Abstract
In this paper, we defined two classes of hypergraphs, hyperbugs and kite hypergraphs. We show that balanced hyperbugs maximize the spectral radii of hypergraphs with fixed number of vertices and diameter and kite hypergraphs minimize the spectral radii of hypergraphs with fixed number of vertices and clique number.
1. Introduction
Hypergraph theory deals extensively with hypergraph invariants, i.e., function represents a certain invariant of the hypergraph as a real number or integer, [1–3]. Well-known invariants are the independence and chromatic numbers, the diameter, and so on. Extremal hypergraph theory deals with the problem of characterizing the families of hypergraph for which an invariant is minimum or maximum. Usual hypergraph classes, such as complete hypergraphs, hyperpaths, hypercycles, and hyperstars, frequently appear as extremal hypergraphs in hypergraph spectral theory problems. Here we want to turn the readers’ attention to two novel, simply defined, hypergraph classes that appear as extremal hypergraphs in several hypergraph spectral theory problems. We call them hyperbugs and kite hypergraphs.
In recent years, a few results are known about the spectral radii of hypergraphs with property , for example, Fan et al. [4] considered the maximum spectral radius of uniform hypergraphs with few edges. Xiao et al. [5] investigated the supertrees whose spectral radii attain the maximum among all uniform supertrees with given degree sequence. Xiao and Wang [6] also determined the unique hypergraph with the maximum spectral radius among all the uniform supertrees and all the connected uniform unicyclic hypergraphs with given number of pendant edges, respectively. Zhang and Li [7] characterized the hypergraph with maximum spectral radius among all connected uniform hypergraphs with given number of pendant vertices. Su et al. [8] determined the largest spectral radius of hypertrees with edges and given size of matching. Xiao et al. [9] determined the supertrees with the first two largest spectral radii among all supertrees in the set of -uniform supertrees with edges and diameter . Su et al. [10] determined the first largest spectral radius of -uniform supertrees with size and diameter . In addition, the first two smallest spectral radii of supertrees with size are also determined. For other related results, readers are referred to [11–20]. In the spirit of the general problem of Brualdi and Solheid [21], one can ask how large or how small can be the spectral radius of hypergraphs with some specific properties. For example: how large can be if is a -uniform hypergraph of order and diameter at least ? Similarly, how small can be if is a -uniform hypergraph of order and clique number at least 2? In fact, for , this question has been answered by Stevanović and Hansen in [22, 23], respectively.
Lemma 1 (see [22, 24]). Let be a graph of order with . If , then . If , thenholds if and only if , where denotes the graph obtained from a complete graph by deleting an edge and attaching paths and to its ends.
Lemma 2 (see [23, 24]). If is a connected graph of order with clique number , then holds if and only if , where denotes the graph obtained by joining an end vertex of the path to a vertex of the complete graph .
In this paper, we mainly generalize the above two results to the -uniform hypergraphs.
2. Notations and Preliminaries
A hypergraph is a pair with and denotes the power set of , is a nonempty finite set (the vertex set), and the elements of are called hyperedges or edges. A hypergraph is -uniform if every edge contains precisely vertices. A sub-hypergraph of is a -uniform hypergraph such that and . For a vertex , we denote by the set of edges containing . We write to express the edges containing the vertices . The cardinality is the degree of , denoted by . A vertex with degree one is called a core vertex. If any two edges in share at most one vertex, then is said to be linear hypergraph. Let be obtained by deleting an edge from , i.e., . is obtained from by deleting vertices of edge , i.e., , where .
A walk in a hypergraph is a finite alternating sequence of vertices and edges, i.e., , satisfying that both and are incident to for . A walk is a path if all the vertices for and all the edges for in are distinct. The length of a path is the number of edges in it. A hypergraph is connected, if there is a path between any pair of vertices of . In this paper, we assume that hypergraphs are -uniform and connected.
Let be a -uniform hypergraph. An edge is called a pendant edge if contains exactly core vertices. If is not a pendant edge, it is called a nonpendant edge. A path of is called a pendant path (attached at ), if all of the vertices are of degree two and the vertex and all the vertices in the set are core vertices in . Let be a -uniform hypergraph of order ; if consists of all -subsets of , then is a complete -uniform hypergraph, denoted by . A clique of a -uniform hypergraph is a complete -uniform sub-hypergraph of . A maximal clique is a clique that cannot be extended to a larger clique. The clique number of a hypergraph is the number of vertices in the maximum clique of .
Definition 1. (see [11]). The adjacency tensor of an -uniform hypergraph on vertices is defined as the tensor of order and dimension whose -entry is
The spectrum, eigenvalues, and spectral radius of are defined to be those of its adjacency tensor .
The following relation between the spectral radius of a -uniform hypergraph and its sub-hypergraph can be found in [11].
Lemma 3 (see [11]). Let be a -uniform hypergraph, and is a sub-hypergraph of ; then, .
The first upper and lower bounds of spectral radius of a -uniform hypergraph are given by Cooper et al. as follows.
Lemma 4 (see [11]). Let be a -uniform hypergraph. Let be the average degree of and be the maximum degree. Then,
In [14], Li et al. proposed an effective method to find a -uniform hypergraph with larger spectral radius.
Definition 2. (see [14]) (general edge-moving operation). Suppose that is a hypergraph with and , such that . Let and . Let be the hypergraph with . Then, we say that is obtained from by moving edges from to .
According to the definition of general edge-moving operation, Li et al. obtained the following relation of spectral radius.
Lemma 5 (see [14]). Suppose that is a -uniform hypergraph and is the hypergraph obtained from by moving edge from to , where contains no multiple edges. If is the perron eigenvector of corresponding to and , then .
In [14], Li et al. also gave two extremal results about upper and lower bounds of the th power of an ordinary tree .
Lemma 6 (see [14]). Let be the th power of an ordinary tree . Suppose that has vertices. Then, we havewhere the former equalities hold if and only if , and the latter equalities hold if and only if .
Let be a -uniform hypergraph. Let be the hypergraph obtained by attaching the paths and to . Similarly, let be the hypergraph obtained by attaching the paths to and to .
In [17], Shan et al. gave an operation to find a -uniform hypergraph with larger spectral radius.
Lemma 7 (see [17]). Let be two non-pendant vertices of hypergraph . If there exist an internal path with length in hypergraph for any , then we have
In [12], Guo and Zhou gave another operation to find a -uniform hypergraph with larger spectral radius.
Lemma 8 (see [12]). For , let be a -uniform hypergraph with and . For , we have
A hypergraph is isomorphic to a hypergraph , if there is a bijection such that if and only if . The bijection is called an isomorphism of . If , then is called an automorphism of . Let be a vector defined on and be an automorphism of . Two vertices and are equivalent in , if there exists an automorphism of such that , . Denote to be the vector such that for each .
Lemma 9 (see [13]). Let be a -uniform hypergraph and be an automorphism of . Let x be an eigenvector of ; then, . Further, for two vertices and in , if , there must be .
Next we discuss two novel hypergraph families, i.e., hyperbugs and kite hypergraphs, which are defined as follows.
Definition 3. A hyperbug is a -uniform hypergraph obtained from a complete -uniform hypergraph by deleting an edge attaching paths and at and . A hyperbug is balanced if (see Figure 1 for an example).
Definition 4. A kite hypergraph is a -uniform hypergraph obtained by joining an end vertex of the path to a vertex of the complete -uniform hypergraph (see Figure 2 for an example).
In this paper, we obtain that if is a -uniform hypergraph of order and diameter at least , thenholds if and only if . Furthermore, we also obtain that if is a -uniform hypergraph of order with clique number , then holds if and only if . These generalize some related results of Nikiforov and Rojo [24] and Hansen and Stevanović [22].
3. Main Results
3.1. The Spectral Radii of Uniform Hypergraphs with Fixed Number of Vertices and Diameter
Let be a -uniform hypergraph containing a path as a sub-hypergraph (see Figure 3). We say that is a pendant path if one of its ends is a cut vertex of ; we call this vertex the root of . Note that a hypergraph can have multiple pendant paths, which may share roots; e.g., the hypergraph has two pendant paths.
From Figure 3, we see that if is a -uniform hypergraph and is a -uniform hypergraph with a pendant path and , then the distribution of the entries of an eigenvector to along is well determined.
In fact, let be a pendant path in with root . Let be the entries of a positive unit eigenvector to corresponding to . The eigenequation of for iswhich implies that
The above equation is equivalent to the following crucial equation:
We can write for the root of (10):and note that is real since ; moreover, , with strict inequality if . Note also that the other root (10) is equal to .
By equations (10) and (11), we can obtain the following theorem.
Theorem 1. Let be a -uniform hypergraph with . Let be a pendant path in with root . Let be the entries of a positive unit eigenvector to corresponding to . If is defined by (11), then for every , we have
Proof. The eigenequation of for isSince , thenBy multiplying the above inequalities, we haveSo, we have By Lemma 8, we haveThus,Proceeding by induction, from the eigenequation of for and the induction assumption, we getThus,By multiplying the above inequalities, we haveSo, we getSupposeSubstituting equation (24) into equation (23), we obtainwhich contradicts equation (25). Hence,The proof is completed.
Corollary 1. Given the hypotheses of Theorem 1., we have
Lemma 10. Let and be fixed positive integers; then,
Proof. According to the definition of , we must have an internal path with 2 lengths in the k-uniform hypergraph . Without loss of generality, we assume . By Lemma 6, we haveContinuing this operation, when and are fixed, we have
Theorem 2. Let be a -uniform hypergraph of order with . If , then . If , thenholds if and only if .
Proof. The statement is clear if , for is the only hypergraph of order and diameter 1. Suppose that ; let be a hypergraph with maximal spectral radius among all -uniform hypergraphs of order and . This choice implies that is edge-maximal, that is, no edge can be added to without diminishing its diameter. According to Lemma 9, we only need to show that for some , satisfying .
Let and be vertices of at distance exactly , for every . Let be the set of the vertices at distance from , and degrees of these vertices are greater than 2. Since is edge-maximal, the set induces a linear complete hypergraph, for every . It is also clear that ; moreover, it is not hard to see that . Indeed, assume for a contradiction that and add all edges between and . These additional edges do not diminish the distance between and ; hence, is not edge-maximal, contradicting its choice; therefore, .
Furthermore, by Lemma 3, we haveand so . Suppose that is vertex of maximum degree in , and let . Clearly, , and in view ofwe find thatHence, if or , then ; furthermore,If , then obviously , so Theorem 2 is proved in this case. Next we will show that all other cases lead to contradictions, by constructing a hypergraph of order and with . Suppose that is a positive unit vector to .
First, consider the case and . If , the proof is completed. So, we suppose that . Let , and suppose by symmetry that . Choose a vertex , obtain from , delete the edge , and add the edge . In other words, is obtained by moving the vertex from into . By symmetry and Lemma 8, for any ; thus, the choice of implies thatimplying that and that x is an eigenvector to . However, the neighborhood of in is a proper subset of the neighborhood of in , so the eigenequations for and for the vertex are contradictory.
The same argument disposes also of the case and ; thus, to complete the proof, it remains to consider the case and .
Let , and . Our first step is to show thatNote that if and , then Theorem 1 gives . Hence, setting , the eigenequation for the vertex implies thatAccording to and Theorem 1, we haveyielding in turnSinceinequality (39) is proved. By symmetry, we also see that . Suppose, again by symmetry, that , which yields . Choose a vertex , obtain from , delete the edges , and add the edges for all . In other words, is obtained by moving the vertex from into . The choice of implies thatThis contradiction completes the proof of Theorem 2.
Remark 1. From the above result, when k = 2, the result of Lemma 1 is obvious [22].
3.2. The Spectral Radii of Uniform Hypergraphs with Fixed Number of Vertices and Clique Number
Lemma 11. For , let be a -uniform hypergraph with and . For , if is the -uniform hypergraph obtained by identifying with a vertex of a hypertree of order , then
Proof. This can be carried out along well-known lines by applying Lemma 7 to recursively flatten until it becomes a path.
Theorem 3. If is a -uniform hypergraph of order with clique number , then holds if and only if .
Proof. Let be a -uniform hypergraph with minimal spectral among all connected -uniform hypergraphs of order and clique number . If , by Lemma 5, must be a path, as the path is the hypergraph with smallest spectral radius among connected -uniform hypergraphs of given order. Thus, we suppose that and let be a complete -uniform sub-hypergraph of of order .
Further, should be edge-minimal, that is, the removal of any edge of either makes disconnected or its clique number diminishes. In particular, if is the hypergraph obtained by removing the edges of , then the components of are supertrees, and each component has exactly one vertex in common with . It follows that is isomorphic to a complete -uniform hypergraph of order with hypertrees attached to some of its common vertices of edges. Moreover, Lemma 10 implies that each of those hypertrees must be a pendant path. To complete the proof, we show that there is only one such path.
Let , and . Suppose that a path is attached to and is attached to . Let be the hypergraph obtained by deleting the edge and adding the edge , that is, is obtained by removing and extending to . To complete the proof, we need to show that .
Let and x be a positive eigenvector of to . Write for the entries of x corresponding to , and let be defined by (2). Now, if , then Theorem 1 implies that , and so Lemma 4 implies that . Thus, we focus on showing that .
Since the eigenequation for is Hence,Likewise, for any , the eigenequation for givesIn particular, we see that for any , since .
Returning to the eigenequation for , we find thatAccording to Theorem 1, Assuming for a contradiction that , after some algebra, we getand therefore,It is known that , since is not a -regular hypergraph. Hence,which is a contradiction if . If , inequality (51) becomesSquaring both sides of this inequality, we getand soTherefore, , an obvious contradiction, and this completes the proof of Theorem 3.
Remark 2. From the above result, when k = 2, the result of Lemma 2 is obvious [23].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was partially supported by the National Natural Science Foundation of China (grant no. 11551003), the Natural Science Foundation of Guangdong Province (grant no. 2021A1515012045), and the Qinghai Natural Science Foundation of China (grant no. 2020-ZJ-924).