Extremal matching energy of complements of trees

# Extremal matching energy of complements of trees

Tingzeng Wu , Weigen Yan, Heping Zhang
School of Mathematics and Statistics, Qinghai Nationalities University,
Xining, Qinghai 810007, P. R. China
School of Sciences, Jimei University, Xiamen 361021, P. R. China
School of Mathematics and Statistics, Lanzhou University,
Lanzhou, Gansu 730000, P. R. China
Corresponding author.
E-mail address: mathtzwu@163.com (T. Wu), weigenyan@263.net (W. Yan), zhanghp@lzu.edu.cn (H. Zhang).
Partially supported by NSFC (11371180).
Partially supported by NSFC (11171134).

Abstract:  The matching energy is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph, which is proposed first by Gutman and Wagner [The matching energy of a graph, Discrete Appl. Math. 160 (2012) 2177–2187]. And they gave some properties and asymptotic results of the matching energy. In this paper, we characterize the trees with vertices whose complements have the maximal, second-maximal and minimal matching energy. Further, we determine the trees with a perfect matching whose complements have the second-maximal matching energy. In particular, show that the trees with edge-independence number number whose complements have the minimum matching energy for .

AMS classification: 05C05; 05C35; 92E10
Keywords: Matching polynomial; Matching energy; Hosoya index; Energy

## 1 Introduction

All graphs only considered in this paper are undirected simple graphs. For notation and terminologies not defined here, see [5]. Let be a graph with the vertex set and the edge set . Denote by or the graphs obtained from by removing or , respectively, where and . Denote by the complement of . The path, star and complete graph with vertices are denoted by , and , respectively. Let be a tree obtained from the star by attaching a path to one of the pendent vertices of , and let be a tree obtained from the star by attaching two paths and to two of the different pendent vertices of respectively. Let be a tree with vertices obtained from the star by attaching a pendent edge to each of pendent vertices in for .

A -matching in is a set of pairwise non-incident edges. The number of -matchings in is denoted by . Specifically, , and for or . For a -matching in , if has no -matching such that , then is called a maximum matching of . The number of edges in a maximum matching is called the edge-independence number of . Let denote the set of trees with vertices and the edge-independence number at least for . The Hosoya index is defined as the total number of matchings of , that is

 Z(G)=⌊n2⌋∑k=0m(G,k).

Recall that for a graph on vertices, the matching polynomial of (sometimes denoted by with no confusion) is given by

 μ(G,x)=∑k≥0(−1)km(G,k)xn−2k.

Its theory is well elaborated [2, 4, 5, 6, 7]. Gutman and Wagner [8] gave the definition of the quasi-order as follows. If and have the matching polynomials in the form (1), then the quasi-order is defined by

 G⪰H⟺m(G,k)≥m(H,k)for all  k=0,1,...,⌊n/2⌋.

Particularly, if and there exists some such that , then we write .
Gutman and Wagner in [8] first proposed the concept of the matching energy of a graph, denoted by , as

 ME=ME(G)=2π∫∞0x−2ln[∑k≥0m(G,k)x2k]dx.

Meanwhile, they gave also an other form of definition of matching energy of a graph. That is,

 ME(G)=n∑i=1|μi|,

where denotes the root of matching polynomial of . Additionally, they found some relations between the matching energy and energy (or reference energy). By (2) and (3), we easily obtain the fact as follows.

 G⪰H⟹ME(G)≥ME(H)andG≻H⟹ME(G)>ME(H).

This property is an important technique to determine extremal graphs with the matching energy.
Note that the energy (or reference energy) of graphs are extensively examined (see [1, 2, 3, 9, 10, 14]). However, the literatures on the matching energy are far less than that on the energy and reference energy. Up to now, we find only few papers about the matching energy published. Gutman and Wagner [8] gave some properties and asymptotic results of the matching energy. Li and Yan [13] characterized the connected graph with the fixed connectivity (resp. chromatic number) which has the maximum matching energy. Ji et al. in [11] determined completely the graphs with the minimal and maximal matching energies in bicyclic graphs. Li et al. [12] characterized the unicyclic graphs with fixed girth (resp. clique number) which has the maximum and minimum matching energy.
In this paper, inspired by the idea [16], we investigate the problem of the matching energy of the complement of trees and obtain the following main theorems.

###### Theorem 1.1.

Let be a tree with vertices. If and , then

 ME(¯¯¯¯T)
###### Theorem 1.2.

Let denote the set of trees with vertices and the edge-independence number at least for . For a tree , then

 ME(¯¯¯¯T)≥ME(¯¯¯¯¯¯¯Tpn)

with equality if and only if .

By Theorems 1.1 and 1.2, we obtain directly the following corollary.

###### Corollary 1.3.

The components of and have the maximum and minimum matching energy in all components of trees, respectively.

###### Theorem 1.4.

Let be a proper subset of containing all trees with a perfect matching. Suppose that , and . If , then

 ME(¯¯¯¯¯¯¯¯Tn2n)

where the equality holds if and only if .

## 2 Some Lemmas

There exists a well-known formula which characterizes the relation between and (see Lovász [15]), which will play a key role in the proofs of the main theorems.

###### Lemma 2.1.

([15]) Let be a simple graph with vertices and the complement of . Then

 m(G,r)=r∑i=0(−1)i(n−2i2r−2i)(2r−2i−1)!!m(¯¯¯¯G,i),

where , and .

The following results about the matching polynomial of can be found in Godsil [5].

###### Lemma 2.2.

([5]) The matching polynomial satisfies the following identities:

if is an edge of ,
if .

###### Lemma 2.3.

([5]) Let and be two positive integers. Then

 μ(Pm+n)=μ(Pm)μ(Pn)−μ(Pm−1)μ(Pn−1).
###### Lemma 2.4.

([16]) If is a tree with vertices and edge-independence number , then has at most vertices of degree one. In particular, if T has exactly vertices of degree one, then every vertex of degree at least two in is adjacent to at least one vertex of degree one.

## 3 Ordering complements of trees with respect to their matchings

For convenience, we use the same definitions of trees which are defined in [16].

###### Definition 3.1.

Let be a tree with vertices shown in Figure 1, where is a tree with vertices () and a vertex of , and . Suppose is a tree with vertices obtained from by attaching a path to u in (see Figure 1). We designate the transformation from to as of type 1 and denote it by : or .

###### Theorem 3.1.

Let and be the trees with vertices defined in Definition 3.1. Then .

###### Proof.

By Lemma 2.2,

 μ(T1)=xμ(T0−u)μ(Pm)μ(Pn)−μ(T0−u)μ(Pm−1)μ(Pn)−μ(T0−u)μ(Pm)μ(Pn−1)−∑v∈V(T0)uv∈E(T0)μ(T0−u−v)μ(Pm)μ(Pn),μ(T2)=xμ(T0−u)μ(Pm+n)−μ(T0−u)μ(Pm+n−1)−∑v∈V(T0)uv∈E(T0)μ(T0−u−v)μ(Pm+n),

where the above sums range over all vertices of adjacent to . Hence

 μ(T1)−μ(T2)=xμ(T0−u)[μ(Pm)μ(Pn)−μ(Pm+n)]−μ(T0−u)[μ(Pm−1)μ(Pn)−μ(Pm+n−1)+μ(Pm)μ(Pn−1)]−[μ(Pm)μ(Pn)−μ(Pm+n)]∑v∈V(T0)uv∈E(T0)μ(T0−u−v)

By (6) and a routine calculation,

 μ(T1)−μ(T2)=−∑v∈V(T0)uv∈E(T0)μ(T0−u−v)μ(Pm−1)μ(Pn−1).

For an arbitrary vertex adjacent to in , let be the forest , which has vertices. By (5), we obtain

 m(¯¯¯¯¯T1,r)−m(¯¯¯¯¯T2,r)=r∑i=0(−1)i(n+m+k−2i2r−2i)(2r−2i−1)!![m(T1,i)−m(T2,i)].

Note that and . Hence

 m(¯¯¯¯¯T1,r)−m(¯¯¯¯¯T2,r)=−∑v∈V(T0)uv∈E(T0)r∑i=2(−1)i(n+m+k−2i2r−2i)(2r−2i−1)!!m(T∗v,i−2).

Note that has vertices. So

 m(¯¯¯¯¯¯¯T∗v,r−2)=r−2∑j=0(−1)j(n+m+k−4−2j2(r−2)−2j)(2(r−2)−2j−1)!!m(T∗v,j)
 =r∑i=2(−1)i(n+m+k−2i2r−2i)(2r−2i−1)!!m(T∗v,i−2).

Hence

 m(¯¯¯¯¯T1,r)−m(¯¯¯¯¯T2,r)=−∑v∈V(T0)uv∈E(T0)m(¯¯¯¯¯¯¯T∗v,r−2).

By the definition of and (10), we have , which implies . Particularly, if , then . By (2), . ∎

###### Remark 1.

By Theorem 3.1 and (4), we obtain immediately a result as follows: If and are the two trees defined in Definition 3.1, then . Additionally, by the definition of the Hosoya index and Theorem 3.1, it is not difficult to see that .

###### Definition 3.2.

Let and be two trees with vertices shown in Fig. 2, where . We designate the transformation from to in Figure 2 as of type 2 and denote it by : or .

###### Theorem 3.2.

Let and be two trees with vertices defined in Definition 3.2. Then .

###### Proof.

Similar to the proof of Theorem 3.1, we can obtain that

 μ(T3)−μ(T4)=−μ(Pm−2)μ(Pn−1)μ(Ps−2).

Furthermore, we also have

 m(¯¯¯¯¯T3,r)−m(¯¯¯¯¯T4,r)=−m(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Pm−2∪Pn−1∪Ps−2,r−3).

By the definition of and (11), we have , which implies . Specially, if then . This means, by (2), that . The proof is complete. ∎

###### Definition 3.3.

Let and be two trees with vertices shown in Fig. 3, where . We designate the transformation from to in Figure 3 as of type 3 and denote it by : or .

###### Theorem 3.3.

Let and be two trees with vertices defined in Definition 3.3. Then .

###### Proof.

Similar to the proof of Theorem 3.1, we have

 μ(T5)−μ(T6)=−μ(Pm−n)

and

 m(¯¯¯¯¯T5,r)−m(¯¯¯¯¯T6,r)=−m(¯¯¯¯¯¯¯¯¯¯¯¯Pm−n,r−n−1).

By the definition of and (12), we have , which indicates . Specially, when , then . By (2), we get that . ∎

###### Definition 3.4.

Suppose that and are two trees with () vertices and with () vertices, respectively. Take one vertex of and one of . Construct two trees and with vertices as follows. The vertex set of is and the edge set of is . is the tree obtained from and by identifying the vertex of and the vertex of and adding a pendent edge to this new vertex (). The result graphs see Fig. 4. We designate the transformation from to as of type 4 and denote it by : or .

###### Theorem 3.4.

Let and be two trees with vertices defined in Definition 3.4. Then .

###### Proof.

By Lemma 2.2,

 μ(T7)=μ(T′1)μ(T′2)−μ(T′1−u)μ(T′2−v),
 μ(T8)=xμ(T8−w)−μ(T′1−u)μ(T′2−v),
 μ(T′1)=xμ(T′1−u)−s∑i=1μ(T′1−u−ui)

and

 μ(T′2)=xμ(T′2−v)−s∑j=1μ(T′2−v−vj).

where the first sum ranges over all vertices () of adjacent to and the second sum ranges over all () of adjacent to . By (15) and (16), we have

 xμ(T8−w)=x2μ(T′1−u)μ(T′2−v)−xt∑j=1μ(T′1−u)μ(T′2−v−vj)−xs∑i=1μ(T′2−v)μ(T′1−u−ui)(17)

and

 μ(T′1)μ(T′2)=x2μ(T′1−u)μ(T′2−v)−xt∑j=1μ(T′1−u)μ(T′2−v−vj)−xs∑i=1μ(T′2−v)μ(T′1−u−ui)+∑1≤i≤s1≤j≤tμ(T′1−u−ui)μ(T′2−v−vj).  (18)

Combining (13), (14), (17) and (18),

 μ(T7)−μ(T8)=∑1≤i≤s1≤j≤tμ(T′1−u−ui)μ(T′2−v−vj).

As in the proof of Theorem 3.1, we can show that

 m(¯¯¯¯¯T7,r)−m(¯¯¯¯¯T8,r)=∑1≤i≤s1≤j≤tm(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯μ(T′1−u−ui)∪μ(T′2−v−vj),r−2),

which implies that

 m(¯¯¯¯¯T7,r)≥m(¯¯¯¯¯T8,r).

Note that when . So, by (2), the theorem holds. ∎

###### Remark 2.

For the trees Fig.5, we note that neither tree nor tree can be transformed into by a single transformation 4. Hence if in Theorem 3.4 is , then . Particularly, for . Similarly, it is easy to show that the statement holds.

###### Definition 3.5.

Suppose that is a tree with vertices and with the edge-independence number shown in Fig. 6 which has exactly pendent vertices, where and . Let be the tree with vertices shown in Fig. 6, which is obtained from . We designate the transformation from to as of type 5 and denote it by : or .

###### Theorem 3.5.

Let and be two trees with vertices defined in Definition 3.5. Then .

###### Proof.

By Lemma 2.2,

 μ(T9)=xμ(T0−u)μ(K1,r)−μ(K1,r)∑v∈V(T0)uv∈E(T0)μ(T0−u−v)−μ(T0−u)μ(P1)r=x2μ(T0−u)μ(P1)r−rxμ(T0−u)μ(P1)r−1−xμ(P1)r∑v∈V(T0)uv∈E(T0)μ(T0−u−v)+rμ(P1)r−1∑v∈V(T0)uv∈E(T0)μ(T0−u−v)−μ(T0−u)μ(P1)r   (20)

and

 μ(T10)=xμ(T0−u)μ(P2)μ(P1)r−1−μ(P2)μ(P1)r−1∑v∈V(T0)uv∈E(T0)μ(T0−u−v)−μ(T0−u)μ(P1)r−(r−1)μ(T0−u)μ(P2)μ(P1)r−2=x2μ(T0−u)μ(P1)r−xμ(T0−u)μ(P1)r−1−xμ(P1)r∑v∈V(T0)uv∈E(T0)μ(T0−u−v)+μ(P1)r−1∑v∈V(T0)uv∈E(T0)μ(T0−u−v)−μ(T0−u)μ(P1)r−(r−1)xμ(T0−u)μ(P1)r−1+(r−1)μ(T0−u)μ(P1)r−2,     (21)

where the sum ranges over all vertices of incident with .
By (20) and (21), we have

 μ(T9)−μ(T10)=−(r−1)μ(T0−u)μ(P1)r−2+(r−1)μ(P1)r−1∑v∈V(T0)uv∈E(T0)μ(T0−u−v).

By Lemma 2.4, there exists at least one pendent vertex in joining vertex of . Hence, , which implies that

 μ(T9)−μ(T10)=(r−1)∑v∈V(T0),v≠v′uv∈E(T0)μ(P1)r−1μ(T0−u−v).

Similar to the proof of Theorem 3.1,

 m(¯¯¯¯¯T9,k)−m(¯¯¯¯¯¯¯T10,k)=(r−1)∑v∈V(T0),v≠v′uv∈E(T0)m(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(r−1)P1∪(T0−u−v),k−2),

where for every vertex () of incident with . Hence . Furthermore, if , then . So . ∎

###### Definition 3.6.

Suppose that is a tree with vertices and with the edge-independence number shown in Figure 7, which has exactly pendent vertices, where , and . Let be the tree with vertices shown in Figure 7, which is obtained from . We designate the transformation from to as of type 6 and denote it by : or .

###### Theorem 3.6.

Let and be two trees with vertices defined in Definition 3.6. Then .

###### Proof.

Suppose . By Lemma 2.2,

 μ(T11)=[x2μ(P1)μ(P2)−sxμ(P2)−txμ(P1)2−μ(P1)μ(P2)]μ(P1)s−1μ(P2)t−1μ(T0−u)−[xμ(P1)μ(P2)−sμ(P2)−tμ(P1)2]μ(P1)s−1μ(P2)t−1∑v∈V(T0)uv∈E(T0)μ(T0−u−v)

and

 μ(T12)=[x2μ(P1)2μ(P2)−xμ(P1)μ(P2)−sxμ(P1)μ(P2)+sμ(P2)−μ(P2)+xμ(P1)μ(P2)−(t+1)μ(P1)2μ(P2)]μ(P1)s−2μ(P2)t−1μ(T0−u)−[xμ(P1)μ(P2)−μ(P2)]μ(P1)s−1μ(P2)t−1∑v∈V(T0)uv∈E(T0)μ(T0−u−v),

where the sum ranges over every vertex of adjacent to .
Combining the above two equations, we obtain that

 μ(T11)−μ(T12)=−[(s+t−1)x2−(s−1)]μ(P1)s−2μ(P2)t−1μ(T0−u)+[(s+t−1)x2−(s−1)]μ(P1)s−1μ(P2)t−1∑v∈V(T0)uv∈E(T0)μ(T0−u−v).

By Lemma 2.4, there exists at least one pendent vertex of adjacent to . Hence . Thus, simplifying the above equation, we have

 μ(T11)−μ(T12)=[(s+t−1)μ(P1)s−1μ(P2)t+tμ(P1)s−1μ(P2)t−1]∑