Abstract
For any and given nonempty subset , the concept of -superhypergraphs is introduced by Florentin Smarandache based on (-th power set of ). In this paper, we present the novel concepts supervertices, superedges, and superhypergraph via the concept of flow. This study computes the number of superedges of any given superhypergraphs, and based on the numbers of superedges and partitions of an underlying set of superhypergraph, we obtain the number of all superhypergraphs on any nonempty set. As a main result of the research, this paper is introducing the incidence matrix of superhypergraph and computing the characteristic polynomial for the incidence matrix of superhypergraph, so we obtain the spectrum of superhypergraphs. The flow of superedges plays the main role in computing of spectrum of superhypergraphs, so we compute the spectrum of superhypergraphs in some types such as regular flow, regular reversed flow, and regular two-sided flow. The new conception of superhypergraph and computation of the spectrum of superhypergraphs are introduced firstly in this paper.
1. Introduction and Preliminaries
The theory of graph is a main and important theory for modeling the real problem in the world, and this theory extends in past years in this regard. The disadvantage of a graph is that it cannot connect more than two elements, so this problem causes weakness in this theory. Berge generalized the theory of graphs to the mathematical concept of theory of hypergraphs with the motivation that hypergraphs solve the conflicts, defects, and shortcomings of graph theory around 1960 [1]. Hypergraphs have some applications in other sciences and the real-world, one of the applications of hypergraphs is a simulation for complex hypernetworks. Today, hypergraphs have a vital role and important performances, so are used in complex hypernetworks such as computer science, wireless sensor hypernetwork, and social hypernetworks. In this regards there has been a lot of research about using hypergraphs to problems in real-world such as hypergraph matching via game-theoretic hypergraph clustering [2], hypergraph matching via game-theoretic hypergraph clustering [3], hypergraph-based centrality metrics for maritime container service networks, a worldwide application [4], clustering ensemble via structured hypergraph learning [5], and hypergraph neural network for skeleton-based action recognition [6]. There is some main connection between graphs and hypergraphs via the mathematical computational tools and basic theorems in which these connections facilitate the modeling of other sciences with mathematics. Further materials regarding graphs and hypergraphs are available such as extending factorizations of complete uniform hypergraphs [7], finding perfect matchings in bipartite hypergraphs [8], graphs and hypergraphs [1], resilient hypergraphs with fixed matching number [9], on the spectrum of hypergraphs [10], on the distance spectrum of minimal cages and associated distance biregular graphs [11], on the spectrum of the perfect matching derangement graph [12], and probabilistic refinement of the asymptotic spectrum of graphs [13]. Recently, Hamidi and Saeid computed eigenvalues of discrete complete hypergraphs and partitioned hypergraphs. They defined positive equivalence relation on hypergraphs that establishes a connection between hypergraphs and graphs, and it makes a connection between a spectrum of graphs and a spectrum of the quotient of any hypergraphs. They studied the construct spectrum of path trees via the quotient of partitioned hypergraphs [14]. A hypergraph on any given set considers a relationship between elements and the set and describes this relationship if it is a weighted hypergraph. It is an ideal condition if proper weights are known, but in most situations, the weights may not be known, and the relationships are hesitant in a natural sense. With the advent of the fuzzy graph, the importance of this theory increased and fuzzy graph as a generalization of a graph provides more information in real-life problems. Based on Zadeh’s fuzzy relations [15], the notion of hypergraph has been extended in the fuzzy theory and the concept of fuzzy hypergraph was provided by Kaufmann [16]. Recently, some researchers investigated the concept of fuzzy hypergraphs and applications such as fuzzy hypergraphs and related extensions [17], an algorithm to compute the strength of competing interactions in the Bering sea based on Pythagorean fuzzy hypergraphs [18] and bipolar fuzzy soft information applied to hypergraphs [19]. Recently, Smarandache introduced a new the concept as a generalization of hypergraphs to -superhypergraph, plithogenic -superhypergraph {with supervertices (that are groups of vertices) and hyperedges {defined on power set of power set…} that is the most general form of a graph as today}, which have several properties and are connected with the real-world [20]. Indeed, -superhypergraphs are a generalization of hypergraphs, with the advantage that they can communicate between the hyperedges.
Regarding these points, we consider a nonempty set and make a partition of the given set into some subsets, and relate this subset together with some maps. Indeed, subsets will call supervertices, the mapping between them will call superedges or flows and the system with supervertices and superedges will call a quasi superhypergraph. The main motivation of this the concept is a generalization of graphs to hypernetworks such that all elements be related together. In hypergraph theory, any hypergraph can relate a set of elements, while without any details that it makes some conflicts, defects, and shortcomings in the hypergraph theory. Thus, by introducing superhypergraph, we try to eliminate defects of graph (sometimes graph structures give very limited information about complex networks) structures and hypergraph structures (although the hypergraph structures are for covering graph defects in the applications but in hypergraphs, the relation between of vertices cannot be described in full details). As a main result of the research, this paper is introducing the incidence matrix of superhypergraph and computing the characteristic polynomial for the incidence matrix of superhypergraph, so we obtain the spectrum of superhypergraphs. Indeed, we computed the number of superedges of any given superhypergraphs and based on superedges and partitions of an underlying set of superhypergraph, we obtained the number of all superhypergraphs on any nonempty set. The flow of superedges plays the main role in computing of spectrum of superhypergraphs, so we computed the spectrum of superhypergraphs in some types regular flow, regular reversed flow, and regular two-sided flow.
Definition 1. [1] Let be a finite set. A hypergraph on is a pair such that for all and . The elements of are called vertices, and the sets are called the hyperedges of the hypergraph . In hypergraphs, hyperedges can contain an element two elements or more than three elements. A hypergraph is called a complete hypergraph, if for any there is such that . A hypergraph is called as a joint complete hypergraph, if for all and element . If for all , , the hypergraph becomes an ordinary graph and rows representing the vertices , where for all and for all , we have if and if .
Definition 2. [20] Let and be a set of vertices, that contains single vertices , indeterminate vertices and null vertices . Consider as the power set of , , and be the -power set of the set . Then, the -superhypergraph is an ordered pair -, where for any , is the set of vertices and is the set of edges. The set contains some type of vertices, such as single vertices , indeterminate vertices , null vertices , and supervertices , i.e., two or more vertices together as a group . An -supervertex is a collection of many vertices such that at least one is a -supervertex and all other supervertices in to the collection if any have the order . The set of edges contains some type of edges such as single edges , indeterminate , null-edge , hyperedge , superedge , -superedge , superhyperedge (containing three or more vertices, at least one being a supervertex, -superhyperedge , multiedges , and loop .
2. On (Quasi) Superhypergraph
In this section, we introduce the concepts of supervertex, superedge, superhypergraph, and investigate their properties. For any given superhypergraph, the lower and upper bound of the set of their superedges is computed and proved in a theorem. Also, we computed and proved the number of all superhypergraphs constructed on any given nonempty set. In the following, for any nonempty set , will denote .
In what follows, based on the concept of -superhypergraph [20], recall, define, and investigate a special case in -superhypergraphs as the notation of quasi superhypergraph.
Definition 3. Let be a nonempty set. Then,(i) is called a quasi superhypergraph, if and , where ,(ii)for all , is called a supervertex and for any , the map (say links to ) is called a superedge,(iii)the quasi superhypergraph is called a superhypergraph, if for any , there exists at least one such that links to .(iv)The superhypergraph is called a trivial superhypergraph, if .
Example 1. Let . Then, is a quasi superhypergraph in Figure 1, whereLet be a superhypergraph. We will denote by the set of all superedges of superhypergraph . In what follows, compute and prove the lower bound and upper bound of , as set of all superedges of superhypergraph.
Theorem 1. Let be a superhypergraph. Then,
Proof. Let be a superhypergraph. Since, by definition, for any , there exists at least one such that links to , we have superedges. In addition, let . For all , in one case, if , then, . Hence, if , then, for all implies thatIn another case, if there exists some such that , since for all , , for all we get that and . So, in any cases, .
Example 2. Let and be a superhypergraph, where . Then, by Theorem 1, , whereThe superhypergraph with minimum superedges is shown in Figure 2.
Let be a nonempty set, and . In what follows, compute and prove the number of , as the set of all superhypergraphs based on any nonempty set , where .
Theorem 2. Let be a nonempty set, and .(i).(ii)If and , then, .
Proof. (i)By definitions is clear.(ii)Let . Since , we get that is a partition of set , where . Consider , then, the numbers of selected vertices in is equal to . Because for any , the number of selected vertices in is equal to . It follows that the numbers of ways to chosen the selected vertices between is equal to for the case and is equal to , for the case . Thus, in the process of doing so and by induction, for all , , where . In addition, by Theorem 1, we have , so .
Theorem 3. Let be a nonempty set and . If and , then,
Proof. It is clear by Theorem 2.
Example 3. Let be an arbitrary set and . Then,
3. Incidence Matrix of Superhypergraphs
In this section, we introduce a square matrix as incidence matrix associate with any given superhypergraph with sign function. Indeed, in the incidence matrix associate to any given superhypergraph the domain and range of any map determine the sign function.
Let be a superhypergraph and . Then, we have the following concepts.
Definition 4. Let be a superhypergraph and . Define as incidence matrix of with columns representing the supervertices , superedges and rows representing the vertices , where and
Example 4. Let . Then, is a superhypergraph in Figure 3, where and .
Then, is the incidence matrix of .
3.1. Characteristic Polynomial for the Incidence Matrix of Superhypergraph
In this subsection, we compute the characteristic polynomial of the incidence matrix of any given superhypergraph. Let be a superhypergraph and so investigate the spectrum of the superhypergraph.
From now on, let be the characteristic polynomial of the incidence matrix corresponding to superhypergraph and . In addition, for any , if , will say flows to and will denote by . In this case, will denote by and by .
Theorem 4. Let , and . Then,(i).(ii)Spec .
Proof. (i)Let . It is easy to see that . Suppose that and . Then, , where . It follows that such that , where that for all . Now, consider , where Thus, , where . Based on induction assumption and computations of determinant based on column in matrix , have(ii)It is clear by item (i).
Theorem 5. Let , and . Then, .
Proof. Let . It is easy to see that . Suppose that and . Then, , where . It follows thatsuch that , where that for all and . Consider , whereThus, , where . Based on induction assumption and computations of determinant based on column in matrix , have
Theorem 6. Let , and . Then, .
Proof. Let . It is easy to see that . Suppose that and .Then, , where . It follows thatsuch that , where that for all and . In addition, , whereThus, , where . Based on induction assumption and computations of determinant based on column in matrix show that
Theorem 7. Let , and . Then, .
Proof. It is similar to Theorem 6.
Corollary 1. Let , and . Then,(i).(ii)Spec , where .(iii) if and only if .
Example 5. Let . Then, is a superhypergraph as shown in Figure 4 and incidence matrix of as follows.Thus, by Theorem 5, we have and so .
3.2. Characteristic Polynomial for the Incidence Matrix of Superhypergraph with Reverse Flows
In this subsection, we compute the characteristic polynomial and spectrum of the superhypergraph in the reverse flows to the previous section.
For any , if , will say flows to and will denote by and so by and by .
Theorem 8. Let , and . Then,(i).(ii)Spec .
Proof. (i)Let . It is easy to see that . Suppose that and . Then, , where . It follows that such that , where that for all . Now, consider , where Thus, , where . Based on induction assumption and computations of determinant based on column in matrix , have(ii)It is clear by item (i).
Theorem 9. Let , and . Then, .
Proof. Let . It easy is to see that . Suppose that and .Then, , where . It follows thatsuch that , where that for all and . Moreover, consider , whereThus, , where . Based on induction assumption and computations of determinant based on column in matrix , have
Theorem 10. Let and . Then, .
Proof. Let . It is easy to see that . Suppose that and .Then, , where . It follows thatsuch that , where that for all , and . Consider whereThus, , where . Based on induction assumption and computations of determinant based on column in matrix show that
Theorem 11. Let , and . Then, .
Proof. It is similar to Theorem 6.
Corollary 2. Let , and . Then,(i).(ii)Spec , where .(iii) if and only if .
Example 6. Let . Then, is a superhypergraph as shown in Figure 5 and incidence matrix of as follows.Thus, by Theorem 9, we have and so Spec .
3.3. Characteristic Polynomial for the Incidence Matrix of Superhypergraph with Two-Sided Flows
In this subsection, we compute the characteristic polynomial and spectrum of the superhypergraph with two-sided flows.
For any , if , will say flows to by two-sided and will denote by and so by and by . When we will show that of type , it means that the first map flows from to , the second map flows from to , the third map flows from to , etc., respectively. We will show it in Figure 6, where, is an odd.
Theorem 12. Let , and of type . Then,(i).(ii)if is an even, then Spec .(iii)if is an odd, then Spec .
Proof. (i) Let and . It is easy to see that and . Suppose that is an odd and . Then, , where for all and where for all . It follows thatsuch that , where , where for all and for all . Now, consider , whereThus, , where and . Based on induction assumption and computations of determinant based on column in matrix , haveIn a similar way, if is an even, we get that .
(ii) and (iii) They are clear by item (i) and (ii).
Theorem 13. Let , and of type . Then,
Proof. Let . It is easy to see that . Suppose that is an odd and . Then, , where for all and where for all . It follows thatsuch that , where , where for all and for all and for . Now, consider , whereThus, , where and . Based on induction assumption and computations of determinant based on column in matrix , we haveIn a similar way, if is an even, we get that .
(ii) and (iii) They are clear by item (i) and (ii).
Theorem 14. Let , and of type . Then,
Proof. Let . It is easy to see that . Suppose that is an odd and . Then, , where for all and where for all . It follows thatsuch that , where , where for all and for and for all . Now, consider , whereThus, , where and . Based on induction assumption and computations of determinant based on column in matrix , we haveIn a similar way, if is an even, we get that .
(ii) and (iii) They are clear by item (i) and (ii).
Theorem 15. Let , and of type . Then,
Proof. It is similar to Theorem 14.
Corollary 3. Let , and of type . If is an odd, then,(i),(ii)If is an odd, then, Spec ,(iii)If is an even, then Spec ,(iv).
Corollary 4. Let , and of type . If is an even, then,(i)if is an even, then, ,(ii)If is an odd, then, Spec ,(iii)If is an even, then, Spec ,(iv).
When we will show that of type , it means that the first map flows from to , the second map flows from to , the third map flows from to , etc., respectively. We will show it in Figure 7, where, is an odd.
Theorem 16. Let , and of type . Then,(i),(ii)if is an even, then, Spec ,(iii)if is an odd, then, Spec .
Proof. (i) Let and . It is easy to see that and . Suppose that is an even and . Then, , where for all and where for all . It follows thatsuch that , where , where for all and for all . Now, consider , whereThus, , where and . Based on induction assumption and computations of determinant based on column in matrix , we haveIn a similar way, if is an odd, we get that .
(ii) and (iii) They are clear by item (i).
Theorem 17. Let , and of type . Then,
Proof. Let . It is easy to see that and Suppose that is an even and . Then, , where for all and where for all . It follows thatsuch that , where , where for all and for all and for . Now, consider , whereThus, , where and . Based on induction assumption and computations of determinant based on column in matrix , we haveIn a similar way, if is an odd, we get that .
(ii) and (iii) They are clear by item (i) and (ii).
Theorem 18. Let , and of type . Then,
Proof. Let . It is easy to see that and for . Suppose that is an odd and . Then, , where for all and where for all . It follows thatsuch that , where , where for all and for and for all and for . Now, consider , whereThus, , where and . Based on induction assumption and computations of determinant based on column in matrix , we haveIn a similar way, if is an even, we get that .
(ii) and (iii) They are clear by item (i) and (ii).
Theorem 19. Let , and of type . Then,
Proof. It is similar to Theorem 18.
Corollary 5. Let , and of type . If is an odd, then,(i),(ii)If is an odd, then, Spec ,(iii)If is an even, then, Spec ,(iv).
Corollary 6. Let , and of type . If is an even, then,(i),(ii)If is an odd, then, Spec ,(iii)If is an even, then, Spec ,(iv).
4. Conclusions and Future Works
The current paper has introduced a novel concept of superhypergraphs as a generalization of graphs. The advantage of the notation of superhypergraphs is that it considers the relationship between a set of elements separately and as a whole, and this helps to eliminate the defects of graphs and superhypergraphs. The notation of superhypergraphs can be useful tools in modeling the real issues in engineering sciences and other sciences, especially network-related issues. For any given superhypergraph, the lower and upper bound of the number of the set of their superedges is computed and so it is computed and proved the number of all superhypergraphs constructed on any given nonempty set. Polynomial characteristics and eigenvalues of a matrix that represents a superhypergraph can provide useful information about the superhypergraph. The concept of the incidence matrix of superhypergraphs is presented and the characteristic polynomial of the incidence matrix of superhypergraphs and spectrum of superhypergraphs is analyzed and computed. It is shown that the spectrum of superhypergraphs depended on to flows of their maps between supervertices and the spectrum of superhypergraphs varies with the change of direction of flows. We presented and computed the spectrum of superhypergraphs with some types of flows such as one-sided flows, left to the right flows, right to left flows, and two-sided reverse flows. We hope that these results are helpful for further studies in the theory of graphs, hypergraphs, and superhypergraphs. In our future studies, we hope to obtain more results regarding domination sets and domination numbers of superhypergraphs, fuzzy superhypergraphs, and obtain some results in this regard and their applications in the real-world.
Data Availability
The data used to support the findings of this study are included within this article and can be obtained from the corresponding author upon request for more details on the data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.