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In the paper, we give a necessary and sufficient condition of matching equivalence of two graphs with the maximum matching root less than or equal to 2.

Let G be a finite simple graph with vertex set

where

Two graphs G and H are called matching-equivalent if

More than 30 years ago E. J. Farrell in [

Throughout the paper, by

Let G be a graph with order n. Since the roots of

Lemma 2.1. [

Lemma 2.2. [

Lemma 2.3. [

1)

2)

Definition 2.1. Let G be a connected graph with a vertex u. The path-tree

Clearly, if G is a tree, then the path tree

Lemma 2.4. [

and

Lemma 2.5. Let G is a connected graph and

Proof. By Lemmas 2.2 and 2.4, we have

□Lemma 2.6. [

1)

2)

Theorem 2.1. Let G be a connected graph. Then

1)

2)

Proof. (1) Since the path-tree of

Necessity:

Case 1. If G is a tree.

Clearly,

Case 2. If G isn’t a tree.

By Lemma 2.5 and 2.6, the path-tree respect to an arbitrary vertex u of G is

Subcase 2.1. If

Subcase 2.2. If

Since G is connected and isn't a tree, then G is

(2) Since the path-tree of

Necessity:

Case 1. If G is a tree.

Clearly,

Case 2. If G isn’t a tree.

By Lemma 2.5, the path-tree respect to an arbitrary vertex u of G is

Subcase 2.1. If

Let u is a 4 degree vertex of G. Since

Subcase 2.2. If

It is clear that the number of 3 degree vertex of path-tree

Subcase 2.3. If

Subcase 2.4. If

It is clear that

By Theorem 2.1 and Lemma 2.1, we can easily obtain the following Theorem 2.2:

Theorem 2.2. Let G be a graph. Then

1)

2)

In this section, the sufficient and necessary condition for matching equivalence of graphs with the maximum matching root less than or equal to 2 is determined. Firstly, we give some lemmas as follows:

Lemma 3.1. [

where

Lemma 3.2. 1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12)

13)

Proof. (1) Let the vertices sequence of path

(2) Let v be the 3 degree vertex and u be a such pendant vertex of

(3)-(12) The results (3)-(12) can easily obtained by the following equalities.

(13) By Lemma 3.1,

Now, by using mathematical induction to prove (13). Firstly, By (8) and

(13) holds for

Hence (13) holds for

Lemma 3.3. 1)

2)

3)

4)

5)

6)

Proof. Clearly, by Lemma 2.3, we obtain Lemma 3.3(1) immediately. And comparing with the maximum root of two sides of equalities in Lemma 3.2, other results in Lemma 3.3 is also obvious. □

Definition 3.1. Let G and

where

Note that some

By Lemma 3.2, the following representations are also obvious.

Lemma 3.4. 1)

3)

5)

7)

9)

11)

Lemma 3.5. If

and the non-vanishing coefficient

Proof. Since

where

Now by transposition terms from side to side of Equations (1) to (2) such that the coefficients of

where

Compare with the maximum root and its multiplicity of graphs in two sides of (2), we shall get

Repeat this proceeding, we shall get

Furthermore, assume that G be represented as a linear combination

and

In fact, assume that

where

it is a contradiction. Thus

compare with the maximum root of

Lemma 3.6. If

Proof. Since

By transposition terms and comparing with the multiplicity of root 2, we have

Furthermore, we can obtain

By Lemmas 3.5, 3.6 and Definition 3.1 we immediately get. □

Theorem 3.1. Let

1) If

2) If

This work is supported by National Natural Science Foundation of China (11561056), National Natural Science Foundation of Qinghai Provence (2011-Z-911), and Scientific Research Fund of Qinghai University for Nationalities (2015G02).

Haicheng Ma,Yinkui Li, (2016) The Matching Equivalence Graphs with the Maximum Matching Root Less than or Equal to 2. Applied Mathematics,07,920-926. doi: 10.4236/am.2016.79082