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 w<sub>i</sub> be a non-isolated vertex of graph G<sub>i</sub> where i=1, 2, …, k . We use G<sub>u</sub>(k) (respectively, H<sub>v</sub>(k) ) to denote the graph which is the coalescence of G (respectively, H ) and G<sub>1</sub>, G<sub>2</sub>,…, G<sub>k</sub> by identifying the vertices u (respectively, v ) and w<sub>1</sub>, w<sub>2</sub>,…, w<sub>k</sub> . In this paper, we first present a new technique of directly comparing the matching energies of G<sub>u</sub>(k) and H<sub>v</sub>(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 2 n with the first to the ninth smallest matching energies for all n≥211 .
Let G be a simple and undirected graph with n vertices and A ( G ) be its adjacency matrix. Let λ 1 , λ 2 , ⋯ , λ n be the eigenvalues of A ( G ) . Then the energy of G, denoted by E ( G ) , is defined as [
E ( G ) = ∑ i = 1 n | λ i | .
A fundamental problem encountered within the study of graph energy is the characterization of the graphs that belong to a given class of graphs having maximal or minimal energy, for example, Trees with extremal energies [
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 m ( G , k ) = 0 for k > ⌊ n / 2 ⌋ or k < 0 . In addition, we assume that m ( G , 0 ) = 1 .
The matching polynomial of a graph G is defined as
α ( G ) = α ( G , x ) = ∑ k = 0 ⌊ n / 2 ⌋ ( − 1 ) k m ( G , k ) x n − 2 k .
Recently, Gutman and Wagner [
Definition 1.1. Let G be a simple graph of order n and μ 1 , μ 2 , ⋯ , μ n be the zeros of its matching polynomial. Then
M E ( G ) = ∑ i = 1 n | μ i | .
Further, Gutman and Wagner [
T R E ( G ) = E ( G ) − M E ( G ) ,
where T R E ( G ) is the so-called topological resonance energy of G, in connection with the chemical applications of matching energy, for more details see [
Similar to the integral formula for the energy of graph, Gutman and Wagner [
M E ( G ) = 2 π ∫ 0 + ∞ 1 x 2 ln ( ∑ k = 0 ⌊ n / 2 ⌋ m ( G , k ) x 2 k ) d x . (1)
Then M E ( G ) is a strictly monotonically increasing function of those numbers m ( G , k ) ( k = 0 , 1 , ⋯ , ⌊ n / 2 ⌋ ) . 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 G 1 and G 2 be two graphs of order n. If m ( G 1 , k ) ≤ m ( G 2 , k ) for all k with 1 ≤ k ≤ ⌊ n / 2 ⌋ , then we write G 1 ≼ G 2 .
Furthermore, if G 1 ≼ G 2 and there exists at least one index j such that m ( G 1 , j ) < m ( G 2 , j ) , then we write G 1 ≺ G 2 . If m ( G 1 , k ) = m ( G 2 , k ) for all k, then we write G 1 ∼ G 2 . According to the integral formula (1), we have for two graphs G 1 and G 2 of order n that
G 1 ≼ G 2 ⇒ M E ( G 1 ) ≤ M E (G2)
G 1 ≺ G 2 ⇒ M E ( G 1 ) < M E ( G 2 ) .
In [
The study on extremal matching energies is very interesting. In [
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 G u ( k ) and H v ( k ) in Section 2 (see
For simplicity, if G 1 is isomorphic to G 2 , then we write G 1 = G 2 . If G 1 is not isomorphic to G 2 , then we write G 1 ≠ G 2 . Let A ( 2 n ) be the set of the unicyclic graphs with perfect matchings of order 2n. Let the unicyclic graphs A 1 , A 2 , A 3 , A 4 , A 4 * , A 5 , A 6 , A 7 , A 8 , A 9 be shown in
Theorem 1.1. Let G ∈ A ( 2 n ) and n ≥ 211 . If G ≠ A 1 , A 2 , A 3 , A 4 , A 4 * , A 5 , A 6 , A 7 , A 8 , A 9 , then M E ( A 1 ) < M E ( A 2 ) < M E ( A 3 ) < M E ( A 4 ) = M E ( A 4 * ) < M E ( A 5 ) < M E ( A 6 ) < M E ( A 7 ) < M E ( A 8 ) < M E ( A 9 ) < M E ( G ) .
By Definition 1.2, we can see that the quasi-order method can be used to compare the matching energies of two graphs. However, if the quantities 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 [
Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let w i be a non-isolated vertex of graph G i where i = 1 , 2 , ⋯ , k . We use G u ( k ) (respectively, H v ( k ) ) to denote the graph which is the coalescence of G (repectively, H) and G 1 , G 2 , ⋯ , G k by identifying the vertices u (respectively, v) and w 1 , w 2 , ⋯ , w k (see
In this paper, we assume that
α ˜ ( G ) = α ˜ ( G , x ) = ∑ k = 0 ⌊ n / 2 ⌋ m ( G , k ) x n − 2 k . (2)
By Equation (2), we can immediately obtain the following results.
Lemma 2.1. If two graphs G and H are disjoint, then
α ˜ ( G ∪ H ) = α ˜ ( G ) ⋅ α ˜ ( H ) .
Lemma 2.2. ( [
α ˜ ( G ) = x α ˜ ( G − u ) + ∑ u v ∈ E ( G ) α ˜ ( G − u − v ) .
The coalescence of two graphs G and H with respect to vertex u in G and vertex v in H, denoted by G u ⋅ H v (sometimes abbreviated as G ⋅ H ), is the graph obtained by identifying the vertices u and v. Zhu and Yang [
Lemma 2.3. ( [
α ˜ ( G ⋅ H ) = α ˜ ( G ) α ˜ ( H − v ) + α ˜ ( G − u ) ( α ˜ ( H ) − x α ˜ ( H − v ) ) .
Proof. Using Lemmas 2.1 and 2.2, we can show
α ˜ ( G ⋅ H ) = x α ˜ ( G ⋅ H − u ) + ∑ u w ∈ E ( G ) α ˜ ( G ⋅ H − u − w ) + ∑ v t ∈ E ( H ) α ˜ ( G ⋅ H − v − t ) = x α ˜ ( G − u ) ⋅ α ˜ ( H − v ) + ∑ u w ∈ E ( G ) α ˜ ( G − u − w ) ⋅ α ˜ ( H − v ) + ∑ v t ∈ E ( H ) α ˜ ( G − u ) ⋅ α ˜ ( H − v − t ) = ( x α ˜ ( G − u ) + ∑ u w ∈ E ( G ) α ˜ ( G − u − w ) ) ⋅ α ˜ ( H − v ) + α ˜ ( G − u ) ⋅ ∑ v t ∈ E ( H ) α ˜ ( H − v − t ) = α ˜ ( G ) ⋅ α ˜ ( H − v ) + α ˜ ( G − u ) ⋅ ( α ˜ ( H ) − x α ˜ ( H − v ) ) .
From Lemma 2.3, we can get the recurrence relations of the graphs α ˜ ( G u ( k ) ) and α ˜ ( H v ( k ) ) which is a generalization of the formula for α ˜ ( G ⋅ H ) .
Lemma 2.4. Let G u ( k ) and H v ( k ) be defined as above (see
1) α ˜ ( G u ( k ) ) = ∏ i = 1 k α ˜ ( G i − w i ) ( α ˜ ( G ) + α ˜ ( G − u ) ( ∑ i = 1 k α ˜ ( G i ) α ˜ ( G i − w i ) − k x ) ) ;
2) α ˜ ( H v ( k ) ) = ∏ i = 1 k α ˜ ( G i − w i ) ( α ˜ ( H ) + α ˜ ( H − v ) ( ∑ i = 1 k α ˜ ( G i ) α ˜ ( G i − w i ) − k x ) ) .
Proof. 1) We prove the result by induction on k. When k = 1 , by Lemma 2.3 we have
α ˜ ( G u ( 1 ) ) = α ˜ ( G ⋅ G 1 ) = α ˜ ( G ) α ˜ ( G 1 − w 1 ) + α ˜ ( G − u ) ( α ˜ ( G 1 ) − x α ˜ ( G 1 − w 1 ) ) ,
which implies that the result holds. We assume that the result holds for k − 1 in what follows. For simplicity, we write h k = ∑ i = 1 k α ˜ ( G i ) α ˜ ( G i − w i ) − k x . By Lemmas 2.1 and 2.3, we can show
α ˜ ( G u ( k ) ) = α ˜ ( G u ( k − 1 ) ⋅ G k ) = α ˜ ( G u ( k − 1 ) ) α ˜ ( G k − w k ) + α ˜ ( G u ( k − 1 ) − u ) ( α ˜ ( G k ) − x α ˜ ( G k − w k ) ) = ∏ i = 1 k − 1 α ˜ ( G i − w i ) ( α ˜ ( G ) + α ˜ ( G − u ) h k − 1 ) α ˜ ( G k − w k ) + ∏ i = 1 k − 1 α ˜ ( G i − w i ) α ˜ ( G − u ) ( α ˜ ( G k ) − x α ˜ ( G k − w k ) ) = ∏ i = 1 k − 1 α ˜ ( G i − w i ) ( ( α ˜ ( G ) + α ˜ ( G − u ) h k − 1 ) α ˜ ( G k − w k )
+ α ˜ ( G − u ) ( α ˜ ( G k ) − α ˜ ( G k − w k ) ) ) = ∏ i = 1 k − 1 α ˜ ( G i − w i ) α ˜ ( G k − w k ) ( α ˜ ( G ) + α ˜ ( G − u ) h k − 1 + α ˜ ( G − u ) ( α ˜ ( G k ) α ˜ ( G k − w k ) − x ) ) = ∏ i = 1 k α ˜ ( G i − w i ) ( α ˜ ( G ) + α ˜ ( G − u ) h k − 1 + α ˜ ( G − u ) ( α ˜ ( G k ) α ˜ ( G k − w k ) − x ) ) = ∏ i = 1 k α ˜ ( G i − w i ) ( α ˜ ( G ) + α ˜ ( G − u ) ( h k − 1 + α ˜ ( G k ) α ˜ ( G k − w k ) − x ) ) = ∏ i = 1 k α ˜ ( G i − w i ) ( α ˜ ( G ) + α ˜ ( G − u ) h k )
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 [
Lemma 2.5. ( [
M E ( G ) − M E ( H ) = 2 π ∫ 0 + ∞ ln α ˜ ( G , x ) α ˜ ( H , x ) d x .
Let x > 0 . For simplicity, we write
h k = ∑ i = 1 k α ˜ ( G i ) α ˜ ( G i − w i ) − k x = ∑ i = 1 k α ˜ ( G i ) − x α ˜ ( G i − w i ) α ˜ ( G i − w i ) .
From Lemma 2.2, we have h k > 0 and h l < h k holds for any positive integer l < k .
In what follows, we define two sets M and M c as follows:
M = { x > 0 | α ˜ ( G − u ) α ˜ ( H ) − α ˜ ( G ) α ˜ ( H − v ) > 0 }
M c = { x > 0 | α ˜ ( G − u ) α ˜ ( H ) − α ˜ ( G ) α ˜ ( H − v ) ≤ 0 } .
It is easily checked that M ∪ M c = ( 0, + ∞ ) . Furthermore, we write
m k ( x ) = α ˜ ( G ) + h k α ˜ ( G − u ) α ˜ ( H ) + h k α ˜ ( H − v )
m ( x ) = α ˜ ( G − u ) α ˜ ( H − v ) .
Combining Lemma 2.4 with Lemma 2.5, we can present a new technique for directly comparing the matching energies of two graphs G u ( k ) and H v ( k ) in the following theorem.
Theorem 2.2. Let M, M c , m k ( x ) and m ( x ) be defined as above. For all positive integers 1 ≤ l < k , we have
∫ M ln m l ( x ) d x + ∫ M c ln m ( x ) d x ≤ π 2 ( M E ( G u ( k ) ) − M E ( H v ( k ) ) ) ≤ ∫ M ln m ( x ) d x + ∫ M c ln m l ( x ) d x .
Proof. By some calculations, we can obtain that
m k ( x ) − m l ( x ) = α ˜ ( G ) + h k α ˜ ( G − u ) α ˜ ( H ) + h k α ˜ ( H − v ) − α ˜ ( G ) + h l α ˜ ( G − u ) α ˜ ( H ) + h l α ˜ ( H − v ) = ( h k − h l ) ( α ˜ ( G − u ) α ˜ ( H ) − α ˜ ( G ) α ˜ ( H − v ) ) ( α ˜ ( H ) + h k α ˜ ( H − v ) ) ( α ˜ ( H ) + h l α ˜ ( H − v ) ) .
m k ( x ) − m ( x ) = α ˜ ( G ) + h k α ˜ ( G − u ) α ˜ ( H ) + h k α ˜ ( H − v ) − α ˜ ( G − u ) α ˜ ( H − v ) = − ( α ˜ ( G − u ) α ˜ ( H ) − α ˜ ( G ) α ˜ ( H − v ) ) ( α ˜ ( H ) + h k α ˜ ( H − v ) ) α ˜ ( H − v ) .
Thus, If x ∈ M , then m l ( x ) ≤ m k ( x ) ≤ m ( x ) . If x ∈ M c , then m ( x ) ≤ m k ( x ) ≤ m l ( x ) .
Moreover, by Lemmas 2.4 and 2.5, we have
π 2 ( M E ( G u ( k ) ) − M E ( H v ( k ) ) ) = ∫ 0 + ∞ ln m k ( x ) d x = ∫ M ln m k ( x ) d x + ∫ M c ln m k ( x ) d x .
Then the result can be obtained immediately.
Next, we use the new technique to compare the matching energies of the quasi-order incomparable graphs A 5 and A 6 , A 8 and A 9 (see
Lemma 2.6. If n ≥ 6 , then M E ( A 5 ) < M E ( A 6 ) .
Proof. Let G be the graph obtained by attaching a pendent edge to a vertex u of C 5 . 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 C 3 , respectively. Let G 1 = G 2 = ⋯ = G n − 3 = P 3 and w i be the pendent vertex of G i . Then G u ( n − 3 ) = A 5 and H v ( n − 3 ) = A 6 (see
α ˜ ( G ) = x 6 + 6 x 4 + 8 x 2 + 1
α ˜ ( G − u ) = x ( x 4 + 3 x 2 + 1 )
α ˜ ( H ) = x 6 + 6 x 4 + 7 x 2 + 1
α ˜ ( H − v ) = ( x 2 + 1 ) ( x 3 + 2 x ) .
It follows that
α ˜ ( G − u ) α ˜ ( H ) − α ˜ ( G ) α ˜ ( H − v ) = − 2 x 7 − 9 x 5 − 9 x 3 − x .
This implies that M = ∅ and M c = ( 0 , + ∞ ) . By Theorem 2.2 and some calculations using the software MATLAB, we have
π 2 ( M E ( A 5 ) − M E ( A 6 ) ) = π 2 ( M E ( G u ( n − 3 ) ) − M E ( H v ( n − 3 ) ) ) ≤ ∫ 0 + ∞ ln m 3 ( x ) d x = ∫ 0 + ∞ ln ( x 2 + 1 ) 2 ( x 8 + 10 x 6 + 23 x 4 + 12 x 2 + 1 ) ( x 2 + 1 ) 3 ( x 6 + 9 x 4 + 13 x 2 + 1 ) d x ≐ − 0.0248 < 0.
Thus, M E ( A 5 ) < M E ( A 6 ) .
Lemma 2.7. If n ≥ 211 , then M E ( A 8 ) < M E ( A 9 ) .
Proof. Let G be the graph obtained by attaching two pendent paths of length 2 to the same vertex of C 4 . Let H be the graph obtained by first attaching a pendent edge to each vertex of C 3 and then attaching a pendent path of length 2 to one vertex of C 3 . Let u be the vertex of degree 4 in G and v be the vertex of degree 3 in H, respectively. Let G 1 = G 2 = ⋯ = G n − 4 = P 3 and w i be the pendent vertex of G i . Then G u ( n − 4 ) = A 8 and H v ( n − 4 ) = A 9 (see
α ˜ ( G ) = x 8 + 8 x 6 + 17 x 4 + 12 x 2 + 2
α ˜ ( G − u ) = ( x 2 + 1 ) 2 ( x 3 + 2 x )
α ˜ ( H ) = x 8 + 8 x 6 + 15 x 4 + 8 x 2 + 1
α ˜ ( H − v ) = x ( x 6 + 5 x 4 + 5 x 2 + 1 ) ,
which implies that
α ˜ ( G − u ) α ˜ ( H ) − α ˜ ( G ) α ˜ ( H − v ) = − x 3 ( x 2 + 1 ) 2 ( x 6 + 8 x 4 + 11 x 2 + 1 ) .
It follows that M = ∅ and M c = ( 0 , + ∞ ) . By Theorem 2.2 and some calculations using the software MATLAB, we have
π 2 ( M E ( A 8 ) − M E ( A 9 ) ) = π 2 ( M E ( G u ( n − 4 ) ) − M E ( H v ( n − 4 ) ) ) ≤ ∫ 0 + ∞ ln m 207 ( x ) d x = ∫ 0 + ∞ ln ( x 2 + 1 ) 208 ( x 6 + 214 x 4 + 424 x 2 + 2 ) ( x 2 + 1 ) 206 ( x 10 + 216 x 8 + 1055 x 6 + 1055 x 4 + 216 x 2 + 1 ) d x ≐ − 7.43 × 10 − 5 < 0.
Consequently, M E ( A 8 ) < M E ( A 9 ) .
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 M ( G ) a perfect matching of a graph G. Let G ^ = G − M ( G ) − S 0 , where S 0 is the set of isolated vertices in G − M ( G ) . We call G ^ the capped graph of G and G the original graph of G ^ . For example, the capped graphs of A 1 , A 2 , A 3 , A 5 are shown in
Let G ∈ A ( 2 n ) . Denote by E ( G ) the edge set of G. It is easy to see that E ( G ) = E ( G ^ ) ∪ M ( G ) . Thus each k-matching Ω of G can be partitioned into two parts: Ω = Φ ∪ Ψ , where Φ ⊆ E ( G ^ ) and Ψ ⊆ M ( G ) . Let r j ( 2 k ) ( G ) be the number of ways to choose k independent edges in G such that just j edges are in G ^ . We agree that r 0 ( 0 ) ( G ) = 1 and r j ( 2 k ) ( G ) = 0 ( k < 0 ) . For example: r 0 ( 2 k ) ( G ) = ( n k ) and r 1 ( 2 k ) ( G ) = n ( n − 2 k − 1 ) .
Then we have
m ( G , k ) = ∑ j = 0 k r j ( 2 k ) ( G ) = p + ∑ j = 2 k r j ( 2 k ) ( G ) , (3)
where
p = ( n k ) + n ( n − 2 k − 1 ) .
This is the main method to compute m ( G , k ) of a graph G in what follows.
Let X n be the star of order n. Let Y n be the graph of order n obtained by attaching n − 3 pendent edges to a pendent vertex of P 3 . Let Z n be the graph of order n obtained from P 4 = v 1 v 2 v 3 v 4 by attaching n − 5 and one pendent edges to v 2 and v 3 , respectively. In [
Lemma 3.1. ( [
Lemma 3.2. ( [
Let S n l be the unicyclic graph of order n obtained by attaching n − l pendent edges to one vertex of C l .
Lemma 3.3. ( [
Let R n 3 be the graph of order n obtained by attaching n − 4 and one pendent edges to v 1 and v 2 of C 3 = v 1 v 2 v 3 v 1 , respectively. Let C 3 ( a , b , c ) be the unicyclic graph obtained by attaching a , b , c pendent edges to v 1 , v 2 , v 3 of C 3 = v 1 v 2 v 3 v 1 , respectively. Let Q n 3 be the graph of order n obtained by attaching n − 4 pendent edges to the pendent vertex of C 3 ( 1,0,0 ) . The graphs S n 3 , R n 3 and Q n 3 are shown in
Lemma 3.4. Let G be a unicyclic graph of order n ≥ 9 . If G ≠ S n 3 , R n 3 , then m ( G ,2 ) ≥ 2 n − 6 .
Proof. Let G be a unicyclic graph with the unique cycle of length l. We consider the following cases.
Case 1: l ≥ 5 .
By Lemma 3.10, we have G ≽ S n l . Then m ( G ,2 ) ≥ m ( S n l ,2 ) ≥ ( n − l ) ( l − 2 ) ≥ 3 ( n − 5 ) ≥ 2 n − 6 .
Case 2: l = 4 .
Using Lemma 3.10, we can show G ≽ S n 4 . So, m ( G , 2 ) ≥ m ( S n 4 , 2 ) = 2 n − 6 .
Case 3: l = 3 .
Denote by d G ( u ) the degree of the vertex u in G. Let C 3 = v 1 v 2 v 3 v 1 be the unique cycle of the unicyclic graph G and N ( G ) = { v i | d G ( v i ) ≥ 3 , i = 1 , 2 , 3 } .
Subcase 3.1: | N ( G ) | = 1 .
Without loss of generality, we can assume that d G ( v 1 ) ≥ 3 . Let T be the rooted tree of order n − 2 with the root v 1 in G. If T = X n − 2 , then G = Q n 3 ( G ≠ S n 3 ). Then m ( G , 2 ) = 3 n − 11 > 2 n − 6 . If T = Y n − 2 , then m ( G , 2 ) ≥ ( n − 3 ) + m ( Y n − 2 , 2 ) + 2 = ( n − 3 ) + ( n − 5 ) + 2 = 2 n − 6 . If T ≠ X n − 2 , Y n − 2 , then by Lemma 3.8 we have
m ( G , 2 ) ≥ ( n − 3 ) + m ( T , 2 ) ≥ ( n − 3 ) + m ( Z n − 2 , 2 ) = n − 3 + 2 ( n − 6 ) = 3 n − 15 ≥ 2 n − 6 .
Subcase 3.2: | N ( G ) | = 2 .
Without loss of generality, we can assume that d G ( v 1 ) ≥ 3 and d G ( v 2 ) ≥ 3 . Let T 1 and T 2 be the rooted tree with the root v 1 and v 2 in G, respectively.
If T 1 = P 2 or T 2 = P 2 , then by Lemma 3.9 we can show m ( G , 2 ) ≥ m ( R n 3 , 2 ) = 2 n − 7 . Since G ≠ R n 3 , we have m ( G ,2 ) ≥ 2 n − 6 .
If T 1 ≠ P 2 and T 2 ≠ P 2 , then by Lemma 3.9 we have G ≽ C 3 ( a , b ,0 ) ( a + b = n − 3 ). Thus, m ( G , 2 ) ≥ m ( C 3 ( a , b , 0 ) , 2 ) ≥ a + b + a b ≥ n − 3 + 2 ( n − 5 ) = 3 n − 13 > 2 n − 6 .
Subcase 3.3: | N ( G ) | = 3 .
According to Lemma 3.9, we have G ≽ C 3 ( a , b , c ) ( a + b + c = n − 3 ). Then we have
m ( G , 2 ) ≥ a + b + c + a b + b c + a c ≥ n − 3 + a + b − 1 + b + c − 1 + a + c − 1 = n − 3 + 2 ( n − 3 ) − 3 = 3 n − 12 > 2 n − 6.
Thus we have completed the proof.
Lemma 3.5. Let G ∈ A ( 2 n ) and n ≥ 9 . If G ≠ A 1 , A 2 , A 3 , A 4 , A 4 * , A 5 , A 6 , A 7 , A 8 , A 9 , then m ( G ^ ,2 ) ≥ 2 n − 6 .
Proof. We consider the following cases.
Case 1: G ^ is a connected graph.
Subcase 1.1: G ^ is a tree.
It can easily be verified that G = A 1 as G ^ = X n + 1 and G = A 3 , A 4 , A 4 * , A 6 as G ^ = Y n + 1 . Thus, G ^ ≠ X n + 1 , Y n + 1 . By Lemma 3.8, we have m ( G ^ , 2 ) ≥ m ( Z n + 1 , 2 ) = 2 n − 6 .
Subcase 1.2: G ^ is a connected unicyclic graph.
It can be shown that G = A 2 as G ^ = S n 3 and G = A 9 as G ^ = R n 3 . Therefore, G ^ ≠ S n 3 , R n 3 . According to Lemma 3.11, we have m ( G ^ ,2 ) ≥ 2 n − 6 .
Case 2: G ^ is a unconnected graph.
Subcase 2.1: G ^ is only composed of trees.
It can be checked that G = A 5 , A 7 , A 8 as G ^ = X n ∪ P 2 . Then, G ^ ≠ X n ∪ P 2 . Let G ^ 1 be the coalescence of all trees in a way such that G ^ 1 ≠ X n + 1 , Y n + 1 . It is clear that m ( G ^ , 2 ) > m ( G ^ 1 , 2 ) . Similar to Subcase 1.1, we have m ( G ^ , 2 ) > 2 n − 6 .
Subcase 2.2: G ^ is composed of trees and unicyclic graphs.
Let G ^ 2 be the coalescence of all trees and unicyclic graphs in a way such that G ^ 2 ≠ S n 3 , R n 3 . It is obvious that m ( G ^ , 2 ) > m ( G ^ 2 , 2 ) . Similar to Subcase 1.2, we have m ( G ^ , 2 ) > 2 n − 6 .
Then we have completed the proof.
From Lemma 3.5, we can immediately derive the following result.
Lemma 3.6. Let G ∈ A ( 2 n ) and n ≥ 9 . If G ≠ A 1 , A 2 , A 3 , A 4 , A 4 * , A 5 , A 6 , A 7 , A 8 , A 9 , then G ≻ A 9 .
Proof. By Equation (3) and Lemma 3.12, when k ≥ 2 , we can get
m ( G , k ) = p + ∑ j = 2 k r j ( 2 k ) ( G ) ≥ p + r 2 ( 2 k ) ( G ) ≥ p + m ( G ^ ,2 ) ( n − 4 k − 2 ) ≥ p + ( 2 n − 6 ) ( n − 4 k − 2 ) .
Furthermore, by some calculations we have
m ( A 9 , k ) = p + ( 2 n − 7 ) ( n − 4 k − 2 ) .
Then we can see that G ≻ A 9 .
Combining Lemma 2.6 with Lemma 2.7, we can show the followings.
Lemma 3.7. If n ≥ 211 , then M E ( A 1 ) < M E ( A 2 ) < M E ( A 3 ) < M E ( A 4 ) = M E ( A 4 * ) < M E ( A 5 ) < M E ( A 6 ) < M E ( A 7 ) < M E ( A 8 ) < M E ( A 9 ) .
Proof. Using Equation (4) and some calculations, we can get
m ( A 1 , k ) = p
m ( A 2 , k ) = p + ( n − 3 ) ( n − 4 k − 2 )
m ( A 3 , k ) = p + ( n − 2 ) ( n − 4 k − 2 )
m ( A 4 , k ) = p + ( n − 3 k − 2 ) + ( n − 3 ) ( n − 4 k − 2 )
m ( A 4 * , k ) = p + ( n − 3 k − 2 ) + ( n − 3 ) ( n − 4 k − 2 )
m ( A 5 , k ) = p + 2 ( n − 3 k − 2 ) + ( n − 3 ) ( n − 4 k − 2 )
m ( A 6 , k ) = p + ( n − 3 ) ( n − 3 k − 2 )
m ( A 7 , k ) = p + ( n − 1 ) ( n − 3 k − 2 )
m ( A 8 , k ) = p + ( n − 2 k − 2 ) + ( n − 2 ) ( n − 3 k − 2 )
m ( A 9 , k ) = p + ( 2 n − 7 ) ( n − 4 k − 2 ) .
It implies that A 1 ≺ A 2 ≺ A 3 ≺ A 4 ≺ A 5 and A 6 ≺ A 7 ≺ A 8 . From Lemmas 2.6 and 2.7, the result can be easily obtained.
Proof of Theorem 1.1:
Proof. The result can follow immediately by Lemmas 3.13 and 3.14.
In this paper, we first present a new technique of directly comparing the matching energies of G u ( k ) and H v ( k ) , which can tackle some quasi-order incomparable problems. As the applications of the technique, we then determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all n ≥ 211 . Furthermore, we can consider characterizing the extremal graphs with maximal or minimal matching energy for other classes of graphs, e.g. graphs with different parameters. These are our work in the future.
The results presented in this paper are for a fixed graph. In reality, most of the graphs or networks are evolving. Some graph invariants have been studied in this setting, e.g. the Estrada index of evolving graphs [
We thank the editor and the referee for their valuable comments. This work is supported by the National Natural Science Foundation of China (No. 11501356) and (No. 11426149).
The author declares no conflicts of interest regarding the publication of this paper.
Zhu, J.M. (2019) Ordering of Unicyclic Graphs with Perfect Matchings by Minimal Matching Energies. Open Journal of Discrete Mathematics, 9, 17-32. https://doi.org/10.4236/ojdm.2019.91004