The Matching Equivalence Graphs with the Maximum Matching Root Less than or Equal to 2

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.


Introduction
Let G be a finite simple graph with vertex set ( )  G H implies that G H ≅ .The union of two graphs G and H, denoted by G H + , is the graph with vertex set ( ) ( ) . kG denotes the union of k graphs G.
More than 30 years ago E. J. Farrell in [1] introduced the concept of matching polynomials.Latterly, Godsil and Gutman in [2] gave another definition.Here we use the definition given by Godsil.Form then on, the research on the properties of matching polynomials has largely been done (see [3]- [13]).But the research on matching-equivalent of graphs is few.In this 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.
Throughout the paper, by ( )

Graphs with the Maximum Matching Root Less than or Equal to 2
Let G be a graph with order n.Since the roots of ( ) are real numbers (see [7]), the maximum root of ( ) and the maximum root of ( ) is also called spectral radius of graph G), respectively.In this section, we determine graphs with the maximum matching root less than or equal to 2, we firstly give some useful lemmas as follows: Lemma 2.1.[7] Let G be a graph with k components Let G be a connected graph with a vertex u.The path-tree ( ) , T G u is a tree with the paths in G which start at u as its vertices, and where two such paths are joined by an edge if one is a maximal subpath of the other.
Clearly, if G is a tree, then the path tree ( ) ( ) ( ) , comparing with the maximum root of ( ) ( )

[14]
Let T be a tree.Then   and 1,2,2 T , respectively.By Lemmas 2.5 and 2.6 the sufficiency is obvious.Necessity: By Lemma 2.5 and 2.6, the path-tree respect to an arbitrary vertex u of G is ( ) 1 , T G u ∈ Γ .Then we get that the maximum degree ( ) or G is a tree).Clearly, the path-tree of G respect to the 3 degree vertex u is , , D and 4,1 D respect to the 3 degree vertex are 2,2,2 T and 1,3,3 T , respectively.By Lemmas 2.5 and 2.6 the sufficiency is clear. Necessity: By Lemma 2.5, the path-tree respect to an arbitrary vertex = ∆ .In order to complete the proof, we will divide four subcases as follows: , and thus It is clear that the number of 3 degree vertex of path-tree ( ) , T G u respect to an arbitrary vertex u of G is also greater than 2. Hence ( ) , , G G G and 4 G (see Figure 2) and the path-tree respect to the 3 degree vertex u is ( ) ( ) , ,

Sufficient and Necessary Condition for Matching Equivalence of Graphs
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.[7] Let G be a connected graph and , , , , , , , , ) ( ) ( ) ) ( ) ) ( ) , , , , , , , , , , , , C + with any one vertex, thus (1) holds.
(2) Let v be the 3 degree vertex and u be a such pendant vertex of 1,1,n T that the distance between u and v is 1.By Lemma 3.1, consider 1,1,n T with u and 2 n C + with any one vertex, thus (2) holds.

( ) ( ) ( ) ( )
, , , Now, by using mathematical induction to prove (13).Firstly, By ( 8) and ( ) k is allowed to be negative.In fact, if all i k are positive, then 1 1 2 2 = , we can denote Ω .According to Lemma 3.4, we get that G can be represented as a linear combination of paths.Next, without loss of generality, assume that G can be represented as and k α is the coefficient of the longest path in (3).Then 0 k α > .
In fact, assume that 0 k α < , then by transposition terms from side to side of Equation ( 3) such that the coefficients of i m P are positive, we can obtain Equation (4).
( )  D and some paths.Next, without loss of generality, assume that G can be represented as 2) If ( ) ( ) , then G H if and only if G and H have the same linear combination repre- sentation of 3,2 D and some paths.
G and edge set ( )E G .A spanning subgraph H is called a matching of G, if every connected component of H is isolated edge or isolated vertex.k-matching of G is a matching with k edges.A matching polynomial of G is defined as is the number of k-matchings of G.
, respectively, denote the path and the cycle with n vertices.( ) G ∆ denotes the maximum degree of graph G.By 1,4 K denote the star graph with 5 vertices.By , , i j k Tdenote the tree which has one 3-degree vertex u and three 1-degree vertices 1 the graph obtained by identifying one end of the path 1 n P + with a vertex of the cycle m C .Let n P be a path with vertices; sequence 1

1 .Figure 1 .
Figure 1.The graphs , , m n n D I and , , i j k T .

( 1 )
Since the path-tree of m C respect to an arbitrary vertex and 3,1 D respect to the 3 degree vertex are 2 1 m P −

2 Ω
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 M G < if and only if every connected component of G belongs to 1 Ω .2) ( ) 2 M G = and 2 is m multiple root of ( ) , G x µ if and only if m connected components of G belong to and others belong to 1 Ω .

( 1 ) 1 m
Let the vertices sequence of path 2

2 M 2 M
G < .Then G can uniquely be represented as a linear combination of the form and the non-vanishing coefficient i α , with the greatest i m , is positive.Furthermore, if k m P is the longest path with the non-vanishing coefficient k α , G < , by Theorem 2.2, every connected component of G belongs to 1

2 M 2 M 2 Ω
G = , then G can uniquely be represented as a linear combination of the form G = , by Theorem 2.2, every connected component of G belongs to 1 ∪ Ω .According to Lemma 3.4, we easily obtain that G can be represented as a linear combination of 3,2 By transposition terms and comparing with the multiplicity of root 2, we have 0 G H if and only if G and H have the same linear combination repre- sentation of paths.
Two graphs G and H are called matching-equivalent if ( ) ( )

Lemma 2.5. Let G is a connected graph and
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