The matching polynomials and spectral radii of uniform supertrees

We study matching polynomials of uniform hypergraph and spectral radii of uniform supertrees. By comparing the matching polynomials of supertrees, we extend Li and Feng's results on grafting operations on graphs to supertrees. Using the methods of grafting operations on supertrees and comparing matching polynomials of supertrees, we determine the first $\lfloor\frac{d}{2}\rfloor+1$ largest spectral radii of $r$-uniform supertrees with size $m$ and diameter $d$. In addition, the first two smallest spectral radii of supertrees with size $m$ are determined.


Introduction
The ordering of graphs by spectral radius was proposed by Collatz and Sinogowitz [7] in 1957. Lovász and Pelikán [21] investigated the spectral radius of trees and determined the first two largest and smallest spectral radii of trees with given order. Brualdi and Solheid [2] proposed the problem of bounding the spectral radius of some class of graphs and characterizing the corresponding extremal graphs. Since then, many authors studied the spectral radius of trees with some given parameters, such as degree, diameter, etc.
A hypergraph H is a pair (V, E), where E ⊆ P(V ) and P(V ) stands for the power set of V .
The elements of V = V (H) are referred to as vertices and the elements of E = E(H) are called hyperedges or edges. A hypergraph H is r-uniform if every edge e ∈ E(H) contains precisely r vertices. For a vertex v ∈ V , we denote by E v the set of edges containing v. The cardinality |E v | is the degree of v, denoted by deg (v). A vertex with degree one is called a core vertex, and a vertex with degree larger than one is called an intersection vertex. If any two edges in H share at most one vertex, then H is said to be a linear hypergraph. In this paper we assume that hypergraphs are linear and r-uniform.
In a hypergraph H, two vertices u and v are adjacent if there is an edge e of H such that {u, v} ⊆ e. A vertex v is said to be incident to an edge e if v ∈ e. A walk of hypergraph H is defined to be an alternating sequence of vertices and edges v 1 e 1 v 2 e 2 · · · v e v +1 satisfying that both In [16] some transformations on hypergraphs such as moving edges and edge-releasing were introduced and the first two spectral radii of supertrees on n vertices were characterized. Yuan et.
al [32] further determined the first eight uniform supertrees on n vertices with the largest spectral radii. Xiao et. al [27] characterized the unique uniform supertree with the maximum spectral radius among all uniform supertrees with a given degree sequence. Recently, the first two largest spectral radii of uniform supertrees with given diameter were characterized in [28].
In this paper, we determine the first d 2 + 1 largest spectral radii of supertrees among all runiform supertrees with size m and diameter d and the first two smallest spectral radii of supertrees with size m. The structure of the remaining part of the paper is as follows: In Section 2, we give some basic definitions and results for tensor and spectra of hypergraphs. Section 3 extends the theory of matching polynomial from graphs to supertrees. By comparing the matching polynomial of supertrees, we generalize Li and Feng's results on grafting operations on graphs to supertrees in Section 4. By using the method of grafting operations on supertrees and comparing matching polynomial of supertrees, we determine the first d 2 + 1 spectral radii of supertrees among all runiform supertrees with size m and diameter d in Section 5. In Section 6, the first two smallest spectral radii of supertrees are determined. We give closing remarks in the last section.

Preliminaries
Let H = (V, E) be an r-uniform hypergraph on n vertices. A partial hypergraph H = (V , E ) of H is a hypergraph with V ⊆ V and E ⊆ E. A proper partial hypergraph H of H is partial hypergraph Let G = (V, E) be an ordinary graph. For every r ≥ 3, the rth power of G, denoted by G r , is an r-uniform hypergraph with vertex set V (G r ) = V ∪ (∪ e∈E {i e,1 , . . . , i e,r−2 }) and edge set E(G r ) = {e ∪ {i e,1 , . . . , i e,r−2 , }| e ∈ E}. The rth power of an ordinary tree is called a hypertree (see [14]). Note that all hypertrees are supertrees by the definition. Let P m and S m denote the path and the star with m edges, respectively. The rth power of P m and S m , denoted by P r m and S r m , are called loose path and hyperstar, respectively.
Let H = (V, E) be an r-uniform hypergraph. An edge e is called a pendent edge if e contains exactly r − 1 core vertices. If e is not a pendent edge, it is called a non-pendent edge. A path P = (v 0 , e 1 , v 1 , . . . , v p−1 , e p , v p ) of H is called a pendent path (attached at v 0 ), if all of the vertices v 1 , . . . , v p−1 are of degree two, the vertex v p and all the r − 2 vertices in the set e i \ {v i−1 , v i } are core vertices in H (i = 1, . . . , p).
For positive integers r and n, a real tensor A = (a i 1 i 2 ···ir ) of order r and dimension n refers to a multidimensional array (also called hypermatrix) with entries a i 1 i 2 ···ir such that a i 1 i 2 ···ir ∈ R for all The following product of tensors, defined by Shao [26], is a generalization of the matrix product.
Let A and B be dimension n, order r 2 and order k 1 tensors, respectively. Define the product AB to be the tensor C of dimension n and order (r − 1)(k − 1) + 1 with entries as where i ∈ [n], α 1 , . . . , α r−1 ∈ [n] k−1 .
From the above definition, if x = (x 1 , x 2 , . . . , x n ) T ∈ C n is a complex column vector of dimension n, then by (1) Ax is a vector in C n whose ith component is given by In 2005, Qi [24] and Lim [18] independently introduced the concepts of tensor eigenvalues and the spectra of tensors.
Let A be an order r dimension n tensor, x = (x 1 , x 2 , . . . , x n ) T ∈ C n a column vector of dimension n. If there exists a number λ ∈ C and a nonzero vector x ∈ C n such that , then λ is called an eigenvalue of A, x is called an eigenvector of A corresponding to the eigenvalue λ. The spectral radius of A is the maximum modulus of the eigenvalues of A.
be an r-uniform hypergraph on n vertices. The adjacency tensor of H is defined as the order r and dimension n tensor A(H) = (a i 1 i 2 ···ir ), whose (i 1 i 2 · · · i r )entry is The spectral radius of hypergraph H is defined as spectral radius of its adjacency tensor, denoted by ρ(H). In [10] the weak irreducibility of nonnegative tensors was defined. It was proved that an r-uniform hypergraph H is connected if and only if its adjacency tensor A(H) is weakly irreducible (see [10] and [31]). Part of the Perron-Frobenius theorem for nonnegative tensors is stated in the following for reference.  The following result can be obtained directly from Theorem 2.3 and will be often used in the sequel.
Theorem 2.4. Suppose that G is a uniform hypergraph, and G is a partial hypergraph of G. Then ρ(G ) ≤ ρ(G). Furthermore, if in addition G is connected and G is a proper partial hypergraph, we have ρ(G ) < ρ(G).
An operation of moving edges on hypergraphs was introduced by Li et. al in [16]. Let H = (V, E) be a hypergraph with u ∈ V and e 1 , . . . , e k ∈ E, such that u / ∈ e i for i = 1, . . . , k. Suppose that The following edge-releasing operation on linear hypergraphs was given in [16].
Let H be an r-uniform linear hypergraph, e be a non-pendent edge of H and u ∈ e. Let The following result was obtained by Zhou et.al [34], we will use it in the sequel. Recently, Zhang et. al [33] obtained the following result.
Based on the result above, Clark and Cooper [6] called the polynomial in Theorem 3.1 as matching polynomial of H. Set m(H, 0) = 1. We redefine the matching polynomial of H as For exmaple, the matching polynomial of N k is ϕ(N k , x) = x k , rather than 1 by Zhang's definition. The definition here seems more appropriate as it guarantees that matching polynomials of hypergraphs of the same order have the same degree and the result in Theorem 3.1 is still valid.
Some classical results on matching polynomial of a graph can be extended to a hypergraph as well. However, the matching polynomial of a hypergraph has its own flavour, e.g. as shown in [6], the roots of matching polynomial of an r-uniform hypergraph with r > 2 need not necessarily be real.
Theorem 3.2. Let G and H be two r-uniform hypergraphs. Then the following statements hold.
Proof. (a) From the fact that each k-matching in G ∪ H consists of an s-matching in G combined with a (k − s)-matching from H for some s, the result follows immediately. By comparing the coefficients of the corresponding matching polynomial in two sides of (b), the result follows.
Repeatedly using (b) of Theorem 3.2, we get Note that u is an isolated vertex of G − ∪ i∈I e i , it follows directly from (2) that Counting the number of the ordered pairs, we obtain that the number of such ordered pairs is equal to m(G, k)(n − rk), which is just the absolute value of the coefficient of On the other hand, if we choose a vertex first, say u, then the number of kmatching not covering u is equal to m(G − u, k). Then, the number of such ordered pairs is equal to The desired result follows. Proposition 3.3. Let T be an ordinary tree on n vertices, r (r ≥ 3) a positive integer. Then the matching polynomials of T and its rth power T r satisfy the following relation: Proof. It is easy to see that m(T, k) = m(T r , k) for any k. Let n denote the order of T r . Then n = n + (n − 1)(r − 2). So we have where a new variable y = x r 2 is used in the second and third equations.
The ordering on forests has been introduced by Lovász and Pelikán in [21]. Now we extend the ordering on forests to superforests. Let T and T be superforests of n vertices. We call T T if does not vanish at the point x = ρ(T ). Note that T ≺ T (T T , resp.) implies ρ(T ) < ρ(T ) (ρ(T ) ≤ ρ(T ), resp.).

Grafting transformations on uniform supertrees
Li and Feng [17] investigated how the spectral radius change when a certain transformation is applied to the graph, and obtained the following result.
the graph obtained from G by attaching a path of length p at u and a path of length q at v. Then ρ(G(u, v; p, q)) > ρ(G(u, v; p + 1, q − 1)) under any of the following conditions Since then, the result has been extensively used in spectral perturbation and proved to be efficient in ordering graphs by spectral radius. The result above is proved by comparing characteristic polynomials of graphs. The characteristic polynomial of a hypergraph is complicated and very little is known about it up to now. However the result of Theorem 3.1 makes it feasible to compare the spectral radii of supertrees by using the matching polynomials of supertrees.
It is known that for any forest, its matching polynomial and characteristic polynomial coincide.
Following a similar proof of Lemma 4 in [21], the following result can be obtained.
Based on Propositions 3.3 and 4.2, the corresponding result for hypertree can be easily obtained. Suppose that T is an r-uniform supertree and v is a vertex in T . Let T (v; p, q) be obtained by attaching two pendent paths of length p and q at v (see Fig. 1(a)).  .
Proof. We first consider the case that p ≥ q = 1. Applying (b) of Theorem 3.2 on T (v; p, 1) and the pendent edge attached at v, we have Similarly, applying (b) of Theorem 3.2 on T (v; p + 1, 0) and the pendent edge of the pendent path of length p + 1 attached at v, we have By (3) and (4), we deduce that Note that (T − v) ∪ P r p−1 is a proper partial hypergraph of T (v; p − 1, 0). By Theorems 2.4 and 4.4, the desired result follows.
When p ≥ q ≥ 2, applying (b) of Theorem 3.2 on T (v; p, q) and the pendent edge of the pendent path of length q attached at v, we have Similarly, By (5) and (6), we deduce that Continue this process, we get Applying Theorem 3.2 once more, we have and Substituting (8) and (9) into (7), we obtain Note that (T − v) ∪ P r p−q is a proper partial hypergraph of T (v; p − q, 0). Applying Theorems 2.4 and 4.4, we get the desired result.
Suppose that T is an r-uniform supertree (with at least two edges) and u and v are two vertices incident with an edge e in T . Let T (1) (u, v; p, q) (see Fig. 1(b)) be obtained by attaching two pendent paths of length p and q at u and v, respectively. Theorem 4.6. If p ≥ q ≥ 1, then In particularly, Proof. Using the similar argument as in the proof of Theorem 4.5, we have Let H 1 and H 2 be the components of T \ e containing vertex u and v respectively, and H be the union of the remaining components. We denote H as the partial hypergraph of H obtained from H by removing r − 2 vertices contained in e.
We may assume that E(H) ∪ E(H 2 ) is not empty. Otherwise, T (1) (u, v; p, q) is isomorphic to H 1 (u; p, q + 1). The result follows from Theorem 4.5.
When p = q ≥ 1, applying (b) of Theorem 3.2 to T (u; 0, 0) and edge e, we have Similarly, applying (b) of Theorem 3.2 to (T − v)(u; 1, 0) and the pendent edge attached at u, Substituting (11) and (12) into (10), we obtain When p > q ≥ 1, applying (b) of Theorem 3.2 to T (u; p − q, 0) and the edge e, we have Similarly, applying (b) of Theorem 3.2 to (T −v)(u; p−q +1, 0) and the pendent edge of the pendent path of length p − q + 1 attached at u, we have Substituting (14) and (15) into (10) yields We consider the following two cases depending on whether or not E(H 1 ) ∪ E(H 2 ) is empty.
Without loss of generality, we assume that E(H 1 ) = ∅. It is easily seen that (H 1 − u) ∪ P r p−q−1 is a proper partial hypergraph of H 1 (u; p − q − 1, 0). By Theorems 2.4, 4.4 and (16), we prove the desired result.   In particularly, Proof. We proceed by induction on s. For the case s = 1, the assertion holds by Theorem 4.6. Let T u and T v denote the components of T \ e s containing u and v, respectively. Using the similar argument as in the proof of Theorem 4.6, we have where the last equality follows from ( Applying (c) of Theorem 3.2 to T (u; p − q, 0) and the edges incident to v in T v , we have Substituting (18) into (17), we obtain By induction hypothesis, T  Then T is a uniform supertree and T ≺ T .
As an application of Theorems 4.5 and 4.6, the minimal supertree can be characterized as follows.
Note that the upper bound and the extremal supertree have been obtained in [16], and they are listed here for completeness.  Then where the left-hand side equality holds if and only if T ∼ = P r m with v as its end vertex whereas the right-hand side equality holds if and only if T ∼ = S r m with v as its center .

Extremal supertrees with given diameter
Let S(m, d, r) be the set of r-uniform supertrees with m edges and diameter d. Xiao et. al [28] determined the first two largest spectral radii of supertress in S(m, d, r). In this section, we determine the first d 2 + 1 largest spectral radii of supertrees in S(m, d, r) by using edge-grafting operations and comparing matching polynomials of supertrees.
Let H be an r-uniform hypergraph and u a vertex of H. Let P r d = (v 1 , e 1 , v 2 , e 2 , . . . , e d , v d+1 ) be a loose path of length d. Denote by P r d (v i , u)H and P r d (e j , u)H the hypergraphs obtained by identifying vertex u of H with vertex v i of P r d and a core vertex of P r d in e j respectively (see Fig. 4). As an immediate application of Theorems 4.5 and 4.6, we have the following result.
Theorem 5.1. Let T be an r-uniform supertree, r ≥ 3. Then Proof. Note that P r d (v i , u)T and P r d (e i , u)T can be depicted as T (u; i−1, d−i+1) and (T ) (1) (v i , v i+1 ; i− 1, d − i) respectively, where T denotes the supertree consists of T and e i . The first two assertions follow directly from Theorem 4.5 and Theorem 4.6 respectively.
Let H 1 and H 2 denote the two supertrees obtained from P r d (e i , u)T by moving all edges in E u ∩ E(T ) from u to v i and moving the edge e i−1 from v i to u, respectively. By Lemma 2.5, we have ρ(P r d (e i , u)T ) < max{ρ(H 1 ), ρ(H 2 )}. However, H 1 ∼ = H 2 ∼ = P r d (v i , u)T and assertion (c) holds. Using the similar approach, we can show the last assertion holds.
In fact, the last two assertions in Theorem 5.1 can be generalized as follows.
Theorem 5.2. Let T be an r-uniform supertree and P r d be a loose path of length d, with d ≥ 3 and r ≥ 3. Then for any 2 ≤ i ≤ d, we have Proof. Suppose that e 1 , e 2 , . . . , e s are all edges incident with vertex u in T . Applying (c) of Theorem 3.2 to P r d (e d 2 , u)T and edges e 1 , e 2 , . . . , e s , we have Similarly, Then It is easy to see that P r i−2 ∪N r−1 ∪P r d−i ≺ P r i−1 ∪P r d−i as P r i−2 ∪N r−1 is a proper partial hypergraph of P For convenience, we adopt the notation from [11]. The following results were obtained in [11] and we shall extend these results from trees to supertrees in this section.
Denote by H 1 and H 2 the supertrees obtained from T by moving all edges in E w 1 ∩ E(T 1 ) from w 1 to v i and v i+1 , respectively. By Theorem 5.1, ρ(T ) < min{ρ(H 1 ), ρ(H 2 )}. The maximality of ρ(T ) implies that H 1 , H 2 ∈ {T r (m,d) ∪ T } and one of them is T . So ρ(T ) < ρ(T ). The proof is finished.
By Theorems 2.6, 5.4 and Lemma 5.6, we have the following results.

The second minimal supertree
Let P r m−1 = (v 1 , e 1 , v 2 , e 2 , . . . , e m−1 , v m ) be a loose path of length m − 1. Denote by D m,r the supertree obtained from P r m−1 by attaching a pendent edge at a core vertex of e 2 (see Fig. 6(a)). LetP r m be the supertree obtained from P r m−1 by attaching a pendent edge at the vertex v 2 (see Fig. 6(b)). We use S(m, r) to denote the set of r-uniform supertrees with m edges. Proof. Choose a supertree T 0 from S(m, r) \ {P r m } such that T 0 T for any T ∈ S(m, r) \ {P r m }. Then T 0 either has a vertex of degree more than two or has an edge with at least three intersection vertices. We consider the two cases as follows.
Case 1. There exists a vertex of degree greater than two, say v ∈ V (T 0 ) with deg(v) ≥ 3. Thus T 0 can be described as a supertree in the form of some supertrees, say T 1 , . . . , T s (s ≥ 3), attached at a single vertex v. Denoted T 0 by T 1 (v)T 2 (v) · · · (v)T s (see Fig. 7(a)). Assume that T i has m i edges for i = 1, . . . , s. Let m = m − (m 1 + m 2 ). By Theorems 4.10 and 4.5, we have T 1 (v)T 2 (v) · · · (v)T s P r m 1 (v)P r m 2 (v) · · · (v)P r ms P r m 1 (v)P t m 2 (v)P r m P r 1 (v)P r 1 (v)P r m−2 , where all loose paths P r m , P r m−2 and P r m j (j = 1, . . . , s) have v as its end vertex. By the minimality of T 0 , T 0 = P r 1 (v)P r 1 (v)P r m−2 =P r m . vertices in e (if there are) are core vertices (see Fig. 7(b)). Then T 0 may be viewed as obtained by attaching supertrees, say T 1 , . . . , T s , at v 1 , . . . , v s respectively.
By Theorems 4.6, 4.10 and the minimality of T 0 , the following conclusions hold.
(3) Two of T 1 , T 2 , T 3 are of length one.
Therefore, T = D m,r .
Combining two cases above, we have shown that T 0 ∈ {P r m , D m,r }. Further by (c) of Theorem 5.1, we haveP r m D m,r . So T 0 = D m,r . Thus we conclude that for any T ∈ S(m, r) \ {P r m }, T D m,r .
Theorem 6.2. The first two smallest spectral radii of supertrees with m (m ≥ 4) edges are P r m , D m,r .

Closing remarks
We conclude this section with some remarks on matching polynomial of a supertree. The work in this paper is based on the relation between the roots of matching polynomial of a supertree and its spectrum developed in [33]. Using the recurrence relations of matching polynomial of supertrees, the effect on the spectral perturbation of supertree by grafting edges in various situations can be explained. The methods are initially used to compare spectral radii of supertrees in this paper. The methods are shown to be efficient in dealing with extremal supertrees with respect to their spectral radii, such as in finding the first two smallest supertrees and the first several largest supertrees with given diameter.
For the corresponding problem on a hypergraph, the characteristic polynomial of adjacency tensor of a hypergraph might be used to compare spectral radii of hypergraphs. However, the degree of characteristic polynomial of a hypergraph is very high relative to its order, and very little is known about it up to now. Finally, we pose the following problem.
Problem 7.1. What kind of polynomial should be associated with a hypergraph satisfying the following conditions: (1) The roots of the associated polynomial consist of the eigenvalues, especially the spectral radius of the hypergraph.
(2) The coefficients of the polynomial reflect certain structural information of the hypergraph, such as matching, cyclic structure or something more complicated.