The largest matching roots of unicyclic graphs with a fixed matching number

In this note, we study the largest matching roots of unicyclic graphs with a given number of fixed matching number. We also characterize the extremal graph with respect to the largest matching roots. In addition, we also study this problem on the trees with a given number of fixed matching number.

ABOUT THE AUTHOR Hailiang Zhang (male 1976.03-) graduated from East China Normal University with a PhD degree in applied mathematics in 2013. With Professor Guangting Chen and some other younger teachers now we have an operational research group in Taizhou University. We mainly study the graph theory and combinatorial problems and their applications. Our manuscript have been published in the Journal of Information Processing Letters, Discrete Mathematics, Ars Combinatorial, Discrete Applied Mathematics, etc. Graph polynomials are intensively studied in recent years, one reason is that it is related to the graph structure and combinational problems, but also some of the index of graphs be used in chemistry to explain the properties of material. In this paper, we study the matching polynomial of unicyclic graphs with a fixed matching number which is a coming results of the last paper published in the "IPL" in which we studied the largest matching roots of unicyclic graphs.

PUBLIC INTEREST STATEMENT
Given a graph we can use several ways to define many kinds of polynomial related to it, such as the characteristic polynomial of its adjacent matrix. The matching polynomial of a graph is defined by the number of matchings with certain number of edges. Graph polynomials are intensively studied in recent years, one reason is that it is related to the graph structure and combinational problems, but also some of the index of graphs be used in chemistry to explain the properties of material. Such as the sum of its matching polynomial are called the Hosoya index, it is widely studied in chemistry. Also the number of unsaturated number of vertices is equal the multiplicity of 0 as a root of matching polynomial. We use the information found in the polynomials of graphs to explain the structural problems. Our research is focused on the matching polynomial by far we have obtained some interesting results, such as we characterized all the graphs which only have six distinct matching roots. The largest matching root of unicyclic graphs, etc.
For a simple connect graph. If m = n − 1, then we call G a tree, denoted by T. Gutman (1982) shows that for trees the largest matching roots and the adjacency spectral radius are the same. Gutman and Zhang (1986) have ordered the graphs by matching numbers. If m = n, then we call G an unicyclic graph. Zhang (2013) gives the largest matching root of unicyclic graphs and characterizes the extremal graph. Let g n (r) be the set of unicyclic graphs on n vertices, which has a cycle of length g and has a matching number r. Let g n (s 1 , t 1 ; … ;s g , t g ) be a graph obtained by attaching s i pendant edges and t i paths of length 2, 1 ≤ 1 ≤ i ≤ g to the cycle of G at v i . In this paper, we give the largest matching roots of unicyclic graphs and trees with a fixed matching number. Furthermore, we characterize the extremal graphs with respect to it.

Preliminaries
Let G − v be the graph obtained by deleting a vertex v with its incident edges form G. G − e be the graph obtained by deleting an edge e from G. The following Lemmas 2.1 and 2.2 are often used to calculate the matching polynomial of a graph.
Lemma 2.1 (Cvetković et al., 1988) Let G be a graph with u ∈ V(G), and suppose the neighborhood of Lemma 2.2 (Cvetković et al., 1988) Let G 1 , G 2 , ⋯ , G k be k components of G. Then Lemma 2.3 (Gutman, 1982) Let G be a connected simple graph, v be a vertex of G and e be an edge of G.
Lemma 2.4 (Gutman, 1982) Let v 1 , … , v n be the vertices of a graph G, Let G − v i be the subgraphs of G obtained by deleting the vertex v i , then (2) ( We need to prove that f (x) ≥ 0, when x ≥ M(G). By applying differentiation on Equation (3) and with Lemma 2.4 (2), we have Now, we apply induction on the number of vertices.
Case 1 When n = 1, the result is trivial.
Case 2 When n = 2, G ≅ K 2 , and G * is empty graph of order 2. Obviously, by the Lemma 2.1 and the Lemma 2.2, we have When x ≥ 1, x 2 > x 2 − 1 holds, so the Lemma 2.5 holds.
Case 3 Assume that the Lemma 2.5 holds when the order of G is less than n. We will show that Lemma 2.5 holds when the order of G is n.
When n ≥ 3, |V(G i )| = n − 1. For every G i there exists a spanning subgraph G * i correspond to it. By our assumption, holds. If G * is a proper spanning subgraph of G, without loss generality, let G * j be the proper spanning subgraph of G j , (1 ≤ j ≤ n) by our assumption, when x = M(G) > M(G j ), holds. Therefore: That is f � (M(G)) > 0, and

Furthermore, we have M(G * ) < M(G). ✷
Since for every subgraph H of G we can add some isolated vertices to H let it be a spanning subgraph of G. Hence we have for any subgraph H of G, we have the following proposition. (4)  Definition 2.7 (Csikvári, 2011) Let u, v be two vertices of the graph G, we obtain the Kelmans transformation of G as follows: we erase all edges between v and N(v) − (N(u) ∪ {u}) and add all edges between u and N(v) − (N(u) ∪ {u}) (see Figure 1). Lemma 2.8 (Csikvári, 2011) Assume that G * is a graph obtained from G by some Kelmans transformation, then M(G * ) ≥ M(G).
In the following proof of our main results, we need to move pendant edges and pendant paths of length two in G to construct another graph G * . This is a special case of the Kelmans transformation.
We give as a Corollary 2.9. Corollary 2.9 Let G be a simple graph, u, v ∈ V(G) with s and t pendant paths of length 1, or pendant paths of length 2 on each. G * denote the graph obtained by moving all pendant paths to one vertex (see Figuer 2). Then Let G be a simple graph. If e = (v 1 v 2 ) is not an edge of C 3 of G, then G ⋄ e denotes the graph obtained by contracting edge e. G * denotes the graph obtained by adding a pendant edge to the contracted vertex u of G ⋄ e (see Figure 3). For the largest matching root of G and G * , we have the following Lemma 2.10.
Lemma 2.10 Let H 1 and H 2 be two graphs with distinguished vertices u 1 , u 2 of H 1 and H 2 , respectively. Let G be the graph connecting u 1 and u 2 by an edge e. Let H 1 uH 2 be the graph obtained from H 1 and H 2 by identifying the vertices of u 1 and u 2 to a new vertex u (see G 1 in Figure 3). Let G * be the graph obtained by attaching a pendant to u of G 1 (see G * in Figure 3). Then Proof By Lemma 2.1 and Lemma 2.2, we calculate the matching polynomial of G and G * as: Corollary 2.9 and Lemma 2.10 guarantee the largest matching root will not decrease under those graph transformations. To finish our proof, we also need to prove that the matching number also does not change under those transformations. Let us first deal with the trees. Proof For a tree T on n vertices, the path P n has the maximal matching of order n / 2 n is even, or (n − 1)∕2 when n is odd, hence we only need to construct an corresponding map from maximal matching M of P n to maximal match M * of S (s, t). For the small number of n we can easily construct.
Assume that n is an even number and n ≥ 4. Let M = {e 1 , e 2 , … , e n∕2 } be a maximal match of P n . We can correspond e 1 , e 2 , … , e n∕2−1 to the pendant n∕2 − 1 edges of P 2 of S(1, n∕2 − 1), and e n∕2 to the one pendant edge of S(1, n∕2 − 1).
Assume that n is an odd number and n ≥ 4. Let M = {e 1 , e 2 , … , e (n−1)∕2 } be a maximal match of P n . We can correspond e 1 , e 2 , … , e (n−1)∕2 to the pendant (n − 1)∕2 edges of P 2 of S(0, (n − 1)∕2). ✷ By Lemma 2.10 and Theorem 3.3, for trees with matching number r, we have Theorem 3.4: Theorem 3.4 Let T be a tree on n vertices and with a matching number r, then S(n − 2r + 1, r − 1) has the largest matching root, which is Now, we study the largest matching root of unicyclic graphs.
Proof Suppose that G ∈ g n (r) and T 1 , T 2 , … , T i be the pendant trees attaching to vertices v 1 , … , v i of C g of G, we use Lemma 2.10 on the T i and move all the pendant paths to one vertex u of C g . After this processing, the vertices of C g are saturated by a pendant edge or an edge of C g , or both a pendant edge and an edge of C g . This situation will happen, if the T i is a path on even number of vertices and g is on an even cycle with perfect matching.
Assume that the vertex v i is saturated by a pendant edge, then the neighbor of v i on the cycle v i−1 and v i+1 are not saturated. We delete v i v i−1 and connect v i−1 and v i+1 . This processing does not change the matching number and creates a pendant path of length 2 on v i+1 . Then move this path to u (actually this processing is a Kelmans transformation).
If the vertex v i is saturated by edges on cycle C g . Let this edge be v i v i+1 . We contract this edge and add a pendant to the path of length 1, this processing shortens the girth of C g by one, and does not change the matching number.