The Quasi-Order of Matching Energy of Circum Graph with Chord

The matching energy of graph G is defined as ( ) 1 n i i ME G λ = = ∑ , where 1 2 , , , n λ λ λ  be the roots of matching polynomial of graph G. In order to compare the energies of a pair of graphs, Gutman and Wager further put forward the concept of quasi-order relation. In this paper, we determine the quasiorder relation on the matching energy for circum graph with one chord.


Introduction
All graphs considered are finite, undirected, loopless and without multiple edges.
The terminology and nomenclature of [1] will be used.Throughout this paper, G will denote a graph with vertex set ( ) { } , m G k be the number of k-matchings in graph G.The matching polynomial of a graph G is defined in [2] as where ( )

( )
ME G of a graph G in [3] and defined in different expressions as follows: (3) and ( ) where 1 2 , , , n λ λ λ  be the roots of matching polynomial of graph G.
The matching energy

( )
ME G of a graph G is an important index, which is widely used in the field of molecular orbital theory.There are many literatures about this parameter.See [4]- [14].
By the above definitions, it is immediately to get In fact, this property provide an important technique to determine the order relation of the matching energy for graphs.In this paper, we discuss the order of the matching energy for circum graph with chord. Let

Preliminaries
Lemma 1. [2] Let e uv = be an edge of graph G and By Lemma 1, it is easy to get Lemma 2. Let e be an edge of graph G. Then , where the summation goes over all vertices v adjacent to the vertex u.Lemma 4. [16] Let { } .
By the definition of graph ~i u v G , we can immediately get Lemma 5.

Main Results
Theorem 6.Let ~i u v G be a circum graph with chord and n is an even.Then Proof.First we consider the graph .By Lemma 3, we obtain that Based on ( 5) and ( 6), we immediately Case 1. s is even.
By Lemma 4, we can obtain that Case 2. s is odd.
By Lemma 4, using a similar argument as in the previous proof we conclude that .Thus, for some k, there be ( , This imply that For graph Repeating the same argument as in the previous proof, combine the fact n is even, we have we get Let n is an odd number.Then Proof.
Case 1. s is even.
By Lemma 4, we have , Based on the above analysis, if 2 2 . Thus for some k, we have . This means , 1 . Thus for some k, we have . This means that By Theorems 6 and 7, we immediately get our main result as follow.

Conclusions and Suggestions
In this paper, we determine the quasi-order relation on the matching energy for circum graph with one chord.If the chord here can be see 2 P .Then the general case, determining the quasi-order relation on the matching energy for circum graph with one generalized chord k P for 2 3 k n ≤ ≤ − is more meaningful.
By G u − denote the induced subgraph obtained from G by deleting vertex u together with its incident edges and by G e − the edge-induced subgraph obtained from G by deleting edge e.As usual, use n P and n C to denote the path and cycle on n vertices, respectively.Let ( ) First consider the graph . s is odd.