ) and | | = 3, if is not connected, then ( ) = 2. If is connected, then ( ) ≥ 3. So if we want to obtain the minimum value of for a tree , then we need to find as less as possible 3-subsets of ( ) whose induced subgraphs in are disconnected. Since the complement of is not connected, it follows that ∗ is our desired. So ∗≤, and hence ( ) +∗≤( ) +. We are now in the position to complete the proof of Theorem 3.1. This will be achieved by combining Lemmas 3.5 and 3.8. Let ∈ ( ). If = 6, 7 and ( ) = 5, then the validity of Theorem 3.1 can be verified by direct checking. Lemma 3.8 Let ∈ ( ). Let ≥ 8, or ≤ 5, or = 6, 7 and ( ) ≠ 5, or = 6, 7 and ≠ 5. Then the lower bounds in parts (1) and (2) of Theorem 3.1 are obeyed. Moreover, these bounds are sharp. Proof. We need to separately examine three cases. 194WANG,MAO,WANG AND WANGCase 1.( ) ≥ 6 or ≥ 6. Without loss of generality, let ( ) ≥
[11] From Corollary 1.1 it is known that = 3, and hence ( ) +≥( ) +(∗). By Lemma 1.7, ( ) =()()− ∑. Note that ∗ is a graph obtained from a clique of order − 1 by deleting an edge and then adding a pendent edge at . Then ∗=++− , and hence S ( ) +≥()()− ∑+++− = − ∑+. Case 2.( ) = 5or= 5. In view of Lemma 3.4, we can assume that ( ) = 5 and ≤ 4. Let ,,⋯, be all the 3-subsets of ( ). Without loss of generality, assume that ,,⋯, are the 3-subsets of ( ) for which ≅ or ≅, where 1 ≤ ≤ . For each (1 ≤ ≤ ), ( ) = 2. For any ( + 1 ≤ ≤), ≅∪or ≅ 3 . Since is connected, it follows that there exists a spanning tree, say . By Lemmas 3.2 and 3.3, there exist at least (− 3) subsets of ( ) whose Steiner 3-distance is 3, and there exist at least (− 2) subsets of ( ) whose Steiner 3-distance is 2. Therefore, there exist at least (2 − 5) subsets of ( ) whose Steiner 3-distance is at most
[12] Without loss of generality, let = 3for ( + 1 ≤ ≤ 2 − 5). Then ≤5 and = 2 for each j (2n − 4 ≤ ≤ (3)). For each (1 ≤ ≤ ), ( ) = 2. By Lemma 3.3, there exist at least (− 3) subsets of V(G) whose Steiner 3-distance is 3. Then there exist at most − ( − 3) subsets of V(G) whose Steiner 3-distance is 4. If ≤ 2 −5, then ( ) ≥+ (2 − 5 − ) +3 − 2 + 5 and ≥ ( − 3) +( − + 3) +3 − , and hence S ( ) +≥3 −+−≥3 +− . If ≥ 2 − 5, then S ( ) ≥+3 − and ≥( − 3) + ( − + 3) +3 − , and hence S ( ) +≥3 ++− ≥3 +− . Case 3. sdia( ) ≤ 4 and ( ) ≤4. Let ,,⋯, be the 3-subsets of ( ). Without loss of generality, let ,,⋯, be the 3-subsets of ( ) for which ≅ or Computing Szeged Index of Graphs on Triples 195 ≅, where 1 ≤ ≤ . For each (1 ≤ ≤ ), ( ) = 2. For any ( + 1 ≤ ≤), ≅∪ or ≅ 3 .Since is connected, there exists a spanning tree, say . By Lemmas 3.2 and 3.3, there exist at least (− 3) subsets of ( ) whose Steiner 3-distance is equal to 3, and there exist at least (− 2) subsets of ( ) whose Steiner 3-distance is 2. Therefore, there exist at least (2 − 5) subsets of ( ) whose Steiner 3-distance is at most 3. Without loss of generality, let = 3 for ( + 1 ≤ ≤ 2 −5). Then ≤ 4 and = 2 for each 2 − 4 ≤ ≤ 3 . For each (1 ≤≤ ), ( ) = 2. By Lemma 3.3, there exist at least ( − 3) subsets of ( ) whose Steiner 3-distance in is 3. Then there exist at most − ( − 3) subsets of whose Steiner 3-distance in is 4. If ≤ 2 − 5, then ( ) ≥+ (2 − 5 − ) +3 − 2 + 5 and ≥ ( − 3) + ( − + 3) +3 − . Thus S ( ) +≥34 3−112+14−23≥34 3+112−312 . If ≥ 2 − 5, then ( ) ≥+3 − and ≥ ( − 3) + ( − +3) +3 − . Thus S ( ) +≥3 +− . For ≥ 6, one can check that − ∑+≤+− and +− ≤+− . So we only need to consider the lower bounds in Cases 1 and 2. From the above argument, we conclude the following:
[13] For ≥ 8, − ∑+≤+− and ( ) +≥− 33− ∑+.
[14] For ≤ 5, the lower bound in Case 2 does not exist. Then S ( ) +≥− 33− ∑+.
[15] If = 6, 7, sdia( ) ≠ 5, and sdia≠ 5, then S ( ) +≥− 33− ∑+. 196WANG,MAO,WANG AND WANG
[16] If = 6, 7 and sdia( ) = 5, or = 6, 7 and sdiam= 5, then S ( ) +≥710 3+1160−12 . This completes the proof. In order to demonstrate the sharpness of the above bounds, we point out the following example. Example 3.4 Let ≅. Then ≅. By Lemma 1.1, S ( ) == , and hence S ( ) +==− ∑+ and S ( ) ∙==+()()()−()()(), which implies that the upper and lower bounds are sharp. Acknowledgment. The authors are very grateful to the referees for their valuable comments and suggestions, which improved the presentation of this paper. This work was supported by the National Science Foundation of China (Nos. 11601254, 11551001, 11161037, 11461054) and the Science Found of Qinghai Province (Nos. 2016-ZJ-948Q, and 2014-ZJ-907). REFERENCES
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