The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two

Yongqiang Bai; Hongping Ma; Xia Zhang

doi:10.1515/math-2024-0120

Article Open Access

The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two

Yongqiang Bai , Hongping Ma and Xia Zhang

Published/Copyright: December 31, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

The number of j -matchings in a graph H is denote by m ( H , j ) . If for two graphs H 1 and H 2 , m ( H 1 , j ) ≥ m ( H 2 , j ) for all j , then we write H 1 ≽ H 2 . If H 1 ≽ H 2 , and m ( H 1 , i ) > m ( H 2 , i ) for some i , then we write H 1 ≻ H 2 . In this article, by utilizing several new graph transformations, we determine the least element with respect to the quasi-order ≽ among all unicyclic graphs with fixed order and number of vertices of degree two. As consequences, we characterize the graphs with minimum matching energy and with minimum Hosoya index in the set of all unicyclic graphs with fixed order and number of vertices of degree two.

Keywords: matching energy; unicyclic graph; quasi-order; Hosoya index

MSC 2010: 05C09; 05C92

1 Introduction

In this article, the graphs mentioned are undirected simple graphs. For notations and terms not included here follow [1]. Let H be a graph with order n and size m . For positive integer j , denote by m ( H , j ) the number of j -matchings of H . Clearly, m ( H , 1 ) = m and m ( H , j ) = 0 for j > ⌊ n ⁄ 2 ⌋ . Define m ( H , 0 ) = 1 . The matching polynomial of H is defined as follows:

α ( H , x ) = ∑ j ≥ 0 ( − 1 ) j m ( H , j ) x n − 2 j .

The energy of a graph H is defined as E ( H ) = ∑ i = 1 n ∣ λ i ∣ , where λ 1 , λ 2 , … , λ n are the eigenvalues of the adjacency matrix of H . The concept of graph energy was put forward by Gutman in 1978. So far, there have been many research results on this topic, see the book [2] and the review [3].

In 2012, the matching energy of a graph H , denoted by ME ( H ) , was introduced by Gutman and Wagner [4]. That is, ME ( H ) is the sum of the absolute values of the roots of α ( H , x ) . Moreover, Gutman and Wagner [4] arrived at the simple relation:

TRE ( H ) = E ( H ) − ME ( H ) ,

where TRE ( H ) is the so-called “topological resonance energy.” The aforementioned equation shows that the matching energy is a quantity of relevance for chemical applications, for more details, see the review in [5].

For matching energy, there also has an equivalent definition as follows [4]:

(1) ME ( H ) = 2 π ∫ 0 + ∞ 1 x 2 ln ∑ j ≥ 0 m ( H , j ) x 2 j d x .

By the formula (1), the so-called quasi-order method is a well-known classic method for comparing the matching energies of two graphs. Here, the quasi-order ≽ is defined as follows: for any two graphs H 1 and H 2 ,

(2) H 1 ≽ H 2 ⇔ m ( H 1 , j ) ≥ m ( H 2 , j ) for all j .

If H 1 ≽ H 2 and m ( H 1 , i ) > m ( H 2 , i ) for some i , then we write H 1 ≻ H 2 . If H 1 ≽ H 2 and H 2 ≽ H 1 , then we write H 1 ∼ H 2 , and the graphs H 1 and H 2 are said to be matching equivalent. Obviously, H 1 ≽ H 2 ⇒ ME ( H 1 ) ≥ ME ( H 2 ) , and H 1 ≻ H 2 ⇒ ME ( H 1 ) > ME ( H 2 ) .

From [4], we know that ME ( H ) = E ( H ) when H is a tree. Since there have been many research results on graph energy for trees, the study on matching energy should be focused more on cycle-containing connected graphs. As well as graph energy, the research on minimum or maximum matching energy over a variety of class of connected graphs (with fixed order n ) has attracted mathematicians into this field. For instance, the class for connected graphs with the size m satisfying m = n (i.e., unicyclic graphs) [4], m = n + 1 (i.e., bicyclic graphs) [6], m = n + 2 (i.e., tricyclic graphs) [7], and n + 2 ≤ m ≤ 2 n − 4 [8–10]; the class of t -apex trees [11]; the class of graphs with fixed connectivity [12], edge connectivity [13], chromatic number [12], and clique number [14]; the class of unicyclic graphs with given girth [14]; and the class of unicyclic, bicyclic, and tricyclic graphs with a give diameter [15,16]. Other results on matching energy, the reader may refer to the review in [5] and papers [17–39]. See [40–44], for some recent extremal results on other topological indices of unicyclic graphs.

Denote by P n , C n , and S n the path, the cycle, and the star of order n , respectively. For g ≥ 3 , let C g ( S l 1 , S l 2 , … , S l g ) be the unicyclic graph with girth g obtained from the cycle C g = v 1 v 2 … v g v 1 by attaching l i − 1 pendent edges to the vertex v i for i = 1 , 2 , … , g . For simplicity, C g ( S n − g + 1 , S 1 , … , S 1 ) is also written as S n g . Denote by P n g the unicyclic graph obtained by joining an edge between a vertex of the cycle C g and one end vertex of the path P n − g . The graphs C 3 ( S n − 4 , S 2 , S 2 ) , C 3 ( S n − 3 , S 2 , S 1 ) , S n g , and P n g are shown in Figure 1.

$Figure 1 The graphs C 3 ( S n ‒ 4 , S 2 , S 2 ) {C}_{3}\left({S}_{n‒4},{S}_{2},{S}_{2}) , C 3 ( S n ‒ 3 , S 2 , S 1 ) {C}_{3}\left({S}_{n‒3},{S}_{2},{S}_{1}) , S n g {S}_{n}^{g} , and P n g {P}_{n}^{g} .$

Figure 1

The graphs C 3 ( S n ‒ 4 , S 2 , S 2 ) , C 3 ( S n ‒ 3 , S 2 , S 1 ) , S n g , and P n g .

Let T n , l + be the set of all trees of order n satisfying that the number of degree two is at least l . We use U n , t to denote the class of all unicyclic graphs with n vertices and exactly t vertices of degree two. Clearly, t ≠ n − 1 and C n is the unique graph in U n , n . In [45], the authors determined the trees in T n , l + attaining minimum energy. Motivated by this, we investigate the minimum matching energy for unicylic graphs in U n , t . The following is the main result obtained in this research.

Theorem 1.1

Let G ∈ U n , t with n ≥ 3 , 0 ≤ t ≤ n − 2 .

if n ≥ 6 and t = 0 , then G ≽ C 3 ( S n − 4 , S 2 , S 2 ) , with equality if and only if G ≅ C 3 ( S n − 4 , S 2 , S 2 ) ;
if n ≥ 5 and t = 1 , then G ≽ C 3 ( S n − 3 , S 2 , S 1 ) , with equality if and only if G ≅ C 3 ( S n − 3 , S 2 , S 1 ) ;
if n ≥ 4 and t = n − 2 , then G ≽ P n 3 ( ∼ S n n − 1 ), with equality if and only if G ≅ P n 3 or S n n − 1 ( P n 3 ≅ S n n − 1 only when n = 4 );
if 2 ≤ t ≤ n − 3 , then G ≽ S n t + 1 , with equality if and only if G ≅ S n t + 1 .

By Theorem 1.1, the minimum graphs in U n , t with respect to matching energy are characterized. Recall that the Hosoya index of a graph G of order n is defined as Z ( G ) = ∑ j = 0 ⌊ n ⁄ 2 ⌋ m ( G , j ) . Clearly, G 1 ≽ G 2 ⇒ Z ( G 1 ) ≥ Z ( G 2 ) , and G 1 ≻ G 2 ⇒ Z ( H 1 ) > Z ( H 2 ) . Therefore by Theorem 1.1, the minimum graphs in U n , t with respect to Hosoya index are also determined.

2 Preliminaries

We will present some known results in this section.

Lemma 2.1

[46] Let e = x y be any edge of a graph H. Then we have

(3) m ( H , j ) = m ( H − x y , j ) + m ( H − x − y , j − 1 ) .

Combining Lemma 2.1 and the definition of quasi-order, we can directly deduce the following lemma.

Lemma 2.2

For arbitrary two graphs H and H ′ , let e = x y and e ′ = x ′ y ′ be two edges of H and H ′ , respectively.

If H − e ≽ H ′ − e ′ and H − x − y ≽ H ′ − x ′ − y ′ , then H ≽ H ′ ;
If H − e ≻ H ′ − e ′ and H − x − y ≽ H ′ − x ′ − y ′ ; or H − e ≽ H ′ − e ′ and H − x − y ≻ H ′ − x ′ − y ′ , then H ≻ H ′ .

Lemma 2.3

[47] If G 1 ≻ G 2 , then for any graph H , G 1 ∪ H ≻ G 2 ∪ H .

Let T n , t be the class of all trees with n ≥ 2 vertices and t vertices of degree two, where 0 ≤ t ≤ n − 2 and t ≠ n − 3 . For 0 ≤ t ≤ n − 2 , let T n , t denote the tree obtained by attaching n − t − 2 pendent edges to the vertex v 1 of the path P t + 2 = v 1 v 2 … v t + 2 , see Figure 2. The tree T n , t is called a rooted broom with the vertex v 1 as its root. It is clear that T n , 0 is the star S n , T n , n − 2 ≅ T n , n − 3 is the path P n , where the root of T n , n − 2 is a pendent vertex and the root of T n , n − 3 is a vertex of degree two. Hence, T n , t ∈ T n , t if t ∈ { 0 , 1 , … , n − 4 , n − 2 } and T n , t ∈ T n , t + 1 if t = n − 3 .

$Figure 2 The rooted broom T n , t {T}_{n,t} with the vertex v 1 {v}_{1} as its root.$

Figure 2

The rooted broom T n , t with the vertex v 1 as its root.

Lemma 2.4

[45] Let T be a tree in T n , t + with n ≥ t + 2 ≥ 2 . If T ≇ T n , t , then T ≻ T n , t ( and so E ( T ) > E ( T n , t ) ) .

Lemma 2.5

[48] Let H be a connected nontrivial graph, and x ∈ V ( H ) . Denote by P k ( x t , x ) H the graph obtained by attaching the vertex x to the vertex x t of the path P k = x 1 x 2 … x k . Then for integer j ∈ { 1 , 3 , … , k − 2 , k } , P k ( x j , x ) H ≻ P k ( x 2 , x ) H .

For a vertex y in a graph H , the degree and open neighborhood of y are written as d H ( y ) and N H ( y ) , respectively.

Lemma 2.6

[34,49] Let H be a connected graph of order n ≥ 4 and x y be an edge of H. From H, we can obtain a graph H ′ by identifying the two vertices x and y and then adding a pendent edge at the common vertex x. If xy is not a pendent edge of H and N H ( x ) ∩ N H ( y ) = ∅ , then H ≻ H ′ .

Lemma 2.7

[49] If H 1 is a subgraph of a graph H, then H ≽ H 1 . Moreover, if H 1 is a proper subgraph of a connected graph H, then H ≻ H 1 .

Lemma 2.8

[48] Let G be any graph, and e = x y be an edge satisfying G − x ≅ G − y . Denote by G ( x , y ) ( r , s ) the graph got from G by adding new r and s pendent edges at the vertices x and y, respectively. Then

G ( x , y ) ( 0 , l ) ≺ G ( x , y ) ( 1 , l − 1 ) ≺ … ≺ G ( x , y ) l 2 , l − l 2 .

Lemma 2.9

[48] Let n and j be two integers with 2 ≤ j ≤ n − 2 . Then P j ∪ P n − j ≻ P 1 ∪ P n − 1 .

3 Some transformations

In Section 3, we will introduce several transformations that decrease the matching energy of graphs in T n , t or U n , t .

Let T ∈ T n , t be the tree depicted in Figure 3, where N T ( u ) = { u 1 , u 2 , … , u d ( u ) } , d ( u ) ≥ 2 , and T j ( j = 1 , 2 , … , d ( u ) ) be the rooted tree with u j as its root in T − u . Suppose that T i ∈ T n i , t i and d T ( u i ) ≥ 4 . We can obtain a tree T ′ , from T , by replacing T i with the rooted broom T n i , t i (i.e., by first deleting the vertices in V ( T i ) \ { u i } , and then identifying the root of T n i , t i with the vertex u i , see Figure 3). We say the transformation from T to T ′ the α 1 -transformation. Since d T ( u i ) ≥ 4 , we have d T i ( u i ) ≥ 3 , and there exist at least three pendent vertices in T i . Therefore, n i ≥ t i + 4 , and d T ′ ( u i ) = n i − t i ≥ 4 . Thus, T ′ ∈ T n , t .

$Figure 3 T T and T ′ T^{\prime} in α 1 {\alpha }_{1} -transformation, where d T ( u i ) ≥ 4 {d}_{T}\left({u}_{i})\ge 4 .$

Figure 3

T and T ′ in α 1 -transformation, where d T ( u i ) ≥ 4 .

Lemma 3.1

Suppose that T ′ can be obtained from T by α 1 -transformation and T ≇ T ′ . Then T ≻ T ′ .

Proof

Without loss of generality, we assume that i = 1 . We apply induction on d ( u ) . If d ( u ) = 2 , by equation (3),

m ( T , l ) = m ( T − u u 2 , l ) + m ( T − u − u 2 , l − 1 ) = m ( T 2 ∪ ( T − V ( T 2 ) ) , l ) + m ( ( T 2 − u 2 ) ∪ T 1 , l − 1 ) , m ( T ′ , l ) = m ( T ′ − u u 2 , l ) + m ( T ′ − u − u 2 , l − 1 ) = m ( T 2 ∪ T n 1 + 1 , t 1 , l ) + m ( ( T 2 − u 2 ) ∪ T n 1 , t 1 , l − 1 ) .

Note that T 1 ∈ T n 1 , t 1 and n 1 ≥ t 1 + 4 . We have T − V ( T 2 ) ∈ T n 1 + 1 , t 1 . Because T ≇ T ′ and d T 1 ( u 1 ) ≥ 3 , we have T 1 ≇ T n 1 , t 1 . By Lemma 2.4, we have T 1 ≻ T n 1 , t 1 and T − V ( T 2 ) ≽ T n 1 + 1 , t 1 . Therefore, T − u − u 2 ≻ T ′ − u − u 2 and T − u u 2 ≽ T ′ − u u 2 by Lemma 2.3. Thus, T ≻ T ′ by Lemma 2.2.

Suppose the result holds when 2 ≤ d ( u ) < p , and consider the case d ( u ) = p . By equation (3),

m ( T , l ) = m ( T − u u p , l ) + m ( T − u − u p , l − 1 ) = m ( T 11 ∪ T p , l ) + m ( T 1 ∪ T 2 ∪ … ∪ T p − 1 ∪ ( T p − u p ) , l − 1 ) , m ( T ′ , l ) = m ( T ′ − u u p , l ) + m ( T ′ − u − u p , l − 1 ) = m ( T 11 ′ ∪ T p , l ) + m ( T n 1 , t 1 ∪ T 2 ∪ … ∪ T p − 1 ∪ ( T p − u p ) , l − 1 ) ,

where T 11 and T 11 ′ are the components containing the vertex u in T − u u p and T ′ − u u p , respectively. Since T ≇ T ′ , we have T 1 ≇ T n 1 , t 1 and T 11 ≇ T 11 ′ . By Lemma 2.4, we have T 1 ≻ T n 1 , t 1 . It is easy to see that T 11 ′ can be obtained from T 11 by α 1 -transformation. Therefore, we have T 11 ≻ T 11 ′ by the induction hypothesis. It follows from Lemmas 2.3 and 2.2 that T ≻ T ′ .

The lemma is thus proved.□

Let T ∈ T n , t be the tree depicted in Figure 4, where N T ( u ) = { u 1 , u 2 , … , u d ( u ) } , d ( u ) ≥ 2 , and T j ( j = 1 , 2 , … , d ( u ) ) be the rooted tree with u j as its root in T − u . Suppose that T i ∈ T n i , t i and d T ( u i ) = 3 ( d T i ( u i ) = 2 ). Let T ′ be the tree obtained from T by replacing T i with the rooted broom T n i , t i − 1 (Figure 4). We say the transformation from T to T ′ the α 2 -transformation. Note that T i ∈ T n i , t i and d T i ( u i ) = 2 implies 1 ≤ t i ≤ n i − 2 and t i ≠ n i − 3 . If t i = n i − 2 , then P n i ≅ T n i , t i − 1 ∈ T n i , t i , d T ′ ( u i ) = 3 and T ′ ∈ T n , t . Otherwise, we have 1 ≤ t i ≤ n i − 4 , and so, d T ′ ( u i ) ≥ 5 and T n i , t i − 1 ∈ T n i , t i − 1 . Therefore, T ′ ∈ T n , t .

$Figure 4 T T and T ′ T^{\prime} in α 2 {\alpha }_{2} -transformation, where d T ( u i ) = 3 {d}_{T}\left({u}_{i})=3 .$

Figure 4

T and T ′ in α 2 -transformation, where d T ( u i ) = 3 .

Lemma 3.2

Suppose that T ′ can be obtained from T by α 2 -transformation and T ≇ T ′ . Then T ≻ T ′ .

Proof

Without loss of generality, we assume that i = 1 . Note that d T 1 ( u 1 ) = 2 and t 1 ≥ 1 . First, we suppose that t 1 = n 1 − 2 . Then T 1 ≅ P n 1 , T n 1 , t 1 − 1 ≅ P n 1 , and one end of T n 1 , t 1 − 1 is a neighbor of u 1 in T ′ . By Lemma 2.5, T ≻ T ′ if T ≇ T ′ . We then suppose that 1 ≤ t 1 ≤ n 1 − 4 and apply induction on d ( u ) . If d ( u ) = 2 , by equation (3),

m ( T , l ) = m ( T − u u 2 , l ) + m ( T − u − u 2 , l − 1 ) = m ( T 2 ∪ ( T − V ( T 2 ) ) , l ) + m ( ( T 2 − u 2 ) ∪ T 1 , l − 1 ) , m ( T ′ , l ) = m ( T ′ − u u 2 , l ) + m ( T ′ − u − u 2 , l − 1 ) = m ( T 2 ∪ T n 1 + 1 , t 1 − 1 , l ) + m ( ( T 2 − u 2 ) ∪ T n 1 , t 1 − 1 , l − 1 ) .

Note that T 1 ∈ T n 1 , t 1 , d T 1 ( u 1 ) = 2 and n 1 ≥ t 1 + 4 . We have T − V ( T 2 ) ∈ T n 1 + 1 , t 1 − 1 . By considering the degrees of u 1 in two trees respectively, we obtain T − V ( T 2 ) ≇ T n 1 + 1 , t 1 − 1 , and so, T − V ( T 2 ) ≻ T n 1 + 1 , t 1 − 1 by Lemma 2.4. It follows from Lemmas 2.4 and 2.6 that T 1 ≽ T n 1 , t 1 ≻ T n 1 , t 1 − 1 . Therefore, T − u u 2 ≻ T ′ − u u 2 and T − u − u 2 ≻ T ′ − u − u 2 by Lemma 2.3, and so, T ≻ T ′ by Lemma 2.2.

Now suppose the result holds when 2 ≤ d ( u ) < p , and consider the case d ( u ) = p . By equation (3),

m ( T , l ) = m ( T − u u p , l ) + m ( T − u − u p , l − 1 ) = m ( T 11 ∪ T p , l ) + m ( T 1 ∪ T 2 ∪ … ∪ T p − 1 ∪ ( T p − u p ) , l − 1 ) , m ( T ′ , l ) = m ( T ′ − u u p , l ) + m ( T ′ − u − u p , l − 1 ) = m ( T 11 ′ ∪ T p , l ) + m ( T n 1 , t 1 − 1 ∪ T 2 ∪ … ∪ T p − 1 ∪ ( T p − u p ) , l − 1 ) ,

where T 11 and T 11 ′ are the components containing the vertex u in T − u u p and T ′ − u u p , respectively. Since T 1 ∈ T n 1 , t 1 , it follows from Lemmas 2.4 and 2.6 that T 1 ≽ T n 1 , t 1 ≻ T n 1 , t 1 − 1 . By considering the degrees of u 1 in two trees respectively, we obtain T 11 ≇ T 11 ′ . It is easy to see that T 11 ′ can be obtained from T 11 by α 2 -transformation. Therefore, we have T 11 ≻ T 11 ′ by the induction hypothesis. It follows from Lemmas 2.3 and 2.2 that T ≻ T ′ .

The lemma is thus proved.□

Denote by U n , t g the subset of graphs with girth g in U n , t . Let C g ( T 1 , T 2 , … , T g ) be the graph obtained from the cycle C g = v 1 v 2 … v g v 1 and g rooted trees T 1 , T 2 , … , T g by identifying the root of T i with the vertex v i in the cycle.

Let G 1 = C g ( T 1 , T 2 , … , T g ) be a unicyclic graph, and T i ′ be a rooted tree. Denote by G the unicyclic graph obtained from G 1 , T i ′ and the path u 1 … u l ( l ≥ 2 ) by identifying the root of T i ′ and v i with the vertices u l and u 1 respectively, see Figure 5. Suppose that G ∈ U n , t g , d G ( u l ) ≥ 3 and T i ′ ∈ T n i ′ , t i ′ . If d G ( u l ) ≥ 4 ( d G ( u l ) = 3 ), and G ′ ( G ″ ) is the unicyclic graph obtained from G by replacing T i ′ with the rooted broom T n i ′ , t i ′ ( T n i ′ , t i ′ − 1 ), see Figure 5 (Figure 6), we say the transformation from G to G ′ ( G ″ ) the β 1 -transformation ( β 2 -transformation). Clearly, G ′ , G ″ ∈ U n , t g .

$Figure 5 G G and G ′ G^{\prime} in β 1 {\beta }_{1} -transformation, where d G ( u l ) ≥ 4 {d}_{G}\left({u}_{l})\ge 4 .$

Figure 5

G and G ′ in β 1 -transformation, where d G ( u l ) ≥ 4 .

$Figure 6 G G and G ″ {G}^{^{\prime\prime} } in β 2 {\beta }_{2} -transformation, where d G ( u l ) = 3 {d}_{G}\left({u}_{l})=3 .$

Figure 6

G and G ″ in β 2 -transformation, where d G ( u l ) = 3 .

Lemma 3.3

Suppose that G ′ can be obtained from G by β 1 -transformation and G ≇ G ′ . Then G ≻ G ′ .

Proof

Without loss of generality, we assume that i = 1 . We divide the following two cases to prove the result.

Case 1. l ≥ 3 . Let H 1 and H 2 be the components containing the vertices u l − 2 and u l − 1 in G − u l − 2 u l − 1 , respectively. By Lemma 2.1,

m ( G , j ) = m ( G − u l − 2 u l − 1 , j ) + m ( G − u l − 2 − u l − 1 , j − 1 ) = m ( H 1 ∪ H 2 , j ) + m ( ( H 1 − u l − 2 ) ∪ T 1 ′ , j − 1 ) , m ( G ′ , j ) = m ( G ′ − u l − 2 u l − 1 , j ) + m ( G ′ − u l − 2 − u l − 1 , j − 1 ) = m ( H 1 ∪ T n 1 ′ + 1 , t 1 ′ , j ) + m ( ( H 1 − u l − 2 ) ∪ T n 1 ′ , t 1 ′ , j − 1 ) .

Since T 1 ′ ∈ T n 1 ′ , t 1 ′ and d T 1 ′ ( u l ) ≥ 3 , we have n 1 ′ ≥ t 1 ′ + 4 . Hence, T n 1 ′ , t 1 ′ ∈ T n 1 ′ , t 1 ′ and H 2 ∈ T n 1 ′ + 1 , t 1 ′ . Because G ≇ G ′ and d T 1 ′ ( u l ) ≥ 3 , we can conclude that T 1 ′ ≇ T n 1 ′ , t 1 ′ and H 2 ≇ T n 1 ′ + 1 , t 1 ′ . By Lemma 2.4, we have T 1 ′ ≻ T n 1 ′ , t 1 ′ and H 2 ≻ T n 1 ′ + 1 , t 1 ′ . Therefore, G − u l − 2 − u l − 1 ≻ G ′ − u l − 2 − u l − 1 and G − u l − 2 u l − 1 ≻ G ′ − u l − 2 u l − 1 by Lemma 2.3. Thus, G ≻ G ′ by Lemma 2.2.

Case 2. l = 2 . Denote H = G 1 − v 1 − v 2 . By Lemma 2.1,

m ( G , j ) = m ( G − v 1 v 2 , j ) + m ( G − v 1 − v 2 , j − 1 ) = m ( G − v 1 v 2 , j ) + m ( H ∪ T 1 ′ , j − 1 ) , m ( G ′ , j ) = m ( G ′ − v 1 v 2 , j ) + m ( G ′ − v 1 − v 2 , j − 1 ) = m ( G ′ − v 1 v 2 , j ) + m ( H ∪ T n 1 ′ , t 1 ′ , j − 1 ) .

Since G ≇ G ′ , we have G − v 1 v 2 ≇ G ′ − v 1 v 2 and T 1 ′ ≇ T n 1 ′ , t 1 ′ . Note that d G ( u 2 ) ≥ 4 , it is easy to see that G ′ − v 1 v 2 can be obtained from G − v 1 v 2 by α 1 -transformation. Therefore, G − v 1 v 2 ≻ G ′ − v 1 v 2 by Lemma 3.1. Note that T 1 ′ , T n 1 ′ , t 1 ′ ∈ T n 1 ′ , t 1 ′ . We have T 1 ′ ≻ T n 1 ′ , t 1 ′ by Lemma 2.4, and so, G − v 1 − v 2 ≻ G ′ − v 1 − v 2 by Lemma 2.3. Therefore, G ≻ G ′ by Lemma 2.2.

The lemma is thus proved.□

Lemma 3.4

Suppose that G ″ can be obtained from G by β 2 -transformation and G ≇ G ″ . Then G ≻ G ″ .

Proof

Without loss of generality, we assume that i = 1 . First, we suppose that t 1 ′ = n 1 ′ − 2 . Then T 1 ′ ≅ P n 1 , T n 1 ′ , t 1 ′ − 1 ≅ P n 1 , and one end vertex of T n 1 , t 1 − 1 is a neighbor of u l . By Lemma 2.5, G ≻ G ″ if G ≇ G ″ . We then suppose that t 1 ′ ≤ n 1 ′ − 4 and divide the following two cases to prove the result.

Case 1. l ≥ 3 . Let H 1 and H 2 be the components containing the vertices u l − 2 and u l − 1 in G − u l − 2 u l − 1 , respectively. By Lemma 2.1,

m ( G , j ) = m ( G − u l − 2 u l − 1 , j ) + m ( G − u l − 2 − u l − 1 , j − 1 ) = m ( H 1 ∪ H 2 , j ) + m ( ( H 1 − u l − 2 ) ∪ T 1 ′ , j − 1 ) , m ( G ″ , j ) = m ( G ″ − u l − 2 u l − 1 , j ) + m ( G ″ − u l − 2 − u l − 1 , j − 1 ) = m ( H 1 ∪ T n 1 ′ + 1 , t 1 ′ − 1 , j ) + m ( ( H 1 − u l − 2 ) ∪ T n 1 ′ , t 1 ′ − 1 , j − 1 ) .

Note that T 1 ′ ∈ T n 1 ′ , t 1 ′ , d T 1 ′ ( u l ) = 2 and n 1 ′ ≥ t 1 ′ + 4 . We have H 2 ∈ T n 1 ′ + 1 , t 1 ′ − 1 . By considering the degrees of u l in two trees respectively, we obtain H 2 ≇ T n 1 ′ + 1 , t 1 ′ − 1 , and so, H 2 ≻ T n 1 ′ + 1 , t 1 ′ − 1 by Lemma 2.4. It follows from Lemmas 2.4 and 2.6 that T 1 ′ ≽ T n 1 ′ , t 1 ′ ≻ T n 1 ′ , t 1 ′ − 1 . Therefore, G − u l − 2 u l − 1 ≻ G ″ − u l − 2 u l − 1 and G − u l − 2 − u l − 1 ≻ G ″ − u l − 2 − u l − 1 by Lemma 2.3, and so, G ≻ G ″ by Lemma 2.2.

Case 2. l = 2 . Denote H = G 1 − v 1 − v 2 . By Lemma 2.1,

m ( G , j ) = m ( G − v 1 v 2 , j ) + m ( G − v 1 − v 2 , j − 1 ) = m ( G − v 1 v 2 , j ) + m ( H ∪ T 1 ′ , j − 1 ) , m ( G ″ , j ) = m ( G ″ − v 1 v 2 , j ) + m ( G ″ − v 1 − v 2 , j − 1 ) = m ( G ″ − v 1 v 2 , j ) + m ( H ∪ T n 1 ′ , t 1 ′ − 1 , j − 1 ) .

Since G ≇ G ″ , we obtain G − v 1 v 2 ≇ G ″ − v 1 v 2 . Note that d G ( u 2 ) = 3 , it is easy to see that G ″ − v 1 v 2 can be obtained from G − v 1 v 2 by α 2 -transformation. Therefore, G − v 1 v 2 ≻ G ″ − v 1 v 2 by Lemma 3.2. Note that T 1 ′ ∈ T n 1 ′ , t 1 ′ . It follows from Lemmas 2.4 and 2.6 that T 1 ′ ≽ T n 1 ′ , t 1 ′ ≻ T n 1 ′ , t 1 ′ − 1 . Therefore, G − v 1 v 2 ≻ G ″ − v 1 v 2 and G − v 1 − v 2 ≻ G ″ − v 1 − v 2 by Lemma 2.3, and so G ≻ G ″ by Lemma 2.2.

The lemma is thus proved.□

Let T ∈ T n , t be a tree with the structure in Figure 7, where N T ( u ) = { v 1 , v 2 , … , v d ( u ) } , T j ( j = 1 , 2 , … , d ( u ) ) be the component containing v j in T − u , d ( u ) ≥ 2 , l ≥ 0 , u 1 = v i , s ≥ 2 and T i ≅ T l + s + 1 , l − 1 when l ≥ 1 , while T i ≅ S s + 1 when l = 0 . From T , we can obtain a tree T ′ by deleting s pendent vertices, which are neighbors of u l + 1 and attaching s pendent edges to the vertex u (Figure 7). We say the transformation from T to T ′ the α 3 -transformation. Clearly, T ′ ∈ T n , t when d ( u ) ≥ 3 and T ′ ∈ T n , t − 1 when d ( u ) = 2 .

$Figure 7 T T and T ′ T^{\prime} in α 3 {\alpha }_{3} -transformation.$

Figure 7

T and T ′ in α 3 -transformation.

Lemma 3.5

If T ′ can be obtained from T by α 3 -transformation, then T ≻ T ′ .

Proof

Without loss of generality, we assume that i = 1 . We apply induction on d ( u ) . If d ( u ) = 2 , by Lemma 2.1,

m ( T , r ) = m ( T − u v 2 , r ) + m ( T − u − v 2 , r − 1 ) = m ( T 2 ∪ T l + s + 2 , l , r ) + m ( ( T 2 − v 2 ) ∪ T 1 , r − 1 ) , m ( T ′ , r ) = m ( T ′ − u v 2 , r ) + m ( T ′ − u − v 2 , r − 1 ) = m ( T 2 ∪ T l + s + 2 , l , r ) + m ( ( T 2 − v 2 ) ∪ P l + 1 ∪ s K 1 , r − 1 ) .

Note that T 1 ≅ T l + s + 1 , l − 1 when l ≥ 1 while T 1 ≅ S s + 1 when l = 0 . Hence, P l + 1 ∪ s K 1 is a proper subgraph of T 1 , and so T 1 ≻ P l + 1 ∪ s K 1 by Lemma 2.7. Therefore, T − u − v 2 ≻ T ′ − u − v 2 by Lemma 2.3. Since T − u v 2 ≅ T ′ − u v 2 implies T − u v 2 ≽ T ′ − u v 2 , we have T ≻ T ′ by Lemma 2.2.

Now suppose the result holds when 2 ≤ d ( u ) < p , and consider the case d ( u ) = p . By Lemma 2.1,

m ( T , r ) = m ( T − u v p , r ) + m ( T − u − v p , r − 1 ) = m ( T 11 ∪ T p , r ) + m ( T 1 ∪ T 2 ∪ … ∪ T p − 1 ∪ ( T p − v p ) , r − 1 ) , m ( T ′ , r ) = m ( T ′ − u v p , r ) + m ( T ′ − u − v p , r − 1 ) = m ( T 11 ′ ∪ T p , r ) + m ( P l + 1 ∪ s K 1 ∪ T 2 ∪ … ∪ T p − 1 ∪ ( T p − v p ) , r − 1 ) ,

where T 11 and T 11 ′ are the components containing the vertex u in T − u v p and T ′ − u v p , respectively. Notice that P l + 1 ∪ s K 1 is a proper subgraph of T 1 . We have T 1 ≻ P l + 1 ∪ s K 1 by Lemma 2.7. It is easy to see that T 11 ′ can be obtained from T 11 by α 3 -transformation. Therefore, we have T 11 ≻ T 11 ′ by the induction hypothesis. It follows from Lemmas 2.3 and 2.2 that T ≻ T ′ .

The lemma is thus proved.□

Let G 1 = C g ( T 1 , T 2 , … , T g ) be a unicyclic graph, and T be the tree obtained from the path u 0 u 1 … u l + r + 1 ( l ≥ 0 , r ≥ 1 ) by attaching s ( s ≥ 1 ) pendent edges to the vertex u l + 1 . By identifying the vertex v i with the vertex u 0 , we can obtain a unicyclic graph G , see Figure 8. Suppose G ∈ U n , t g . Let G ′ be the unicyclic graph obtained from G by deleting s + 1 vertices in { u l + r + 1 } ∪ ( N G ( u l + 1 ) \ { u l , u l + 2 } ) and attaching s + 1 pendent edges to the vertex v i (Figure 8). We say the transformation from G to G ′ the β 3 -transformation. Obviously, G ′ ∈ U n , t g .

$Figure 8 G G and G ′ G^{\prime} in β 3 {\beta }_{3} -transformation.$

Figure 8

G and G ′ in β 3 -transformation.

Lemma 3.6

If G ′ can be obtained from G by β 3 -transformation, then G ≻ G ′ .

Proof

Without loss of generality, we assume that i = 1 . We divide the following two cases to prove the result.

Case 1. r = 1 . By Lemma 2.1,

m ( G , j ) = m ( G − v 1 v 2 , j ) + m ( G − v 1 − v 2 , j − 1 ) = m ( G − v 1 v 2 , j ) + m ( ( G 1 − v 1 − v 2 ) ∪ ( T − u 0 ) , j − 1 ) , m ( G ′ , j ) = m ( G ′ − v 1 v 2 , j ) + m ( G ′ − v 1 − v 2 , j − 1 ) = m ( G ′ − v 1 v 2 , j ) + m ( ( G 1 − v 1 − v 2 ) ∪ ( s + 1 ) K 1 ∪ P l + 1 , j − 1 ) .

Note that G ′ − v 1 v 2 can be obtained from G − v 1 v 2 by α 3 -transformation. According to Lemma 3.5, G − v 1 v 2 ≻ G ′ − v 1 v 2 . Because ( s + 1 ) K 1 ∪ P l + 1 is a proper subgraph of T − u 0 , we obtain T − u 0 ≻ ( s + 1 ) K 1 ∪ P l + 1 by Lemma 2.7, and so G − v 1 − v 2 ≻ G ′ − v 1 − v 2 by Lemma 2.3. According to Lemma 2.2, G ≻ G ′ .

Case 2. r ≥ 2 . Denote H = G − { u l + 3 , … , u l + r + 1 } , H ′ = G ′ − { u l + 2 , … , u l + r } , K = H − u l + 2 , and K ′ = H ′ − u l + 1 . By Lemma 2.1,

m ( G , j ) = m ( G − u l + 2 u l + 3 , j ) + m ( G − u l + 2 − u l + 3 , j − 1 ) = m ( H ∪ P r − 1 , j ) + m ( K ∪ P r − 2 , j − 1 ) , m ( G ′ , j ) = m ( G ′ − u l + 1 u l + 2 , j ) + m ( G ′ − u l + 1 − u l + 2 , j − 1 ) = m ( H ′ ∪ P r − 1 , j ) + m ( K ′ ∪ P r − 2 , j − 1 ) .

Note that H ′ can be obtained from H by β 3 -transformation with the condition r = 1 . Therefore, H ≻ H ′ . Similarly, denote by K ″ the graph obtained from K by β 3 -transformation (with the condition r = 1 ). We then have K ≻ K ″ . Furthermore according to Lemma 2.6, we can conclude that K ″ ≻ K ′ . Thus, K ≻ K ′ . Therefore, the result G ≻ G ′ holds by Lemmas 2.3 and 2.2.

The proof is thus finished.□

Let G = C g ( T 1 , T 2 , … , T g ) ∈ U n , t g be a unicyclic graph (with the cycle C g = v 1 v 2 … v g v 1 ). Suppose d G ( v i ) ≥ 3 and T i ∈ T n i , t i . From G , we can obtain a unicyclic graph G ′ by replacing T i with the rooted broom T n i , t i when d G ( v i ) ≠ 4 or T n i , t i − 1 when d G ( v i ) = 4 . We say the transformation from G to G ′ the β 4 -transformation. Obviously, G ′ ∈ U n , t g .

Lemma 3.7

If G ′ can be obtained from G by β 4 -transformation and G ≇ G ′ , then G ≻ G ′ .

Proof

Suppose that G ∈ U n , t g , T i ∈ T n i , t i , d G ( v i ) = p + 2 ( p ≥ 1 ) , and N G ( v i ) = { v i − 1 , v i + 1 , x 1 , … , x p } . Let T i j ( j = 1 , … , p ) denote the rooted tree with x j as its root in G − v i . We divide the following two cases to prove the result.

Case 1. For each j ∈ { 1 , … , p } , T i j is a rooted path.

Suppose T i j ≅ P n i j , then n i j ≥ 1 and n i 1 + ⋯ + n i p = n i − 1 .

Case 1.1. n i 1 = ⋯ = n i p = 1 . That is, T i is a star S n i with v i as its center. Then we have that t i = 0 when d G ( v i ) ≠ 4 , and t i = 1 when d G ( v i ) = 4 . Since S n i ≅ T n i , 0 , it is clear that G ≅ G ′ .

Case 1.2. p = 1 and n i 1 ≥ 2 . That is, T i is a path P n i with v i as one end vertex. Then we have that d G ( v i ) = 3 , t i = n i − 2 and T i ≅ T n i , t i . Clearly, G ≅ G ′ .

Case 1.3. p ≥ 2 and there exists exactly one k such that n i k ≥ 2 . It is easy to see that T i ≅ T n i , t i when d G ( v i ) ≠ 4 or T i ≅ T n i , t i − 1 when d G ( v i ) = 4 . At any case, we have G ≅ G ′ .

Case 1.4. There exist r ≥ 2 numbers in { n i 1 , … , n i p } with at least 2. Suppose that n i 1 , … , n i r ≥ 2 . Note that T i 1 ∪ T i 2 ∪ v i x 1 ∪ v i x 2 is a path of G with length at least 4. Let H be the graph obtained from G by replacing T i 1 and T i 2 with P 1 and P n 1 + n 2 − 1 ( d H ( x 2 ) = 2 ) , respectively. Clearly, H ∈ U n , t g . Hence, G ≻ H by Lemma 2.5. If r > 2 , we can still carry out the aforementioned procedure repeatedly r − 2 times and finally obtain G ′ . Therefore, G ≻ H ≻ … ≻ G ′ .

Case 2. There exists at least one k ∈ { 1 , … , p } such that T i k is not a rooted path.

Suppose that T i 1 , … , T i r ( r ≥ 1 ) are not rooted paths. Since T i 1 is not a path, there exists a path v i u 2 ( = x 1 ) … u l such that u 2 , … , u l ∈ V ( T i 1 ) , l ≥ 2 , d G ( u 2 ) = ⋯ = d G ( u l − 1 ) = 2 and d G ( u l ) ≥ 3 . Let G 1 be the graph obtained from G by β 1 -transformation (or β 2 -transformation, respectively) if d G ( u l ) ≥ 4 (or d G ( u l ) = 3 , respectively). Furthermore, let G 2 be the graph obtained from G 1 by β 3 -transformation. Clearly, G 1 , G 2 ∈ U n , t g . By combining Lemmas 3.3, 3.4, and 3.6, we can conclude that G ≻ G 2 since G ≇ G 2 . If r ≥ 2 , we can still carry out the aforementioned procedure repeatedly r − 1 times and finally obtain a graph H , where H is of the form of the case 1. Therefore, G ≻ H . If H ≇ G ′ , then by the argument of the case 1.4, we can obtain H ≻ G ′ . Therefore, G ≻ G ′ .

The proof is thus finished.□

4 Proof of Theorem 1.1

We assume that G ∈ U n , t g with the unique cycle C g = v 1 v 2 … v g v 1 for some g ≥ 3 . Note that t ≠ n , we obtain g ≠ n . By applying β 4 -transformation on each vertex v i with degree at least 3 repeatedly, we can finally obtain a graph G 1 = C g ( T 1 , T 2 , … , T g ) such that each T i is a star or a broom. Let n i be the order of T i . Then n 1 + ⋯ + n g = n . According to Lemma 3.7, we obtain G ≻ G 1 if G ≇ G 1 .

(1) If t = 0 , then T i is a star with n i ≥ 2 for i = 1 , 2 , … , g . That is, G 1 = C g ( S n 1 , S n 2 , … , S n g ) . If g > 3 , then by choosing the edge v 1 v g and applying Lemma 2.6 to G 1 , we obtain the graph G 2 = C g − 1 ( S n 1 + n g , S n 2 , … , S n g − 1 ) such that G 1 ≻ G 2 . By carrying out the aforementioned procedure repeatedly, we can finally obtain the graph G 3 = C 3 ( S n 1 + n 4 + … + n g , S n 2 , S n 3 ) such that G ≻ G 3 . Thus, G 1 ≻ G 3 if G 1 ≇ G 3 .

We may assume that n 1 + n 4 + ⋯ + n g = n 1 ′ ≥ n 2 ≥ n 3 . If n 2 > 2 , then by Lemma 2.8, we have G 3 = C 3 ( S n 1 ′ , S n 2 , S n 3 ) ≽ C 3 ( S n 1 ′ , S n 2 + n 3 − 2 , S 2 ) ≻ C 3 ( S n − 4 , S 2 , S 2 ) . If n 2 = 2 , then n 3 = 2 , and so G 3 ≅ C 3 ( S n − 4 , S 2 , S 2 ) . Thus, G 3 ≻ C 3 ( S n − 4 , S 2 , S 2 ) if G 3 ≇ C 3 ( S n − 4 , S 2 , S 2 ) .

Therefore, G ≻ C 3 ( S n − 4 , S 2 , S 2 ) if G ≇ C 3 ( S n − 4 , S 2 , S 2 ) .

(2) If t = 1 , we divide the following two cases to prove the result.

Case 1. d G 1 ( v i ) ≥ 3 for i = 1 , 2 , … , g . clearly, n i ≥ 2 . We may assume that the unique vertex with degree 2 of G 1 is in V ( T 1 ) . Then T i is a star for i = 2 , … , g , and the longest path of T 1 is P 3 , say v 1 x y . Hence, by choosing the edge v 1 x and applying Lemma 2.6 to G 1 , we obtain the graph G 2 = C g ( S n 1 , S n 2 , … , S n g ) such that G 1 ≻ G 2 . Note that G 2 ∈ U n , 0 g . By the result (1), we obtain G 2 ≽ C 3 ( S n − 4 , S 2 , S 2 ) . It follows from Lemma 2.8 that C 3 ( S n − 4 , S 2 , S 2 ) ≻ C 3 ( S n − 3 , S 2 , S 1 ) .

Case 2. The unique vertex with degree 2 of G 1 is in V ( C g ) , say, d G ( v g ) = 2 . Then G 1 = C g ( S n 1 , … , S n g ) with n g = 1 and n i ≥ 2 for i = 1 , … , g − 1 . By an analogous argument, we have that G 1 ≽ C g − 1 ( S n 1 + n 2 , S n 3 , … , S g 1 , S 1 ) ≽ C 3 ( S n − 3 , S 2 , S 1 ) , and G 1 ≻ C 3 ( S n − 3 , S 2 , S 1 ) if G 1 ≇ C 3 ( S n − 3 , S 2 , S 1 ) .

Therefore, G ≻ C 3 ( S n − 3 , S 2 , S 1 ) if G ≇ C 3 ( S n − 3 , S 2 , S 1 ) .

(3) If t = n − 2 , then G 1 ≅ G ≅ P n g . We may assume that d G ( v 1 ) = 3 . By Lemma 2.1,

m ( G 1 , j ) = m ( G 1 − v 1 v 2 , j ) + m ( G 1 − v 1 − v 2 , j − 1 ) = m ( P n , j ) + m ( P g − 2 ∪ P n − g , j − 1 ) .

Obviously, m ( P n 3 , j ) = m ( P n n − 1 , j ) for all j , that is, P n 3 ∼ P n n − 1 . It is easy to see that P n n − 1 ≅ S n n − 1 and P n 3 ≅ S n n − 1 only when n = 4 . From Lemma 2.9, we deduce that P g − 2 ∪ P n − g ≽ P 1 ∪ P n − 3 , with equality if and only if g = 3 or n − 1 . It follows from Lemma 2.2 that G ≽ P n 3 ( ∼ S n n − 1 ) , with equality if and only if G ≅ P n 3 or S n n − 1 .

(4) If 2 ≤ t ≤ n − 3 , we may assume that n i 1 , … , n i l ≥ 2 , n i l + 1 = ⋯ = n i g = 1 ( { i 1 , … , i g } = { 1 , … , g } ) and the longest path of subtree T i j is P k j for j = 1 , … , l . It is clearly that l ≥ 1 and k j ≥ 2 for j = 1 , … , l . Then we have t = g − l + ∑ i = 1 l ( k i − 2 ) . Let u 1 … u t + 1 u 1 be the unique cycle of S n t + 1 with u 1 having the maximum degree. By Lemma 2.1,

m ( G 1 , r ) = m ( G 1 − v i 1 v i 1 + 1 , r ) + m ( G 1 − v i 1 − v i 1 + 1 , r − 1 ) , m ( S n t + 1 , r ) = m ( S n t + 1 − u 1 u 2 , r ) + m ( S n t + 1 − u 1 − u 2 , r − 1 ) = m ( T n , t − 1 , r ) + m ( P t − 1 ∪ ( n − t − 1 ) P 1 , r − 1 ) .

Since d G 1 ( v i 1 ) ≥ 3 and d G 1 ( v i 1 + 1 ) ≥ 2 , the number of vertices with degree 2 of G 1 − v i 1 v i 1 + 1 is at least t − 1 . According to Lemma 2.4, G 1 − v i 1 v i 1 + 1 ≽ T n , t − 1 , with equality if and only if G 1 − v i 1 v i 1 + 1 ≅ T n , t − 1 . On the other hand, it is easy to see that P g − 2 ∪ P k 1 − 1 ∪ … ∪ P k l − 1 is a subgraph of G 1 − v i 1 − v i 1 + 1 . By Lemma 2.9, we have

P g − 2 ∪ P k 1 − 1 ∪ … ∪ P k l − 1 ≽ P 1 ∪ P g + k 1 − 4 ∪ P k 2 − 1 ∪ … ∪ P k l − 1 ≽ 2 P 1 ∪ P g + k 1 + k 2 − 6 ∪ P k 3 − 1 ∪ … ∪ P k l − 1 ≽ … ≽ l P 1 ∪ P g + k 1 + … + k l − 2 l − 2 .

Note that P 1 ∪ H ∼ H for any graph H , and g + k 1 + … + k l − 2 l − 2 = t + l − 2 . Thus, P g − 2 ∪ P k 1 − 1 ∪ … ∪ P k l − 1 ≽ P t − 1 , and P g − 2 ∪ P k 1 − 1 ∪ … ∪ P k l − 1 ≽ P t ≻ P t − 1 if l ≥ 2 . Now suppose l = 1 . Then t = g + k 1 − 3 , and if g ≠ 3 or k 1 ≠ 2 , we still have P g − 2 ∪ P k 1 − 1 ≻ P t − 1 ∪ P 1 by Lemma 2.9. If l = 1 and g = 3 , then it is easy to check that G 1 ≅ C 3 ( S 1 , S 1 , T n − 2 , t − 2 ) . Thus, G 1 ≅ S n 3 when t = 2 , or G 1 − v i 1 v i 1 + 1 ≇ T n , t − 1 when t ≥ 3 , and so G 1 − v i 1 v i 1 + 1 ≻ T n , t − 1 when t ≥ 3 . If l = 1 and k 1 = 2 , then t = g − 1 , and it is easy to see that G 1 ≅ S n t + 1 . It follows from Lemmas 2.2 and 2.7 that G 1 ≻ S n t + 1 if G 1 ≇ S n t + 1 .

Therefore, G ≻ S n t + 1 if G ≇ S n t + 1 .

The theorem is thus proved.

5 Conclusion

Lin and Guo [45] proved that the broom T n , t is the tree with minimum matching energy or Hosoya index in T n , t + for 0 ≤ t ≤ n − 2 . In this article, we characterize the graphs with minimum matching energy or Hosoya index in U n , t for 0 ≤ t ≤ n − 2 . It is interesting to further investigate the graphs with minimum matching energy or Hosoya index in the class of all bicyclic and tricyclic graphs with fixed order and number of vertices of degree two, respectively. The new graph transformations and the main result of this article can be applied in the research of next works.

Acknowledgments

The authors are grateful to the referees for valuable comments and suggestions.

Funding information: This work was supported by the National Natural Science Foundation of China (No. 12371348).
Author contributions: All authors contributed equally to this work. All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
Conflict of interest: The authors state no conflict of interest.

References

[1] J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, New York, 2008. 10.1007/978-1-84628-970-5Search in Google Scholar

[2] X. Li, Y. Shi, and I. Gutman, Graph Energy, Springer, New York, 2012. 10.1007/978-1-4614-4220-2Search in Google Scholar

[3] M. Ghorbani, B. Deng, M. Hakimi-Nezhaad, and X. Li, A survey on borderenergetic graphs, MATCH Commun. Math. Comput. Chem. 84 (2020), no. 2, 293–322. Search in Google Scholar

[4] I. Gutman and S. Wagner, The matching energy of a graph, Discrete Appl. Math. 160 (2012), no. 15, 2177–2187, DOI: https://doi.org/10.1016/j.dam.2012.06.001. 10.1016/j.dam.2012.06.001Search in Google Scholar

[5] I. Gutman, Matching energy, in: I. Gutman, X. Li (Eds.), Energies of Graphs - Theory and Applications, Univ. Kragujevac, Kragujevac, 2016, pp. 167–190. Search in Google Scholar

[6] S. Ji, X. Li, and Y. Shi, Extremal matching energy of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 70 (2013), no. 2, 697–706. Search in Google Scholar

[7] L. Chen and Y. Shi, Maximal matching energy of tricyclic graphs, MATCH Commun. Math. Comput. Chem. 73 (2015), no. 1, 105–119. Search in Google Scholar

[8] S. Ji and H. Ma, The extremal matching energy of graphs, Ars Combin. 115 (2014), 343–355. Search in Google Scholar

[9] K. Xu, K. C. Das, and Z. Zheng, The minimal matching energy of (n,m)-graphs with a given matching number, MATCH Commun. Math. Comput. Chem. 73 (2015), no. 1, 93–104. Search in Google Scholar

[10] W. So and W. Wang, Finding the least element of the ordering of graphs with respect to their matching numbers, MATCH Commun. Math. Comput. Chem. 73 (2015), no. 1, 225–238. Search in Google Scholar

[11] K. Xu, Z. Zheng, and K. C. Das, Extremal t-apex trees with respect to matching energy, Complexity 21 (2016), no. 5, 238–247, DOI: https://doi.org/10.1002/cplx.21651. 10.1002/cplx.21651Search in Google Scholar

[12] S. Li and W. Yan, The matching energy of graphs with given parameters, Discrete Appl. Math. 162 (2014), 415–420, DOI: https://doi.org/10.1016/j.dam.2013.09.014. 10.1016/j.dam.2013.09.014Search in Google Scholar

[13] S. Ji, H. Ma, and G. Ma, The matching energy of graphs with given edge connectivity, J. Inequal. Appl. 2015 (2015), 415, DOI: https://doi.org/10.1186/s13660-015-0938-3. 10.1186/s13660-015-0938-3Search in Google Scholar

[14] H. Li, Y. Zhou, and L. Su, Graphs with extremal matching energies and prescribed parameters, MATCH Commun. Math. Comput. Chem. 72 (2014), no. 1, 239–248. Search in Google Scholar

[15] L. Chen, J. Liu, and Y. Shi, Matching energy of unicyclic and bicyclic graphs with a given diameter, Comlexity 21 (2015), no. 2, 224–238, DOI: https://doi.org/10.1002/cplx.21599. 10.1002/cplx.21599Search in Google Scholar

[16] L. Chen and J. Liu, Extremal values of matching energies of one class of graphs, Appl. Math. Comput. 273 (2016), 976–992, DOI: https://doi.org/10.1016/j.amc.2015.10.025. 10.1016/j.amc.2015.10.025Search in Google Scholar

[17] H. Chen and H. Deng, Extremal bipartite graphs of given connectivity with respect to matching energy, Discrete Appl. Math. 239 (2018), 200–205, DOI: https://doi.org/10.1016/j.dam.2018.01.003. 10.1016/j.dam.2018.01.003Search in Google Scholar

[18] H. Chen and Y. Liu, Extremal graphs with respect to matching energy for random six-membered ring spiro chains, Acta Math. Appl. Sin. Engl. Ser. 35 (2019), no. 2, 319–326, DOI: https://doi.org/10.1007/s10255-019-0820-z. 10.1007/s10255-019-0820-zSearch in Google Scholar

[19] L. Chen and J. Liu, The bipartite unicyclic graphs with the first ⌊n‒34⌋ largest matching energies, Appl. Math. Comput. 268 (2015), 644–656, DOI: https://doi.org/10.1016/j.amc.2015.06.115. 10.1016/j.amc.2015.06.115Search in Google Scholar

[20] L. Chen, J. Liu, and Y. Shi, Bounds on the matching energy of unicyclic odd-cycle graphs, MATCH Commun. Math. Comput. Chem. 75 (2016), no. 2, 315–330. Search in Google Scholar

[21] X. Chen, X. Li, and H. Lian, The matching energy of random graphs, Discrete Appl. Math. 193 (2015), 102–109, DOI: https://doi.org/10.1016/j.dam.2015.04.022. 10.1016/j.dam.2015.04.022Search in Google Scholar

[22] X. Chen and H. Lian, Extremal matching energy and the largest matching root of complete multipartite graphs, Complexity 2019 (2019), 9728976, DOI: https://doi.org/10.1155/2019/9728976. 10.1155/2019/9728976Search in Google Scholar

[23] F. Huang and J. Liu, Bipartite graphs with extremal matching energies with given matching number, MATCH Commun. Math. Comput. Chem. 81 (2019), no. 2, 425–442. Search in Google Scholar

[24] G. Huang, M. Kuang, and H. Deng, Extremal graph with respect to matching energy for a random polyphenyl chain, MATCH Commun. Math. Comput. Chem. 73 (2015), no. 1, 121–131. Search in Google Scholar

[25] Y. Jia, Q. Du, B. Liu, Z. Deng, and B. Huo, The energy of trees with diameter five under given conditions, in: D. Volchenkov, A. C. J. Luo (Eds.), New Perspectives on Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity, Vol. 35, Springer, Cham, 2023, pp. 181–193. 10.1007/978-3-030-97328-5_12Search in Google Scholar

[26] S. Khalashi Ghezelahmad, Lower bounds on matching energy of graphs, Discrete Appl. Math. 307 (2022), 153–159, DOI: https://doi.org/10.1016/j.dam.2021.10.018. 10.1016/j.dam.2021.10.018Search in Google Scholar

[27] S. Khalashi Ghezelahmad, Matching energy of graphs with maximum degree at most 3, MATCH Commun. Math. Comput. Chem. 89 (2023), no. 3, 687–697, DOI: https://doi.org/10.46793/match.89-3.687G. 10.46793/match.89-3.687GSearch in Google Scholar

[28] H. Li and M. Shi, On minimal matching energy of unicyclic graphs with prescribed girth and pendent vertices, Util. Math. 108 (2018), 293–306. Search in Google Scholar

[29] H. Li and Q. Wu, On ordering of complements of graphs with respect to matching numbers. II, MATCH Commun. Math. Comput. Chem. 76 (2016), no. 3, 669–692. Search in Google Scholar

[30] X. Liu, L. Wang, and P. Xiao, Ordering of bicyclic graphs by matching energy, MATCH Commun. Math. Comput. Chem. 79 (2018), no. 2, 341–365. Search in Google Scholar

[31] G. Ma, S. Ji, Q. Bian, and X. Li, The maximum matching energy of bicyclic graphs with even girth, Discrete Appl. Math. 206 (2016), 203–210, DOI: https://doi.org/10.1016/j.dam.2016.01.020. 10.1016/j.dam.2016.01.020Search in Google Scholar

[32] G. Ma, S. Ji, Q. Bian, and X. Li, The minimum matching energy of tricyclic graphs with given girth and without K4-subdivision, Ars Combin. 132 (2017), 403–419. Search in Google Scholar

[33] G. Ma, S. Ji, and J. Wang, Corrigendum to The minimum matching energy of bicyclic graphs with given girth, Rocky Mountain J. Math. 48 (2018), no. 6, 1983–1992, DOI: https://doi.org/10.1216/rmj-2018-48-6-1983. 10.1216/RMJ-2018-48-6-1983Search in Google Scholar

[34] W. Wang and W. So, On minimum matching energy of graphs, MATCH Commun. Math. Comput. Chem. 74 (2015), no. 2, 399–410. Search in Google Scholar

[35] T. Wu, H. Lü, and X. Zhang, Extremal matching energy of random polyomino chains, Entropy 19 (2017), no. 12, 684, DOI: https://doi.org/10.3390/e19120684. 10.3390/e19120684Search in Google Scholar

[36] T. Wu, W. Yan, and H. Zhang, Extremal matching energy of complements of trees, Discuss. Math. Graph Theory 36 (2016), no. 3, 505–521, DOI: https://doi.org/10.7151/dmgt.1869. 10.7151/dmgt.1869Search in Google Scholar

[37] Y. Xue and W. Wang, The matching energy of k-trees, Ars Combin. 155 (2021), 81–93. Search in Google Scholar

[38] H. Zhang and L. Liu, Two graph transformations and their applications to matching theory of graphs, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 3, 621–633. Search in Google Scholar

[39] Y. Zhou and H. Li, On the tricyclic graphs with three disjoint 6-cycle and maximum matching energy, Tamkang J. Math. 46 (2015), no. 4, 389–399, DOI: https://doi.org/10.5556/j.tkjm.46.2015.1768. 10.5556/j.tkjm.46.2015.1768Search in Google Scholar

[40] N. Dehgardi and M. Azari, Trees, unicyclic graphs and their geometric Sombor index: an extremal approach, Comput. Appl. Math. 43 (2024), no. 5, 271, DOI: https://doi.org/10.1007/s40314-024-02793-5. 10.1007/s40314-024-02793-5Search in Google Scholar

[41] K. C. Das, Open problems on Sombor index of unicyclic and bicyclic graphs, Appl. Math. Comput. 473 (2024), 128647, DOI: https://doi.org/10.1016/j.amc.2024.128647. 10.1016/j.amc.2024.128647Search in Google Scholar

[42] S. Zhang, X. Chen, Z. Ma, X. Zhang, and Y. Chen, The minimum Wiener index of unicyclic graphs with maximum degree, Appl. Math. Comput. 470 (2024), 128581, DOI: https://doi.org/10.1016/j.amc.2024.128581. 10.1016/j.amc.2024.128581Search in Google Scholar

[43] P. Nithya, S. Elumalai, S. Balachandran, and S. Mondal, Smallest ABS index of unicyclic graphs with given girth, J. Appl. Math. Comput. 69 (2023), no. 5, 3675–3692, DOI: https://doi.org/10.1007/s12190-023-01898-0. 10.1007/s12190-023-01898-0Search in Google Scholar

[44] M. Azari, N. Dehgardi, and T. Došlić, Lower bounds on the irregularity of trees and unicyclic graphs, Discrete Appl. Math. 324 (2023), 136–144, DOI: https://doi.org/10.1016/j.dam.2022.09.022. 10.1016/j.dam.2022.09.022Search in Google Scholar

[45] X. Lin and X. Guo, On the minimal energy of trees with a given number of degree two, MATCH Commun. Math. Comput. Chem. 62 (2009), no. 3, 473–480. Search in Google Scholar

[46] E. J. Farrell, An introduction to matching polynomials, J. Comb. Theory B 27 (1979), no. 1, 75–86, DOI: https://doi.org/10.1016/0095-8956(79)90070-4. 10.1016/0095-8956(79)90070-4Search in Google Scholar

[47] I. Gutman and D. Cvetković, Finding tricyclic graphs with a maximal number of matchings - another example of computer aided research in graph theory, Publ. Inst. Math. (Beograd) (N.S.) 35 (1984), no. 49, 33–40. Search in Google Scholar

[48] I. Gutman and F. Zhang, On the ordering of graphs with respect to their matching numbers, Discrete Appl. Math. 15 (1986), no. 1, 25–33, DOI: https://doi.org/10.1016/0166-218X(86)90015-6. 10.1016/0166-218X(86)90015-6Search in Google Scholar

[49] I. Gutman, Graphs with greatest number of matchings, Publ. Inst. Math. (Beograd) (N.S.) 27 (1980), no. 41, 67–76. Search in Google Scholar

Received: 2024-05-16

Revised: 2024-10-16

Accepted: 2024-12-11

Published Online: 2024-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0120

Keywords for this article

matching energy; unicyclic graph; quasi-order; Hosoya index

Creative Commons

BY 4.0