The Ordering of the Unicyclic Graphs with respect to Largest Matching Root with Given Matching Number

The matching roots of a simple connected graph G are the roots of the matching polynomial which is defined as M G ( x ) (cid:29) (cid:31) n /2 k (cid:29) 0 (− 1 ) k m ( G,k ) x n − 2 k , where m ( G,k ) is the number of the k matchings of G . Let λ 1 ( G ) denote the largest matching root of the graph G . In this paper, among the unicyclic graphs of order n , we present the ordering of the unicyclic graphs with matching number 2 according to the λ 1 ( G ) values for n ≥ 11 and also determine the graphs with the first and second largest λ 1 ( G ) values with matching number 3.


Introduction
In this paper, all of the graphs considered are connected and nite. Let G (V(G), E(G)) be a graph of order n, where V(G) and E(G) denote the vertex set and edge set of G, respectively. Let N(u) and d G (u) be the neighbor set and the degree of the vertex u ∈ V(G) in G, respectively. A k-edge matching of G is a set of k mutually independent edges, and the number of all the k-edge matchings of G is denoted by m(G, k). Obviously, m(G, k) 0 if k > n/2. For convenience, we set m(G, 0) 1. e original de nition of the matching polynomial was introduced in [9] as k m(G, k)x k , which is now de ned in [14] as (1) Each root of the matching polynomial M G (x) is called a matching root of G. e largest matching root of a graph G which is denoted by λ 1 (G) is the largest root of M G (x). In particular, λ 1 (G) has been proved to be positive real numbers excepting for the edgeless graphs in [3,4]. e 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. is graph invariant has recently attracted much attention (see [2,3,[5][6][7][8][9][10][11]). In [15], Gutman and Zhang have ordered the graphs by matching numbers. Zhang et al. [18] investigated the largest matching root λ 1 (G) 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 xed 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 λ 1 (G) values.
Motivated by the previous research, in this paper, we focus on the ordering of the unicyclic graph with the xed matching number with respect to the largest matching root λ 1 (G). Among the unicyclic graphs of order n, we order the unicyclic graphs with matching number 2 according to the λ 1 (G) values for n ≥ 11 and also determine the graphs with the rst and second largest λ 1 (G) values with matching number 3.

Preliminaries
Let H be a subgraph of G. If V(H) � V(G), then H is said to be a spanning subgraph of G. e following lemma about the largest matching root is well known.
Lemma 1 (see [8]). Let H be a spanning subgraph of G and λ 1 (G) the largest matching root of G.
Definition 1. Let u, v denote two vertices of G. e Kelmans transformation of G is defined as follows (cf. Figure 1): delete all edges which are with one end v and another end in N(v)/(N(u) ∪ u) and add all edges between u and Let G ′ denote the graph obtained from G by a series of Kelmans transformation without referring to the vertices u, v. Obviously, G ′ and G have the same size. e relationship between the largest matching roots of G ′ and G is given as follows.
In [3], Liu et al. characterized the unicyclic graphs with a triangle as given in Figure 2 and obtained the following lemma.
Let S n be the star graph of order n and S + n the unicyclic graph obtained by adding a new edge to S n . Let C n− 1 be a cycle of length n − 1 and C 2 n the graph obtained from C n− 1 by adding a new pendent edge. Let D 2 n be the graph obtained by identifying one end vertex of the path P n− 2 and one vertex of the triangle C 3 .
Lemma 4 (see [3]). Among all connected unicyclic graphs with n(n ≥ 8) vertices, the first four largest matching roots are , and the last three largest matching roots are λ 1 (C 2 n ) or λ 1 (D 2 n ) and λ 1 (C n ), where S 2 n , S 3 n , and S 4 n are shown in Figure 3.
Lemma 5 (see [11]). Let G be a graph of order n. en, Corollary 1. Let G be a connected unicyclic graph with n vertices. en, λ 1

Main Results
In this section, we first investigate the largest matching root of unicyclic graphs with matching number 2.
For k ≥ 2, let S k be a star graph with center u. Let v 1 , v 2 be two vertices of S + n− k+1 which are of degree 1 and 2, respectively. We write D n,k (C n,k ) denote the graph obtained by identifying u with v 1 (v 2 ). In fact, C n,k is the graph D k− 1 in Figure 2. All the connected unicyclic graphs with matching number 2 are S + n , C n,k , D n,n− 3 , S 4 n , and T k n as shown in Figure 4.
(2) erefore, (3) is strictly monotonically increasing for x > 0. □ Theorem 2. Among all connected unicyclic graphs of order n(n ≥ 11) with matching number 2, λ 1  2 Journal of Mathematics Proof. It is obvious that m(G, 1) � n for any connected unicyclic graph G of order n. By simple calculations, we have Since Putting x � λ 1 (C n, 3 ) and x � λ 1 (D n,n− 3 ) to the above two equations, respectively, we have Since S + n can be obtained from C n,k by Kelmans transformations, then λ 1 (S + n ) > λ 1 (C n,k ) by Lemma 3. By Lemma 3, we have λ 1 (C n,k− 1 ) > λ 1 (C n,k ). Combining the discussions above and eorem 1, the proof is completed.

□
We now investigate the largest matching root of graphs with matching number 3.

Theorem 3.
Let the graphs D n,k+1 and D n,k be defined above.
Proof. Let G 1 , G 2 denote the graphs G(a, b, c) and G(a, b + 1, c − 1), respectively. Without loss of generality, assume that a ≥ n/3 − 1 and b ≥ c. It is easy to get that en,
By eorem 5, G(1, n − 5, 1) is the graph with the first largest matching root among the graphs of type G(a, b, c). Moreover, G(1, n − 5, 1) � H 1 . us, by eorem 6, the graph with the second largest matching root and girth 3 is H 1 . □ 3.2. Case 2: g � 4. In this case, G possibly is the graph T i (i � 1, 2, 3, 4, 5) in Figure 8. Obviously, T 1 can be obtained by a series of Kelmans transformations from the graph T i (i � 2, 3, 4, 5). Moreover, T 1 is the graph L k in eorem 4. By eorem 4, the graph with the largest matching roots among graphs of type L k is L 1 which is isomorphic to H 3 . erefore, the graph with the largest matching root among the graphs with matching number 3 and girth 4 is H 3 .

Further Remarks
In the introduction, we already talk about that the matching energies have been getting a lot of attention recently. e graphs with maximum and minimum values of matching roots are the graphs with minimum and maximum matching energy. erefore, 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. erefore, the results of this paper have a direct application value [4,16].

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.