Abstract
The matching roots of a simple connected graph are the roots of the matching polynomial which is defined as , where is the number of the matchings of . Let denote the largest matching root of the graph . In this paper, among the unicyclic graphs of order , we present the ordering of the unicyclic graphs with matching number 2 according to the values for and also determine the graphs with the first and second largest values with matching number 3.
1. Introduction
In this paper, all of the graphs considered are connected and finite. Let be a graph of order , where and denote the vertex set and edge set of , respectively. Let and be the neighbor set and the degree of the vertex in , respectively. A -edge matching of is a set of mutually independent edges, and the number of all the -edge matchings of is denoted by . Obviously, if . For convenience, we set .
The original definition of the matching polynomial was introduced in [9] as , which is now defined in [14] as
Each root of the matching polynomial is called a matching root of . The largest matching root of a graph which is denoted by is the largest root of . In particular, has been proved to be positive real numbers excepting for the edgeless graphs in [3, 4].
The matching polynomial of a graph has been used in various branches of physics and chemistry. In analogy with the traditional graph energy, the matching energy of a graph has been conceived as the sum of the absolute values of the roots of the matching polynomial. This graph invariant has recently attracted much attention (see [2, 3, 5–11]). In [15], Gutman and Zhang have ordered the graphs by matching numbers. Zhang et al. [18] investigated the largest matching root for unicyclic graphs and characterized the extremal graph. Zhang and Chen [10] studied the largest matching roots of unicyclic graphs and trees with a given number of fixed matching numbers and characterized the extremal graph with respect to the largest matching roots. In [3], Liu et al. determined the graphs with the four largest and two smallest values.
Motivated by the previous research, in this paper, we focus on the ordering of the unicyclic graph with the fixed matching number with respect to the largest matching root . Among the unicyclic graphs of order , we order the unicyclic graphs with matching number 2 according to the values for and also determine the graphs with the first and second largest values with matching number 3.
2. Preliminaries
Let be a subgraph of . If , then is said to be a spanning subgraph of . The following lemma about the largest matching root is well known.
Lemma 1. (see [8]). Let be a spanning subgraph of and the largest matching root of . If , then ; if is a proper subgraph of and , then .
Definition 1. Let denote two vertices of . The Kelmans transformation of is defined as follows (cf. Figure 1): delete all edges which are with one end and another end in and add all edges between and .
Let denote the graph obtained from by a series of Kelmans transformation without referring to the vertices . Obviously, and have the same size.
The relationship between the largest matching roots of and is given as follows.
Lemma 2. (see [7]). Let be the graph obtained from by some Kelmans transformation. Then, .
In [3], Liu et al. characterized the unicyclic graphs with a triangle as given in Figure 2 and obtained the following lemma.
Lemma 3. (see [3]). The graphs of order and are shown in Figure 2. If , then , with equality holding if and only if .
Let be the star graph of order and the unicyclic graph obtained by adding a new edge to . Let be a cycle of length and the graph obtained from by adding a new pendent edge. Let be the graph obtained by identifying one end vertex of the path and one vertex of the triangle .
Lemma 4. (see [3]). Among all connected unicyclic graphs with vertices, the first four largest matching roots are , and the last three largest matching roots are or and , where , , and are shown in Figure 3.
Lemma 5. (see [11]). Let be a graph of order . Then, , where is the number of edges of .
Corollary 1. Let be a connected unicyclic graph with vertices. Then, .
3. Main Results
In this section, we first investigate the largest matching root of unicyclic graphs with matching number 2.
For , let be a star graph with center . Let be two vertices of which are of degree 1 and 2, respectively. We write denote the graph obtained by identifying with . In fact, is the graph in Figure 2. All the connected unicyclic graphs with matching number 2 are , , , and as shown in Figure 4.
Theorem 1. , .
Proof. We can easily get and , , then the matching polynomial of graphs and isTherefore,Then , , and , so , since the is strictly monotonically increasing for .
Theorem 2. Among all connected unicyclic graphs of order with matching number 2, , with .
Proof. It is obvious that for any connected unicyclic graph of order . By simple calculations, we haveThen,Since , then . Since the matching polynomial is strictly monotonically increasing for , . Similarly, since , then by putting , and so . Similarly, we havePutting and to the above two equations, respectively, we haveSince can be obtained from by Kelmans transformations, then by Lemma 3. By Lemma 3, we have . Combining the discussions above and Theorem 1, the proof is completed.
We now investigate the largest matching root of graphs with matching number 3.
Theorem 3. Let the graphs and be defined above. Then, if and if .
Proof. By simple calculations, it is obtained thatThen,If , thenPutting . Since is a connected unicyclic graph, then by Lemma 5. Thus, if , then and so . If , then and so .
If , then and so .
Theorem 4. Let and be the graphs as shown in Figure 5. If , then , with equality holding if and only if .
Proof. By simple calculations, it is obtained thatThen,It follows thatBy Lemma 5, . Therefore, putting , if , then and so , with equality holding if and only if .
Theorem 5. The graphs and with vertices are shown in Figure 6, where . If and , then . If and , then .
Proof. Let denote the graphs and , respectively. Without loss of generality, assume that and . It is easy to get thatThen,Putting to , thenSince , thenSince and , thenTherefore, , and so . Moreover, . Since , then for any . Thus, if and , thenthat is, . Therefore, .
We now discuss the case for . Without loss of generality, assume that . Obviously, we haveThen,By Corollary 1, . Thus, if , then . Hence, .
Theorem 6. The graphs with vertices are shown in Figure 7. Then, .
Proof. Obviously, (for ) andThen,Putting to matching polynomial of , we haveSince , then and so . Since , then for any . Therefore, , and thus, .
Since , then , and thus, . It is obvious that . Since and , then and so . Therefore, .
Theorem 7. Among all unicyclic graphs of order with matching number 3, the graphs with the first second largest matching root are and .
Proof. Since , then the graph with the largest matching root among all unicyclic graphs with matching number 3 is by Lemma 4.
Let be the unicyclic graph of order with the second largest matching root among all graphs with matching number 3. Since is a unicyclic graph of order with matching number 3, then the girth of is possibly , or 5.
3.1. Case 1:
Under the Kelmans transformation, is or in Figure 6. By Theorem 3, if and if . Therefore, among all the graphs of type , is or . Since and , then by Theorem 6.
By Theorem 5, is the graph with the first largest matching root among the graphs of type . Moreover, . Thus, by Theorem 6, the graph with the second largest matching root and girth 3 is . □
3.2. Case 2:
In this case, possibly is the graph in Figure 8. Obviously, can be obtained by a series of Kelmans transformations from the graph . Moreover, is the graph in Theorem 4. By Theorem 4, the graph with the largest matching roots among graphs of type is which is isomorphic to . Therefore, the graph with the largest matching root among the graphs with matching number 3 and girth 4 is .
3.3. Case 3:
In this case, the unicyclic graphs with matching number 3 isomorphism to the following four families graphs denoted by , and where denote the unicyclic graphs with girth 5, such that there are pendent edges attached to the vertex of the cycle. Obviously, is isomorphic to in Theorem 5, and the rest three families can arrive at graphs of Case 1 or 2 by a series of Kelmans transformations. Thus, is in this case.
Therefore, by Theorem 6, the graph with the second largest matching root is .
4. Further Remarks
In the introduction, we already talk about that the matching energies have been getting a lot of attention recently. The graphs with maximum and minimum values of matching roots are the graphs with minimum and maximum matching energy. Therefore, any result for the largest matching roots can reveal the structural dependence of the matching energy or at least provide guidance for its research. Recently, References [2,5,6] showed some new findings on the matching energy of unicyclic graphs. Therefore, the results of this paper have a direct application value [4, 16].
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 12071484 and 11701046) and the Natural Science Foundation of Hunan Province, China (nos. 2020JJ4675, 2017JJ2280 and 2018JJ2450).