Power Graphs of Finite Groups Determined by Hosoya Properties
Abstract
1. Introduction
2. Basic Notions and Notations
3. Hosoya Properties
3.1. Main Results
3.2. Hosoya Polynomial
4. Reciprocal Status Hosoya Polynomial
Proof of Theorems 1 and 2
Type of Edge | Edge Set’s Partition | Edges Count |
---|---|---|
Type of Edge | Edge Set’s Partition | Edges Count |
---|---|---|
5. Hosoya Index
⋯ | ||||||
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⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |
⋯ |
- Type 1:
- , for
- Type 2:
- , for
- Type 3:
- , for
- T1:
- For each type, the number of matchings may be calculated as follows: Due to the fact that the edges of Type 1 and Type 2 are the edges of a complete graph , which is induced by the nodes in , so the number of matchings in this type can be obtained by counting the matchings in , which are given in Table 4, where denotes the number of matchings of order i, for
- T2:
- Every matching of this kind may be generated by substituting one edge of Type 3 for any edge of Type 1. Given that every Type-1 edge is also an edge of , which is induced by the nodes of , so every matching of Type-1 edges is also a matching of . The total number of certain matchings is described in Table 5, where for any , signifies the total number of matchings of order i.
- Matchings having one order: These are the m such matches that correspond to the m Type-3 edges;
- Matchings having two orders: All of these matchings may be achieved by inserting a Type-3 edge through every other matching having one order in . Using Table 5, there are m edges of Type 3, as well as matchings having one order in . As a result of the product rule, the number of matchings having two order is: ;
- Matchings having three orders: Every one of these matchings may be achieved by inserting a Type-3 edge into every other matching of order two in . Thus, there are m Type-3 edges, as well as matchings having two order in , by Table 5. As a result of the product rule, the number of order three matchings is:
- Order four matchings: Any of these matchings may be generated by inserting one edge of Type 3 through every order three matching in . Thus, there are m Type-3 edges, as well as matchings having three orders in , by Table 5. As a consequence of the product rule, the number of matchings of order four is given by:
- Order i matchings: In general, every order i matching may be generated by adding one edge of Type 3 to every order matching in . According to Table 5, there are m possible edges of Type 3 and matchings of order in . Therefore, by the product rule, the number of matchings of order i is:
⋯ | ||||||
---|---|---|---|---|---|---|
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |
⋯ |
⋯ | ||||||
---|---|---|---|---|---|---|
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |
⋯ |
- Type 1:
- , for any
- Type 2:
- , for any
- Type 3:
- , for any
- Type 4:
- , for any
- Type 5:
- , for any where
- Matchings amongst the Type-1, Type-2, as well as Type-3 edges;
- Matchings amongst the Type-4 edges;
- Matchings amongst the Type-5 edges;
- Matchings amongst the Type-1 and Type-4 edges;
- Matchings amongst the Type-3 and Type-4 edges;
- Matchings amongst the Type-4 and Type-5 edges;
- Matchings amongst the Type-1, Type-2, Type- and Type-5 edges.
- As we know that the subgraph induced by is complete, that is, , so all the Type-1, Type-2, and Type-3 edges are exactly the edges of , and all the matchings between these edges are counted in Table 6, where denotes the number of matchings of order i, where ;
- For , let indicate the number of order i matchings:
- For
- : The number of Type-4 edges that are which is equal to the number of order one matchings. Consequently,
- For
- : Suppose is a Type-4 edge with and for a fixed Then, in addition to the edge e, every edge of Type 4 with one end in and another in forms a matching of order two. As a result,Hence, there is no order larger than two matching in this situation;
- Type 5 has n edges, none of which have a similar node. Thus, for any order i, there exists a matching such that . Suppose represents the number of order i matchings. Then,
- Assume that represents the number of order i matchings, where Thus, in this context, . There are no Type-1 edges that connect a node to any Type-4 edge in . Hence, we may obtain a matching in this situation by joining every matching of Type-1 edges to any matching of the Type-4 edges. The edges of Type 1 are also the edges of , and there are matchings of order ℓ between them, where Every can be found in Table 6. In between the edges of Type 4, there are and matchings having one and two orders, respectively.As a result of the product rule, we obtain:When then:Furthermore, when then:
- For , represents the total matchings of order i. Then, . We can only utilize matchings of order one between the edges of Type 4 in this case. Otherwise, we are unable to employ any Type-3 edge, since both kinds of edges often share the nodes in . As a result, we can only obtain matchings of order two in this case. Assume that is the order one matching amongst the Type-4 edges with , for Then, any Type-3 edge that is non-adjacent with can lead to construct an order two matchings. Given that there are such Type-3 edges, every of which may be utilized in every one of matchings of order one amongst Type-4 edges, so we obtain:
- For , , denote the number of order i matchings, then . To find matchings, both matchings of order one and two between the edges of Type 4 will be considered and any matching of order ℓ between the edges of Type 5, where . Thus, by counting these matchings using the product rule, we obtain:
- Given that the edges of Type 1, Type 2, as well as Type 3 are also the edges of , which is induced by , so we may utilize them to identify matchings between the edges of Type 5 and the edges of . Let be the number of order i matchings. Then, . Because no edge of Type 5 shares a node with an edge of , this corresponds to each matching of the edges of Type 5. Therefore, each matching of the edges of can be used to determine a match in this situation. Since there exist matchings of the cardinality among the edges of , as shown in Table 6, as well as matchings of the order among the Type 5 edges, thus, in this example, the greatest order of a matching is . As a result, we may determine , for , as follows:
⋯ | ||||||
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⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |
⋯ |
6. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Ali, F.; Rather, B.A.; Din, A.; Saeed, T.; Ullah, A. Power Graphs of Finite Groups Determined by Hosoya Properties. Entropy 2022, 24, 213. https://doi.org/10.3390/e24020213
Ali F, Rather BA, Din A, Saeed T, Ullah A. Power Graphs of Finite Groups Determined by Hosoya Properties. Entropy. 2022; 24(2):213. https://doi.org/10.3390/e24020213
Chicago/Turabian StyleAli, Fawad, Bilal Ahmad Rather, Anwarud Din, Tareq Saeed, and Asad Ullah. 2022. "Power Graphs of Finite Groups Determined by Hosoya Properties" Entropy 24, no. 2: 213. https://doi.org/10.3390/e24020213
APA StyleAli, F., Rather, B. A., Din, A., Saeed, T., & Ullah, A. (2022). Power Graphs of Finite Groups Determined by Hosoya Properties. Entropy, 24(2), 213. https://doi.org/10.3390/e24020213