Ordering of Unicyclic Graphs with Perfect Matchings by Minimal Matching Energies

In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let i w be a non-isolated vertex of graph i G where 1,2, , i k =  . We use ( ) u G k (respectively, ( ) v H k ) to denote the graph which is the coalescence of G (respectively, H) and 1 2 , , , k G G G  by identifying the vertices u (respectively, v) and 1 2 , , , k w w w  . In this paper, we first present a new technique of directly comparing the matching energies of ( ) u G k and ( ) v H k , which can tackle some quasi-order incomparable problems. As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all 211 n ≥ .

which can tackle some quasi-order incomparable problems.As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all 211 n ≥ .

Introduction
Let G be a simple and undirected graph with n vertices and ( ) A G be its adja- cency matrix.Let 1 2 , , , n λ λ λ  be the eigenvalues of ( ) A G .Then the energy of G, denoted by ( ) E G , is defined as [1] ( ) .
A matching in a graph G is a set of pairwise nonadjacent edges.A matching is called k-matching if its size is k.Let ( ) , m G k be the number of k-matching of G, where ( ) In addition, we assume that ( ) The matching polynomial of a graph G is defined as Recently, Gutman and Wagner [31] generalized the concept of graph energy and defined the matching energy of a graph G based on the zeros of its matching polynomial.
Definition 1.1.Let G be a simple graph of order n and 1 2 , , , n µ µ µ  be the zeros of its matching polynomial.Then ( ) Further, Gutman and Wagner [31] pointed out that the matching energy is a quantity of relevance for chemical applications.They arrived at the simple relation: ( ) ( ) ( ),

TRE G E G ME G = −
where ( ) TRE G is the so-called topological resonance energy of G, in connection with the chemical applications of matching energy, for more details see [32] [33] [34].
Similar to the integral formula for the energy of graph, Gutman and Wagner [31] have shown a beautiful integral formula for the matching energy of a graph G as follows: ( ) ( ) . In the followings, the method of the quasi-order relation "  " is an important tool of comparing the matching energies of a pair of graphs.
Definition 1.2.Let 1 G and 2 G be two graphs of order n.If and there exists at least one index j such that ( ) ( ) According to the integral formula (1), we have for two graphs 1 G and 2 G of order n that ( ) ( ) [31], Gutman and Wagner shown that its matching energy coincides with its energy if T is a forest.Many properties of the matching energy are analogous to those of the graph energy.However, there are some notable differences.Then they raised a question: is it true that the matching energy of a graph G coincides with its energy if and only if G is a forest?Up to now, the question is still open.
The study on extremal matching energies is very interesting.In [31], Gutman and Wagner characterized the unicyclic graphs with the minimal and maximal matching energy.Zhu and Yang [35] determined the unicyclic graphs with the first eight minimal matching energies.In [36], Chen and Liu characterized the bipartite unicyclic graphs with the first ( )  largest matching energies.Moreover, Chen et al. [37] determined the unicyclic odd-cycle graphs with the second to the fourth maximal matching energies.For bicyclic graph, Ji et al. [38] obtained the graphs with the minimal and maximal matching energy.In [39], Liu et al. further determined the bicyclic graphs with first five minimal matching energies and the second maximal matching energies, respectively.Chen and Shi [40] characterized tricyclic graph with maximal matching energy, for more results about extremal matching energies, see [41]- [47].
A fundamental problem encountered within the study of the matching energy is the characterization of the graphs that belong to a given class of graphs having maximal or minimal matching energy.One of the graph classes that are quite interestingly studied is the class of all unicyclic graphs with perfect matchings.As far as we are concerned, no results are on this topic.In this paper, we first present a new technique of directly comparing the matching energies of ( ) u G k and ( ) v H k in Section 2 (see Figure 2).As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all 211 n ≥ in Section 3.
For simplicity, if 1 G is isomorphic to 2 G , then we write be the set of the unicyclic graphs with perfect matchings of order 2n.Let the unicyclic graphs 1 A , A be shown in Figure 1.The following theorem is the main result of this paper., , , , , , , , , , m G k cannot be compared uniformly, then the common comparing method is invalid, and this happens quite often.Recently much effort has been made to tackle these quasi-order incomparable problems [35] [39] [40].

A New
Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively.Let i w be a non-isolated vertex of graph i G where 1, 2, , i k =  .We use by identifying the vertices u (respectively, v) and 1 2 , , , k w w w  (see Figure 2).In [14], He et al.In this paper, we assume that By Equation ( 2), we can immediately obtain the following results.
Lemma 2.1.If two graphs G and H are disjoint, then The coalescence of two graphs G and H with respect to vertex u in G and vertex v in H, denoted by u v G H ⋅ (sometimes abbreviated as G H ⋅ ), is the graph obtained by identifying the vertices u and v. Zhu and Yang [35] shown the recurrence relation of ( ) in the following.For convenience of the reader, we present a full proof.
Lemma 2.3.( [35]) Let G H ⋅ be the coalescence of two graphs G and H with respect to vertex u in G and vertex v in H. Then Proof.Using Lemmas 2.1 and 2.2, we can show with the first to the ninth smallest matching energies.
For each graph i A , the dashed lines denote the copies of 3 P attached to the maximal degree vertex.) H k be defined as above (see Figure 2).Then we have the followings. ; .
We prove the result by induction on k.When which implies that the result holds.We assume that the result holds for 1 k − in what follows.For simplicity, we write ( ) ( ) ) ) Then we can see that the result holds.
2) The proof is similar to 1).
The following lemma illustrates an integral formula for the difference of the matching energies of two graphs with the same order which was obtained by Zhu and Yang [35].
Lemma 2.5.( [35]) Let ( ) be the matching polynomials of two graphs G and H with the same order, respectively.Then .
From Lemma 2.2, we have 0 k h > and l k h h < holds for any positive integer In what follows, we define two sets M and c M as follows: . Furthermore, we write Proof.By some calculations, we can obtain that ( .
Moreover, by Lemmas 2.4 and 2.5, we have Then the result can be obtained immediately.
Next, we use the new technique to compare the matching energies of the quasi-order incomparable graphs 5 A and 6 A , 8 A and 9 A (see Figure 1), re- spectively.Denote by k C and k P the cycle of length k and the path of length Proof.Let G be the graph obtained by attaching a pendent edge to a vertex u of 5 C .Let H be the graph obtained by attaching a pendent edge and a pendent path of length 2 to the vertices w and v of 3 C , respectively.Let  1).By some calculations, we can show This implies that M = ∅ and ( ) 2 and some calculations using the software MATLAB, we have It follows that M = ∅ and ( ) 2 and some calcu- lations using the software MATLAB, we have ME A ME A < .

Minimal Matching Energies of Unicyclic Graphs with Perfect Matchings of Order 2n
In this section, we will determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies (i.e., to prove Theorem 1.1).
In what follows, we denote by call Ĝ the capped graph of G and G the original graph of Ĝ .For example, the capped graphs of 1 2 3 5 , , , A A A A are shown in Figure 3. Let ( ) G be the number of ways to choose k independent edges in G such that just j edges are in Ĝ .We agree that ( ) ( ) Then we have where 2 . 1 This is the main method to compute ( ) v and 3 v , respectively.In [2] and [31], the following results were shown.
Lemma 3.1.( [2]) Let T be a tree of order , and the equalities do not hold for all k, where , , [31]) Suppose that G is a connected graph and T is an induced subgraph of G such that T is a tree and T is connected to the rest of G only by a cut vertex v.If T is replaced by a star of the same order, centered at v, then the quasi-order decreases (unless T is already such a star).Proof.Let G be a unicyclic graph with the unique cycle of length l.We consider the following cases.
Without loss of generality, we can assume that ( ) Without of generality, we can assume that ( ) T and 2 T be the rooted tree with the root 1 v and 2 v in G, respectively. If T P = , then by Lemma 3.9 we can show ( ) ( )

, , , G A A A A A A A A A A ≠
, then ( ) Proof.Using Equation ( 4) and some calculations, we can get to denote the graph which is the coalescence of G (respectively, H) and 1 2 , , , k G G G  by identifying the vertices u (respectively, v) and 1 2 , , , k w w w  .In this paper, we first present a new technique of directly comparing the matching energies of

presented a new method of directly comparing the energies of the
In this section, we apply the main idea of this me- thod to present a new technique of comparing the matching energies of the graphs can be used to tackle these quasi-order in- comparable problems.

Figure 1 .
Figure 1.The graphs in ( ) 2n with the first to the ninth smallest matching energies.

Figure 2 From
Figure 2. The graphs

(
1 k = , by Lemma 2.3 we have Open Journal of Discrete Mathematics

w
be the pendent vertex of i G .Then Open Journal of Discrete Mathematics

Figure 3 .X
Figure 3.The capped graphs of 1 2 3 , , A A A and 5A .For each graph, the dashed lines denote the copies of 2 P attached to the maximal degree vertex.

Let l n S 3 nRand 2 v of 3
be the unicyclic graph of order n obtained by attaching n l − pendent edges to one vertex of l C .Lemma 3.3.([43]) Let G be a unicyclic graph of order n with a cycle of length lbe the graph of order n obtained by attaching 4 n − and one pen- dent edges to 1 v c be the unicyclic graph obtained by attaching , , a b c pendent edges to 1 2 3 , , v v v of

3 nQ
be the graph of order n obtained by attaching 4 n − pendent edges to the pendent vertex of

and 3 nQ
are shown in Figure 4. Lemma 3.4.Let G be a unicyclic graph of order 9 n degree of the vertex u in G. Let 3

Figure 4 ., 3 n R and 3 nQ
Figure 4.The graphs 3 n S , 3 n R and 3 n Q in Lemma 3.4.For each graph, the dashed lines denote the copies of 2 P attached to the maximal degree vertex.

Technique of Directly Comparing the Matching Energies of
) We consider the following cases.Case 1: Ĝ is a connected graph.Subcase 1.1: Ĝ is a tree.It can easily be verified that Subcase 2.2: Ĝ is composed of trees and unicyclic graphs.Let 2 Ĝ be the coalescence of all trees and unicyclic graphs in a way such that Proof.