Sharp Bounds on the Spectral Radii of Uniform Hypergraphs concerning Diameter or Clique Number

Hypergraph theory deals extensively with hypergraph invariants, i.e., function iH 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 H for which an invariant iH 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 P, 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 r 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 m-uniform supertrees with r edges and diameter d. Su et al. [10] determined the first ⌊d/2⌋ + 1 largest spectral radius of k-uniform supertrees with size m and diameter d. In addition, the first two smallest spectral radii of supertrees with size m 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 ρ(H) can be if H is a k-uniform hypergraph of order n and diameter at least r? Similarly, how small ρ(H) can be if H is a k-uniform hypergraph of order n and clique number at least 2? In fact, for k � 2, this question has been answered by Stevanović and Hansen in [22, 23], respectively.


Introduction
Hypergraph theory deals extensively with hypergraph invariants, i.e., function i H represents a certain invariant of the hypergraph as a real number or integer, [1][2][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 H for which an invariant i H 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 P, 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 r 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 m-uniform supertrees with r edges and diameter d. Su et al. [10] determined the first ⌊d/2⌋ + 1 largest spectral radius of k-uniform supertrees with size m and diameter d. In addition, the first two smallest spectral radii of supertrees with size m are also determined. For other related results, readers are referred to [11][12][13][14][15][16][17][18][19][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 ρ(H) can be if H is a k-uniform hypergraph of order n and diameter at least r? Similarly, how small ρ(H) can be if H is a k-uniform hypergraph of order n and clique number at least 2? In fact, for k � 2, this question has been answered by Stevanović and Hansen in [22,23], respectively. Lemma 1 (see [22,24]). Let G be a graph of order n with diam(G) ≥ r. If r � 1, then ρ(G) � ρ(K n ). If r ≥ 2, then holds if and only if G � B n− r+2,⌊r/2⌋,⌊r/2⌋ , where B p,q,r denotes the graph obtained from a complete graph K p by deleting an edge and attaching paths P q and P r to its ends.
Lemma 2 (see [23,24]). If G is a connected graph of order n with clique number ω ≥ 2, then ρ(G) ≥ ρ(PK n− ω,ω ) holds if and only if G � PK n− ω,ω , where PK p,q denotes the graph obtained by joining an end vertex of the path P p to a vertex of the complete graph K q .
In this paper, we mainly generalize the above two results to the k-uniform hypergraphs.

Notations and Preliminaries
A walk W in a hypergraph H is a finite alternating sequence of vertices and edges, i.e., W � (v 0 , e 1 , v 1 , e 2 , . . . , e k , v k ), satisfying that both v i− 1 and v i are incident to e i for 1 ≤ i ≤ k. A walk W is a path if all the vertices v i for i � 0, 1, . . . , k and all the edges e j for j � 1, 2, . . . , k in W are distinct. e 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 H. In this paper, we assume that hypergraphs are k-uniform and connected.
Let Definition 1 (see [11]). e adjacency tensor of an r-uniform hypergraph H on n vertices is defined as the tensor A(H) of order r and dimension n whose (i 1 , . . . , i r )-entry is e spectrum, eigenvalues, and spectral radius ρ(H) of H are defined to be those of its adjacency tensor A.
e following relation between the spectral radius of a k-uniform hypergraph and its sub-hypergraph can be found in [11].
Lemma 3 (see [11]). Let H be a k-uniform hypergraph, and e first upper and lower bounds of spectral radius of a k-uniform hypergraph are given by Cooper et al. as follows.
Lemma 4 (see [11]). Let H be a k-uniform hypergraph. Let d be the average degree of H and Δ be the maximum degree. en, In [14], Li et al. proposed an effective method to find a k-uniform hypergraph with larger spectral radius.
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 H is a k-uniform hypergraph and H * is the hypergraph obtained from H by In [14], Li et al. also gave two extremal results about upper and lower bounds of the kth power of an ordinary tree T.
Lemma 6 (see [14]). Let T k be the kth power of an ordinary tree T. Suppose that T k has n vertices. en, we have 2 Discrete Dynamics in Nature and Society where the former equalities hold if and only if T k � P k n , and the latter equalities hold if and only if T k � S k n .
Let H be a k-uniform hypergraph. Let H p,q (u) be the hypergraph obtained by attaching the paths P p and P q to u ∈ H. Similarly, let H p,q (u, v) be the hypergraph obtained by attaching the paths P p to u ∈ H and P q to v ∈ H.
In [17], Shan et al. gave an operation to find a k-uniform hypergraph with larger spectral radius.
Lemma 7 (see [17]). Let u, v be two non-pendant vertices of hypergraph H. If there exist an internal path P with s length in hypergraph H p,q (u, v) for any p ≥ q ≥ 1, then we have In [12], Guo and Zhou gave another operation to find a k-uniform hypergraph with larger spectral radius.
Lemma 9 (see [13]). Let H be a k-uniform hypergraph and σ be an automorphism of H. Let x be an eigenvector of A(H); Next we discuss two novel hypergraph families, i.e., hyperbugs and kite hypergraphs, which are defined as follows.
Definition 3. A hyperbug B p,q,s is a k-uniform hypergraph obtained from a complete k-uniform hypergraph K k p by deleting an edge u, u 2 , . . . , u k− 2 , v attaching paths P q and P s at u and v. A hyperbug is balanced if |q − s| ≤ 1 (see Figure 1 for an example).

Definition 4.
A kite hypergraph PK p,q is a k-uniform hypergraph obtained by joining an end vertex of the path P p to a vertex of the complete k-uniform hypergraph K k q (see Figure 2 for an example).
In this paper, we obtain that if H is a k-uniform hypergraph of order n and diameter at least r, then holds if and only if H � B n− (r− 2)(k− 1),⌊r/2⌋,⌈r/2⌉ . Furthermore, we also obtain that if H is a k-uniform hypergraph of order n with clique number ω ≥ 2, then ρ(H) ≥ ρ(PK n− ω,ω ) holds if and only if H � PK n− ω,ω . ese generalize some related results of Nikiforov and Rojo [24] and Hansen and Stevanović [22].

e Spectral Radii of Uniform Hypergraphs with Fixed
Number of Vertices and Diameter. Let H be a k-uniform hypergraph containing a path P as a sub-hypergraph (see Figure 3). We say that P is a pendant path if one of its ends is a cut vertex of H; we call this vertex the root of P. Note that a hypergraph can have multiple pendant paths, which may share roots; e.g., the hypergraph B p,q,s has two pendant paths.
From Figure 3, we see that if H ′ is a k-uniform hypergraph and H is a k-uniform hypergraph with a pendant path P and ρ(H) � ρ ≥ 2, then the distribution of the entries of an eigenvector to ρ along P is well determined.
In fact, let P � (u 1 , . . . , u i , . . . , u i+k− 1 ) be a pendant path in H with root u 1 . Let x 1 , . . . , x i , . . . , x i+k− 1 (i ∈ N) be the entries of a positive unit eigenvector to ρ(H) corresponding to u 1 , . . . , u i , . . . , which implies that Discrete Dynamics in Nature and Society e above equation is equivalent to the following crucial equation: We can write c for the root of (10): and note that c is real since ρ ≥ 2; moreover, c ≥ 1, with strict inequality if ρ > 2. Note also that the other root (10) is equal to c − 1 . By equations (10) and (11), we can obtain the following theorem. N) be the entries of a positive unit eigenvector to ρ(H) corresponding to u 1 , . . . , u i , . . . , u i+k− 1 , (i � n(k − 1) + 1, n ∈ N). If c is defined by (11), then for every x j (1 ≤ j ≤ i + k − 1), we have Proof. e eigenequation of A(H) for Since ρ > c, then By multiplying the above k − 1 inequalities, we have So, we have By Lemma 8, we have us, Proceeding by induction, from the eigenequation of A(H) for x j (i − k + 2 ≤ j ≤ i) and the induction assumption, we get us, By multiplying the above k − 1 inequalities, we have So, we get Suppose x i− k+1 < cx i .
(25) Substituting equation (24) into equation (23), we obtain which contradicts equation (25). Hence, e proof is completed. □ Corollary 1. Given the hypotheses of eorem 1., we have Lemma 10. Let p and q + r be fixed positive integers; then, Proof. According to the definition of B p,q,r , we must have an internal path with 2 lengths in the k-uniform hypergraph B p,q,r . Without loss of generality, we assume q ≥ r ≥ 1. By Lemma 6, we have Continuing this operation, when p and q + r are fixed, we have ρ B p,q,r < ρ B p,⌊q+r/2⌋,⌈q+r/2⌉ .
Proof. e statement is clear if r � 1, for K k n is the only hypergraph of order n and diameter 1. Suppose that r ≥ 2; let H be a hypergraph with maximal spectral radius among all k-uniform hypergraphs of order n and diam(H) ≥ r. is choice implies that H is edge-maximal, that is, no edge can be added to H without diminishing its diameter. According to Lemma 9, we only need to show that H � B n− (r− 2)(k− 1),p,r− p for some p, satisfying 1 ≤ p ≤ r − 1.
Let ρ ≔ ρ(H) and u, v be vertices of H at distance exactly r, for every i � 0, . . . , r. Let V i be the set of the vertices at distance i from u, and degrees of these vertices are greater than 2. Since H is edge-maximal, the set V i ∪ V i+1 induces a linear complete hypergraph, for every i � 0, . . . , r − 1. It is also clear that |V 0 | � 1; moreover, it is not hard to see that |V r | � 1. Indeed, assume for a contradiction that |V r | ≥ 2 and add all edges between V r− 2 and V\ v { }. ese additional edges do not diminish the distance between u and v; hence, H is not edge-maximal, contradicting its choice; therefore, |V r | � 1.
Furthermore, by Lemma 3, we have and so Δ(H) ≥ C k− 1 n− 1− (r− 2)(k− 1) . Suppose that w is vertex of maximum degree in H, and let w ∈ V i . Clearly, 0 < i < r, and in view of we find that Hence, if j < i − 1 or j > i + 1, then |V i | � 1; furthermore, so eorem 2 is proved in this case. Next we will show that all other cases lead to contradictions, by constructing a hypergraph H * of order n and diam(H * ) � r with ρ(H * ) > ρ. Suppose that x: � (x 1 , . . . , x n ) is a positive unit vector to ρ(H).
{ }, and suppose by symmetry that x b ≥ x a . Choose a vertex w ∈ V i , obtain H ′ from H, delete the edge e wa , and add the edge e wb . In other words, H ′ is  Discrete Dynamics in Nature and Society 5 obtained by moving the vertex w from V i into V i+1 . By symmetry and Lemma 8, x w′ � x w for any w ′ ∈ V i ; thus, the choice of H implies that implying that ρ(H * ) � ρ(H) and that x is an eigenvector to ρ(H). However, the neighborhood of a in H * is a proper subset of the neighborhood of a in H, so the eigenequations for ρ(H * ) and ρ(H) for the vertex a are contradictory. e same argument disposes also of the case |V i− 1 | ≥ 2 and |V i+1 | � 1; thus, to complete the proof, it remains to consider the case |V i− 1 | ≥ 2 and |V i+1 | ≥ 2. Let Note that if i ≥ 3 and V i− 3 � z { }, then eorem 1 gives x z < x a . Hence, setting l : � |V i− 1 |, the eigenequation for the vertex a implies that According to x z < x a and eorem 1, we have yielding in turn Since inequality (39) is proved. By symmetry, we also see that is contradiction completes the proof of eorem 2. □ Remark 1. From the above result, when k � 2, the result of Lemma 1 is obvious [22].

e Spectral
Proof.
is can be carried out along well-known lines by applying Lemma 7 to recursively flatten T until it becomes a path. Proof. Let H be a k-uniform hypergraph with minimal spectral among all connected k-uniform hypergraphs of order n and clique number ω. If ω � 2, by Lemma 5, H must be a path, as the path is the hypergraph with smallest spectral radius among connected k-uniform hypergraphs of given order. us, we suppose that ω ≥ 3 and let H ′ be a complete k-uniform sub-hypergraph of H of order ω.
Further, H should be edge-minimal, that is, the removal of any edge of H either makes H disconnected or its clique number diminishes. In particular, if H ′′ is the hypergraph obtained by removing the edges of H ′ , then the components of H ′′ are supertrees, and each component has exactly one vertex in common with H ′ . It follows that H is isomorphic to a complete k-uniform hypergraph of order ω 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 S � v: d(v) ≥ 2 { } ⊆ V(H′), and u, v ∈ S. Suppose that a path P p � (v 1 � v, . . . , v p ) is attached to v and P q � (u 1 � u, . . . , u q ) is attached to u. Let H * be the hypergraph obtained by deleting the edge e u 2 u 1 and adding the edge e u 2 v p , that is, H * is obtained by removing P q and extending P p to P p+q− 1 . To complete the proof, we need to show that ρ(H) > ρ(H * ).