3.1. Results on ℓ-Connected Graphs
In this subsection, we show the results on ℓ-connected graphs.
For these graphs in with the connectivity ℓ, the following theorem provides a lower bound for their Wiener-type indices with a monotone increasing function and identifies the extremal graph where the equality holds.
Theorem 1. Let , and with connectivity ℓ. If has monotonic increasing property, thenwith equality iff . Proof. Let with connectivity ℓ and minimum Wiener-type indices. Then, there is a vertex cut Y of with . Let be the components of the graph .
Claim 1. For , is a complete graph.
Suppose that there exists such that is not a complete graph and . Let . Then with connectivity ℓ. Since is strictly increasing, then , which is a contradiction.
By a similar analysis, we have
Claim 2. is a complete graph.
Suppose that . Let , where , and . Then, with connectivity ℓ. Since is strictly increasing, then , which is a contradiction.
Combining Claims 1–3, we have that
, where
.
When or , we obtain the minimum value of , i.e., and .
By the above arguments, the conclusion holds. □
In Theorem 1, let (where ). Then, we can derive the minimal values of these Wiener-type indices for ℓ-connected graphs (it should be noted that is a monotone increasing function when ). The results are shown below.
When , we have . The corresponding result on the Wiener index is as follows:
Corollary 1 ([
22]).
Let , , and with connectivity ℓ. Then,the equality holds iff . When , we have . The corresponding result on the modified Wiener index is as follows:
Corollary 2. Let , , , and with connectivity ℓ. Then,the equality holds iff . When , we have . The corresponding result on the hyper-Wiener index is as follows:
Corollary 3. Let , , and with connectivity ℓ. Then,the equality holds iff . For graphs in , the following theorem provides an upper bound for their Wiener-type indices with a monotone increasing function and gives some graphs that make the equality hold.
Theorem 2. Let , and . When is a monotone increasing function, then The bound is the best possible when , and Proof. Let and . Since , by Lemma 5, diameter . Since , then for . Since is strictly increasing, we have . Therefore, by Lemma 8, .
Now, we show that, when
, the bound can be obtained by giving an example. It is obvious that
Then, by Lemma 4, when , we have ; when , since ℓ is even, we have .
Let and . Let for . Then, we have for . Therefore, for , , and if , if . Then, .
This concludes the proof. □
In Theorem 2, let (where ). Then, we can derive the maximal values of these Wiener-type indices for ℓ-connected graphs. The results are shown below.
When , we have . The corresponding result on the Wiener index is as follows:
Corollary 4 ([
19]).
Let , , and . Then,The bound is the best possible when , and When , we have . The corresponding result on the modified Wiener index is as follows:
Corollary 5. Let , , , and . Then, The bound is the best possible when , and When , we have . The corresponding result on the hyper-Wiener index is as follows:
Corollary 6. Let , , and . Then, The bound is the best possible when , and Theorem 3. Let such that , . When is a monotone increasing function, then Proof. Let
. Then, it is well known that
. Since
, then
. By Lemma 2, we have
. By Lemma 1,
. By Lemma 3 and the monotonicity of
, we have
, where
Based on the above arguments, the conclusion holds. □
For graphs in where ℓ and n satisfy that n is even, and , the following theorem provides an upper bound for their Wiener-type indices with a monotone increasing function and gives some graphs that make the equality hold.
Theorem 4. Let , , , and . If has monotonic increasing property, thenthe bound is obtained if , where is a perfect matching in the graph . Proof. Let and . Since , then by Lemma 5. By and the monotonicity of , we have . Thus, we have . If , then the equality holds. □
3.2. Results on Pancyclic Graphs
In this subsection, we show the results on pancyclic graphs.
For graphs in , the following theorem provides a lower bound (or an upper bound) for their Wiener-type indices with a monotone increasing (or decreasing) function and identifies the extremal graph where the equality holds.
Lemma 9. Let .
(i)
When has monotonic increasing property, thenwith equality iff .(ii)
When has monotonic decreasing property, thenwith equality iff . Proof. Let
have two disjoint independent sets, denoted by
and
such that
,
and
. If
is strictly increasing, then we have
with equality iff
.
The proof for the case where is strictly decreasing is similar, and thus it is omitted here. This completes the proof. □
Let
[
20].
Yu et al. showed Lemma 6 in [
20]. Based on Lemmas 6 and 9, we show the following results.
Theorem 5. Let , and .
(i)
If has monotonic increasing property, andthen the graph G has pancyclic property if it is not an element of the set .(ii)
If has monotonic decreasing property, andthen the graph G has pancyclic property if it is not an element of the set . Proof. Let
,
and
. If
is strictly increasing, then
Since , then . By Lemma 6, G is pancyclic unless or G is a graph in the set .
If G is a pancyclic graph, the conclusion holds. Otherwise, we have or . By Lemma 9, we have if G is a bipartite graph. Since , then G is not a bipartite graph. For all , we can easily verify that .
The proof for the case where is strictly decreasing is similar, and thus it is omitted here.
By the above arguments, we have completed the proof. □
In Theorem 5, let (where ). Then, we can derive some conditions that ensure graphs to have the pancyclic property with respect to these Wiener-type indices. The results are shown below.
When , we have . The corresponding result on the Wiener index is as follows.
Corollary 7 ([
23]).
Let , and . Ifthen the graph G has pancyclic property if it is not an element of the set . When , we have . The corresponding result on the modified Wiener index is as follows.
Corollary 8. Let , , and .
(i)
If andthen the graph G has pancyclic property if it is not an element of the set .(ii)
If andthen the graph G has pancyclic property if it is not an element of the set . When , we have . The corresponding result on the hyper-Wiener index is as follows.
Corollary 9. Let , and . Ifthen the graph G has pancyclic property if it is not an element of the set . When , we have . The corresponding result on the Harary index is as follows.
Corollary 10 ([
23]).
Let , and . Ifthen the graph G has pancyclic property if it is not an element of the set . For convenience, we say G is an NCB-graph if G is a non-complete bipartite graph.
Theorem 6. Let , and .
(i)
If has monotonic increasing property, andthen the graph G has pancyclic property if it is not an NCB-graph.
(ii)
If has monotonic decreasing property, andthen the graph G has pancyclic property if it is not an NCB-graph.
Proof. For convenience, let and . Then, .
Let
and
. If
has monotonic increasing property, then
Since , then . By Lemma 6, G has the pancyclic property unless it is in the set or the set .
If G is a pancyclic graph, the conclusion holds. Otherwise, we have or . If or G is a complete bipartite graph, then . Hence, G is an NCB-graph.
The proof for the case where is strictly decreasing is similar, and thus it is omitted here.
The proof is complete by the above arguments. □
In Theorem 6, let (where ). Then, we can derive some conditions that ensure graphs to have the pancyclic property with respect to these Wiener-type indices of complement graphs. The results are shown below.
When , we have . The corresponding result on the Wiener index is as follows.
Corollary 11 ([
23]).
Let , and . If andthen the graph G has pancyclic property if it is not an NCB-graph.
When , we have . The corresponding result on the modified Wiener index is as follows.
Corollary 12. Let , and .
(i)
If , andthen the graph G has pancyclic property if it is not an NCB-graph.
(ii)
If , andthen the graph G has pancyclic property if it is not an NCB-graph.
When , we have . The corresponding result on the hyper-Wiener index is as follows.
Corollary 13. Let , and . If and then the graph G has pancyclic property if it is not an NCB-graph.
When , we have . The corresponding result on the Harary index is as follows.
Corollary 14 ([
23]).
Let , and . If andthen the graph G has pancyclic property if it is not an NCB-graph.
Let
[
21].
Xu et al. showed Lemma 7 in [
21]. Based on Lemmas 7 and 9, we show the following results.
Theorem 7. Let , and .
(i)
If has monotonic increasing property andthen the graph G has pancyclic property if it is not an element of the set .(ii)
If has monotonic decreasing property andthen the graph G has pancyclic property if it is not an element of the set . Proof. Let
,
and
. If
is strictly increasing, then
Since , then . By Lemma 7, G is pancyclic unless G is a graph in the set or .
If G is a pancyclic graph, the conclusion holds. Otherwise, we have or G is a bipartite graph. By Lemma 9, we have if G is a bipartite graph. Since , then G is not a bipartite graph. For all , we can easily verify that .
The proof for the case where is strictly decreasing is similar, and thus it is omitted here.
Based on the above arguments, the proof is complete. □
In Theorem 7, let (where ). Then, we can derive some conditions that ensure graphs to have the pancyclic property with respect to these Wiener-type indices. The results are shown below.
When , we have . The corresponding result on the Wiener index is as follows.
Corollary 15. Let , and . Ifthen the graph G has pancyclic property if it is not an element of the set . When , we have . The corresponding result on the modified Wiener index is as follows.
Corollary 16. Let , and .
(i)
If andthen the graph G has pancyclic property if it is not an element of the set .(ii)
If andthen the graph G has pancyclic property if it is not an element of the set . When , we have . The corresponding result on the hyper-Wiener index is as follows.
Corollary 17. Let , and . Ifthen the graph G has pancyclic property if it is not an element of the set . When , we have . The corresponding result on the Harary index is as follows.
Corollary 18. Let , and . Ifthen the graph G has pancyclic property if it is not an element of the set . Example 1. Let , , , , where and . Then, if is a monotone increasing function, and if is a monotone decreasing function.
Proof. Clearly, G is connected, and .
Let . Clearly, . Since there exists exactly one edge between and , G contains no . Let be a monotone increasing function. If , then by Theorem 5, G is pancyclic, a contradiction. Hence, .
Similarly, if is strictly decreasing, we have . □
Example 2. Let , , , , where and . Then, G has the pancyclic property if .
Proof. Clearly, and . Let . Then, . If , then . If , then . By Corollaries 7 and 15, G has the pancyclic property if . □