We use standard graph-theoretical notation and terminology. For concepts and notations not defined here, we refer the reader to  .
By a graph we always mean a simple undirected graph G with the vertex set and the edge set .We denote the complement of G by . The degree of a vertex is denoted by , abbreviated as . Let be the union of two graphs G and H which have no common vertices. For any positive integer l, let lG be the union of l disjoint copies of graph G. An acyclic graph, containing no cycles, is called a forest. A connected forest is called a tree. The complete bipartite graph with vertices is denoted by . The path, star and complete graph with vertices are denoted by , and , respectively.
A r-matching in G is a set of r pairwise non-incident edges. The number of r- matchings in G is denoted by . Specifically, and
for . It is both consistent and convenient to define . The matching polynomial of the graph G is defined as
Matching polynomials have some important applications in statistical physics and structural chemistry. Up to now, the matching polynomials of graphs are extensively examined, we refer the reader to  -  and the references therein.
Two nonisomorphic graphs G and H are said to be matching equivalent, symbolically , if . Godsil and Gutman  first proposed the question to determine matching equivalent graphs. This seems difficult in the theory of graph polynomials. By now, little about the matching equivalent graphs have been published     . In this paper, we plan to investigate the problem which graphs are matching equivalent.
2. Some Lemmas
In the section, we will present some lemmas which are required in the proof of the main results.
Lemma 1.  
Corollary 1. if and only if .
Proof. Corollary 1 follows directly by Lemma 1.
Lemma 2.  The matching polynomial satisfies the following identities:
2) if is an edge of G;
3) , if .
Let G be a graph with a vertex u. The path tree is the 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.
Lemma 3.  Suppose G is a connected graph with , and suppose is the path tree associated with G with root u. Then has a subforest such that
Remark 1. It is not difficult to derive the following description of an appropriate subforest : find a spanning tree of G by depth-first search from u, and delete from T all vertices corresponding to a path contained in .
Wu and Zhang  determined all connected graphs with matching number 2 as follows.
Lemma 4. (  ) Let G be a connected graph with n vertices. Then if and only if G is one of 22 graphs as shown in Figure 1, where in , in , in , in , in , and in , and and in , and and in .
The eigenvalues of the adjacency matrix of G, denoted by are the eigenvalues of G, and form the spectrum of G  . Two nonisomorphic
Figure 1. All connected graphs with the matching number 2.
graphs of the same order are cospectral if they have the same spectrum.
Lemma 5. (  ) If G is a forest then , where denotes the characteristic polynomial of G.
Lemma 5 implies the following result.
Corollary 2. Let G and H be two forests. If , then G and H are cospectral.
3. Main Results
For convenience, and in Figure 1 are replaced by and , respectively, where and denote the degree of vertices u and v, respectively, and k denotes the number of common neighbors of u and v.
Theorem 1. .
Proof. By (2) of Lemma 2, we have
By Corollary 1 and Theorem 1, we have
Corollary 3. .
Theorem 2. and are matching equivalent, and their complements are also matching equivalent, where .
Proof. By (2) of Lemma 2, we have
Checking and , it can be seen that and are isomorphic. So, . By Corollary 1, .
Theorem 3. Let G be a graph obtained by identifying a cycle to vertex u of , and let H be a graph obtained by attaching a path to a vertex of degree two in , where (see Figure 2). Then and .
Proof. By (2) of Lemma 2, we obtain that and . Checking G and H, it can be known that and are isomorphic. So, . By Corollary 1, again we have .
Checking , it is easy to find that and are matching equivalent. Based on the result above, we construct a pair of matching equivalent graphs as follows.
Theorem 4. Let G be a graph obtained by joining two single vertices to in , and let H be a graph obtained by joining a single vertex to and in , respectively, where and are both in the bipartition of q vertices in (see Figure 3). Then and .
Proof. By (2) of Lemma 2, we obtain that and . Checking G and H, it can be known that and are isomorphic. So, . By Corollary 1, we have .
Figure 2. Graphs G and H in Theorem 3.
Figure 3. Graphs G and H in Theorem 4.
Theorem 5. Let G be a graph obtained by attaching a path to in , and let H be a graph obtained by identifying the center of to a pendant vertex of (see Figure 4). Then and .
Proof. By (2) of Lemma 2, we obtain that and . Checking G and H, it can be known that is isomorphic to . So . By Corollary 1, we have .
Theorem 6. Let G and H be two graphs which are defined in Theorem 5. Then and , where I denotes a graph obtained by attaching a single vertex to a pendant vertex of , and and denote the path-trees of G and H, respec- tively (see Figure 5). In particular, and are cospec- tral.
Proof. By Lemma 3, we have and . By Theorem 5, we have . So, . This implies that .
By Corollary 1, we have . Furthermore, by Corollary 5, we know that and are cospectral.
In this paper, we constructed some families matching equivalent graphs in Theorems 1,..., and 4. Based on these results, using the same method in Theorem 6, we can construct some pairs of matching equivalent forests and cospectral forests, respectively.
Figure 4. Graphs G and H in Theorem 5.
Figure 5. Path trees of G and H in Theorem 6.
We thank the editor and referees for their comments. Research of Tingzeng Wu is funded by NSF of Qinghai (2016-ZJ-947Q), and high-level personnel of scientific research projects of QHMU(2016XJG07). These supports are greatly appreciated.