On the largest matching roots of graphs with a given number of pendent vertices

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On the largest matching roots of graphs with a given number of pendent vertices✩ Hailiang Zhang a, *, Guangting Chen a , Guanglong Yu b,c, * a b c

Department of Mathematics, Taizhou University, Linhai, Zhejiang, 317000, PR China Department of Mathematics, Lingnan Normal University, Zhanjiang, 524048, Guangdong, PR China Department of Mathematics, Yancheng Teachers University, Yancheng, 224002, Jiangsu, PR China

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Article history: Received 10 July 2017 Received in revised form 12 June 2018 Accepted 23 July 2018 Available online xxxx

a b s t r a c t In this note, we study the bounds of the largest matching roots of graphs with 0, 1, 2, 3, 4, n − 2, and n − 1 pendent vertices. We also determine the extremal graphs with respect to these bounds. © 2018 Elsevier B.V. All rights reserved.

MSC: 05C50 Keywords: Matching polynomial Largest matching root Pendent vertex Extremal graph

1. Introduction Let G be a simple graph, i.e., no loops or multiple edges are allowed. Denote by V (G) and E(G) the vertex set and edge set, respectively. A matching M is a subset of pairwise nonadjacent edges of E(G). If |M | = k, then M is called a k-matching. By m(G, k) we denote the number of k-matchings. Cvetković etc. [2] have defined the matching polynomial as: MG (x) =

n/2 ∑

(−1)k m(G, k)xn−2k .

(1)

k≥0

For convenience, we set m(G, 0) = 1. Obviously, m(G, 1) = |E(G)|, is the number of edges. Let θ (G) be the largest matching root of MG (x). The matching polynomial and some related indices on this polynomial such as Hosoya index [8], which is defined as the sum of absolute values of the coefficients of MG (x), are used in chemistry to study certain physio-chemical properties of alkane (saturated hydrocarbons). Matching polynomials are also independently discovered in statistical physics and mathematical context. In statistical physics they were first considered by Heilman and Lieb in 1970 [7]. It is used in mathematical chemistry for quantifying relevant details of molecular structure [8,6]. In recent years, a lot of work has been done on studying the extremal problems with respect to the largest matching root of graphs. i.e., on determining the graphs within a prescribed class ✩ Supported by NSFC (No. 11571252 & No. 11771376), ‘‘333’’ Project of Jiangsu (2016). Corresponding authors. E-mail addresses: [email protected] (H. Zhang), [email protected] (G. Yu).

*

https://doi.org/10.1016/j.dam.2018.07.028 0166-218X/© 2018 Elsevier B.V. All rights reserved.

Please cite this article in press as: H. Zhang, et al., On the largest matching roots of graphs with a given number of pendent vertices, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.07.028.

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Fig. 2.1. DS(p, q) and Yn .

that minimize or maximize the value of the largest matching roots, such as [11] and [9]. For more fundamental properties of matching polynomials of graphs, the readers are referred to [2] and [5]. The largest matching root of a graph and the extremal problems related to it are intensively studied in [11,5,4]. A pendent vertex of G is a vertex of degree 1, and a pendent edge of G is an edge incident to a pendent vertex. Denote by Pnr the set of all connected graphs of order n with r pendent vertices. In this paper, we discuss the bounds and extremal graphs with respect to the largest matching roots of graphs in Pnr . The layout of this paper is as follows: Section 2 introduces some notations and working lemmas; Section 3 represents the extremal graphs with respect to the largest matching root and the bounds for the largest matching roots of these extremal graphs. 2. Preliminary In this section, we introduce some notations and some basic lemmas. We denote by Pn , Cn , Kn a path, a cycle, and a complete graph of order n, respectively. Denote by K1,n−1 a star of order n, namely a graph obtained from a K1 by attaching n − 1 pendent edges to the unique vertex of K1 . For a graph G and a subset U ⊂ V (G), we denote by G[U ] the subgraph induced by U. For positive integers p and q, denote by DS(p, q) a double star, namely a graph obtained from a K2 by attaching respectively p and q pendent edges to the two end vertices of K2 (see Fig. 2.1). Clearly, the order of the graph DS(p, q) is p + q + 2. According to this definition, the path P4 is isomorphic to DS(1, 1). Denote by Yn the graphs obtained by connecting two central vertices of two K1,2 by a path Pn−4 , n ≥ 6 (see Fig. 2.1). Lemma 2.1 ([2]). Let G be a graph with u ∈ V (G). Suppose that the neighborhood of u is Γ (u) = {v1 , v2 , . . . , vk }. Then, MG (x) = xMG\u (x) −

k ∑

MG\{uvi } (x).

(2)

i=1

Lemma 2.2 ([2]). Let G1 , G2 , . . . , Gk be k components of G. Then, MG (x) =

k ∏

MGi (x).

(3)

i=1

Lemma 2.3 ([2]). Let G be any graph and suppose uv ∈ E(G), k ≥ 1 integer. Then, m(G, k) = m(G \ uv, k) + m(G \ {u, v}, k − 1) and MG (x) = MG\uv (x) − MG\{u,v} (x).

(4)

Lemma 2.4 ([5]). Let G be a connected simple graph. If v is an arbitrary vertex and e is an arbitrary edge of G, H is a subgraph of G, then,

θ (G) > θ (G − v ), θ (G) > θ (G − e), θ (G) > θ (H). By using Lemma 2.3 and other polynomial properties, Csikvári [1] gave the change of the largest matching roots of graphs under Kelmans transformation of graphs. Definition 2.5 ([1]). 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 each vertex in N(v ) − (N(u) ∪ {u}) (see Fig. 2.2). Lemma 2.6 ([1]). Assume that G∗ is a graph obtained from G by the Kelmans transformation, then θ (G∗ ) ≥ θ (G). For a graph G, denote by A(G) the adjacency matrix. The characteristic polynomial of G is defined to be φ (x) = det(xI − A(G)), where I is the identity matrix. The spectral radius of graph G, denoted by ρ (G), is the largest root of φ (x). In [5], it is shown that MG (x) = φ (x) for any tree G, which implies the following Lemma 2.7. Please cite this article in press as: H. Zhang, et al., On the largest matching roots of graphs with a given number of pendent vertices, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.07.028.

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Fig. 2.2. Kelmans transformation.

Fig. 3.1. Knr−r .

Lemma 2.7. For graphs Yn , n ≥ 6, ρ (Yn ) = 2. Therefore, θ (Yn ) = 2.

3. Main results In this section, we determine the extremal graphs respectively with respect to the maximal largest matching roots and the minimal largest matching roots of graphs in Pnr . 3.1. Graph with maximal largest matching root Note that P31 = ∅, there is only one graph C3 in P30 , and there is only one graph P3 in P32 . Hence, it is trivial for n = 3. Next, in what follows, we consider the graphs in Pnr for n ≥ 4. For r = n − 1 and r = n − 2 the graphs are uniquely determined. We give as Lemma 3.1. Lemma 3.1. For n ≥ 4, Pnr ̸ = ∅ if and only if 0 ≤ r ≤ n − 1. Furthermore, Pnn−1 has only one graph, namely, K1,n−1 ; Pnn−2 consists of precisely all double stars of order n. About the largest matching roots of the graphs with r = n − 1 and r = n − 2, we obtain the following Theorem 3.2. Theorem 3.2. Among Pnr , when r = n − 1, graph K1,n−1 has the maximal largest matching root; when r = n − 2, DS(⌈(n − 2)/2⌉, ⌊(n − 2)/2⌋) has the minimal largest matching root, and DS(1, n − 3) has the maximal largest matching root. Proof. When r = n − 1, the result follows from Lemma 3.1 directly. Now we prove it when r = n − 2. Assume that the double stars are DS(y, n − 2 − y). Then by Lemmas 2.1 and 2.2, the matching polynomial of DS(y, n − 2 − y) is MDS(y,n−2−y) (x) = xn − (n − 1)xn−2 + y(n − 2 − y)xn−4 . The largest matching root is

√ θ (DS(y, n − 2 − y)) =

n−1+

√

(n − 1)2 − 4y(n − 2 − y) 2

.

(5)

Therefore,

θ (DS(1, n − 3)) > θ (DS(2, n − 4)) ≥ . . . ≥ θ (DS(⌊(n − 2)/2⌋, ⌈(n − 2)/2⌉)).

□

Theorem 3.3. For n ≥ 4 and 0 ≤ r ≤ n − 3, in Pnr , the maximal largest matching root attains uniquely at graph Knr−r , which is obtained from Kn−r by attaching r pendent edges to one vertex of it (see Fig. 3.1). Please cite this article in press as: H. Zhang, et al., On the largest matching roots of graphs with a given number of pendent vertices, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.07.028.

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Fig. 3.2. Five forbidden graphs.

Fig. 3.3. Four types of graph in Pnr .

Proof. Let G ∈ Pnr , U be the set of its pendent vertices. We add edges to V (G) − U, until G[V (G) − U ] becomes a complete graph. Let G′ be the resulting graph. Next by applying Kelmans transformation on G′ , we move all the pendent vertices to one vertex of Kn−r . The resulting graph would be Knr−r . By Lemmas 2.4 and 2.6,

θ (G) < θ (G′ ) ≤ θ (Knr−r ). Therefore, Knr−r has the maximal largest matching root in Pnr .

□

3.2. Graphs with minimal largest matching root In this section, we characterize the graphs in Pnr with minimal largest matching roots for each r ∈ {0, 1, 2, 3, 4, n − 2, n − 1}. When r = n − 1, by Lemma 3.1, there is exactly one graph in Pnn−1 , which is isomorphic to K1,n−1 . For r = n − 2 readers may refer to Theorem 3.2. For r = 0, 1, 2, 3, 4, we need some forbidden subgraphs and lemmas. By computing directly, we get the following Lemma 3.4. Lemma 3.4. For the graphs shown in Fig. 3.2, we have the matching polynomials and the largest matching roots are: (0) MG0 (x) = x5 − 5x3 + 3x, θ (G0 ) ≈ 2.0743; (1) MG1 (x) = x6 − 5x4 + 3x2 , θ (G1 ) ≈ 2.0743; (2) MG2 (x) = x5 − 5x3 + 4x, θ (G2 ) = 2.0; (3) MG3 (x) = x6 − 6x4 + 7x2 − 1, θ (G4 ) ≈ 2.1192; (4) MG4 (x) = x6 − 6x4 + 7x2 , θ (G5 ) ≈ 2.1010. The following fact Remark 3.5 is essential in the proof of Theorems 3.6, 3.8, 3.9, and 3.10, we give as a remark. By directly computing and with the bounds for θ (G) in [7] we have: Remark 3.5. For K1,4 and Cn , we have MK1,4 (x) = x5 − 4x3 , θ (K1,4 ) = 2; θ (Cn ) < 2. Let D(r , s, t) be the graphs obtained by connecting two cycles Cr and Ct with a path Ps , where s ≥ 2, and r , t ≥ 3. θ (r , s, t) are θ -graphs in which three paths with length r, s, t, respectively, share two common end vertices, where r , s, t ≥ 3. T (r , s) be the graphs obtained by jointing Cr and Cs at one vertex, where r , s ≥ 3. These graphs are shown in Fig. 3.3. Theorem 3.6. Among all graphs in Pn0 (n ≥ 3) (i.e. graphs without pendent vertices), the cycle Cn is the unique one with minimal largest matching root. Proof. If we delete edges in a connected graph, by Lemma 2.4, it makes the largest matching root of a graph decrease. Therefore, the graphs in Pn0 (n ≥ 3) which have the minimal θ (G) should have the following type of subgraphs shown in Fig. 3.3. In particular, in graphs D(r , s, t), θ (r , s, t), r + s + t ≤ n + 4 and T (r , s), r + s ≤ n + 1. We know that θ (Cn ) < 2, [5]. Suppose that G ∈ Pn0 has the minimal largest matching root. Case 1. G contains D(r , s, t) as a subgraph. Note that Ys+4 be a subgraph of D(r , s, t). By Lemmas 2.4 and 2.7, θ (G) > θ (D(r , s, t)) > θ (Ys+4 ) = 2. Case 2. Let θ (r , s, t) be a subgraph of G. Subcase 2.1. If r = 3, s = 2. Then G0 in Fig. 3.2 will be a subgraph of θ (r , s, t). By Lemma 2.4, and Lemma 3.4,

θ (G) > θ (θ (r , s, t)) > θ (G0 ) > 2. In particular, θ (θ (3, 2, 3)) =

√ (5 +

√

17)/2 > 2.

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Subcase 2.2. If r = 3, s = 3. Then G2 in Fig. 3.2 will be a subgraph of θ (r , s, t). By Lemma 2.4, and Lemma 3.4, θ (G) > θ (θ (r , s, t)) > θ (G2 ) > 2. Subcase 2.3. If r > 2, s > 2. Then Ys+5 in Fig. 2.1. will be a subgraph of θ (r , s, t). By Lemma 2.4, and Lemma 2.7, θ (G) > θ (θ (r , s, t)) > 2. Case 3. Let T (r , s) be a subgraph of G. Then K1,4 will be a subgraph of T (r , s). Hence by Lemma 2.4, θ (G) > θ (T (r , s)) > θ (K1,4 ) = 2 (Remark 3.5). But the bounds given by Heilman and Lieb [7]: √ √ ∆ ≤ θ (G) ≤ 4∆ − 4. (6) We have

√

2 ≤ θ (Cn ) < 2,

From Case 1, Case 2 and Case 3, for any graph G ∈

(7) Pn0 ,

if G ≇ Cn , then we have

θ (G) > θ (Cn ). Therefore, Cn has the minimal largest matching root among graphs in Pn0 . □ Next we study the largest matching roots of graphs in Pn1 . Denote by Q (s, n − s + 1) the set of graphs by identifying a vertex of Cs and a vertex of Pn−s+1 , n = 4k + i, i ∈ 0, 1, 2, 3. The matching polynomial of Q (s, n − s + 1) are: MQ (s,n−s+1) (x) = MPn (x) − MPs−2 ∪(Pn−s ) (x).

(8)

Theorem 3.7. For graphs in Pn1 , Q (3, n − 2) and Q (n − 1, 2) have the minimal largest matching roots. For Q (s, n − s + 1), s ≥ 3, s < n/2, we have

θ (Q (3, n − 2)) < θ (Q (4, n − 3)) < · · · < θ (Q (⌈n/2⌉, n − ⌊n/2⌋)). Proof. For graphs in Pn1 , we delete the proper edges, and let the graphs have induced subgraphs like Q (s, n − s + 1), which have exactly one pendent vertex. By Lemma 2.4, θ (G) > θ (Q (s, n − s + 1)). By Eq. (8) we have MQ (3,n−2) (x) − MQ (4,n−3) (x) = MP2 ∪Pn−4 (x) − MP1 ∪Pn−3 (x). Since Pn−4 is a subgraph Pn−3 , and Pn−3 is a subgraph of Q (4, n − 3) and Q (3, n − 3). Then by Lemma 2.4, we have

θ (Pn−4 ) < θ (Pn−3 ) < θ (Q (4, n − 3)), and with the fact that MG (x) are monotone increasing function on [θ (G), +∞). Then MQ (3,n−2) (x) − MQ (4,n−3) (x) ≥ 0 on interval [θ (Pn−3 ), +∞) ⊂ [θ (Q (4, n − 3)), +∞) holds. Hence

θ (Q (3, n − 2)) < θ (Q (4, n − 3)) holds. □ More results on Q (s, n − s + 1) readers may refer to [10]. Actually, by Eq. (8), the matching polynomials of Q (3, n − 2) and Q (n − 1, 2) are same. Therefore they have the same largest matching roots. Theorem 3.8. Among all graphs in Pn2 (n ≥ 4), the path Pn is the unique one with minimal largest matching root. Proof. Let G ∈ Pn2 . If G is a tree, then since it has exactly two pendent vertices, necessarily G = Pn . If G is not a tree, then G must contain a cycle Cl , 3 ≤ l ≤ n − 2 as a proper subgraph. Case 1 For l = 3, then G contains a subgraph G0 . Then θ (G) > θ (G0 ) = 2.0743. Case 2 For l = 4, then G will contain G2 or G3 or G4 as its subgraph. Anyway θ (G) > θ (Gi ) > 2, i = 2, 3, 4. Case 3 For l ≥ 5, we have three subcases. Subcase 3.1 If Cl contains two vertices of degree 3, then these two vertices with their neighbors and the path between them induce a subgraph Yi , i ≥ 6. Hence, θ (G) > θ (Yi ) > 2 (Remark 3.5). Subcase 3.2 If Cl contains exactly one vertex of degree 3, then, there is one vertex w of degree 3 adjacent to v that does not belong to the cycle. Since G has two pendent vertices, then the vertex degree 3 with the neighbors on the cycle,and with the other two pendent vertices induce a subgraph Yi , i ≥ 6. Hence θ (G) > θ (Yi ) = 2 (Remark 3.5); Please cite this article in press as: H. Zhang, et al., On the largest matching roots of graphs with a given number of pendent vertices, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.07.028.

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Subcase 3.3 If Cl has a vertex vj , d(vj ) ≥ 4, then G contains a subgraph K1,4 ⊂ G1 in Fig. 3.2, θ (G) > θ (G1 ) > θ (K1,4 ) = 2 (Remark 3.5). Since θ (Pn ) = 2 cos nn+π1 < 2 ([3] p. 47), and with Lemma 2.4, and Lemma 2.7 θ (G) > θ (Yi ) = 2 (Remark 3.5), and

θ (G) > θ (G3 ) > 2 . Then, Theorem 3.8 holds. □ Theorem 3.9. Among all graphs in Pn3 (n ≥ 6), the tree T (1, 1, n − 3) is the unique one with minimal largest matching root. Proof. Suppose that G ∈ Pn3 , which has the minimal θ (G). If G is a tree and only have one vertex of degree 3, that is G = T (a, b, c). By properly arranging the length of a, b, c, and with the Theorem 3.11.2 p. 93 in [3], and T (1, 1, n − 3) is a subgraph Yn − v ,where v is a pendant vertex of Yn , and with Lemmas 2.4 and 2.7, we have

θ (T (1, 1, n − 3)) = ρ (T (1, 1, n − 3)) < 2. If G is a tree and has more than one vertex of degree 3, then the number of the pendent vertices is greater than 3, which means that G ̸ ∈ Pn3 . Now we consider G has cycles. Case 1 Suppose that G contains a C3 as its subgraph and has three pendent vertices, then G must has G0 as its subgraph. Hence θ (G) > θ (G0 ) = 2.0743. Case 2 Suppose that G has a cycle C4 , and G has three pendant vertices, then G must have a subgraph like G2 , G3 and G4 showed in Fig. 3.2, then θ (G) > θ (G2 ), θ (G3 ), θ (G4 ) > 2. Case 3 If G has a cycle Cl , l ≥ 5, and has three pendant vertices, then by a similar discussion as in Theorem 3.8 case 3, θ (G) > 2. Therefore, T (1, 1, n − 3) minimize the largest matching roots in Pn3 . □ Theorem 3.10. Among all graphs in Pn4 , n ≥ 6, Yn is the unique graph with minimal largest matching root and θ (Yn ) = 2. In particular, K1,4 is the minimal graph when n = 5. Proof. Let G ⊂ Pn4 . Since G has four pendent vertices, then G ∼ = K1,4 or G contains a subgraph G1 in Fig. 3.2, or contains a Yi , i ≥ 6. By Lemmas 2.4, and 2.7 θ (G) > θ (G1 ) = 2.0743, θ (G) > θ (Yi ) = 2. Therefore, Theorem 3.10 holds. □ Acknowledgments We are very grateful to anonymous referees for their valuable, detailed comments and thoughtful suggestions, which lead to a great improvement on the presentation and proofs of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

Péter Csikvári, Graph polynomial and graph transformations (Ph.D. Thesis), April 2012. D.M. Cvetković, M. Doob, I. Gutman, A. Torgašev, Recent Results in the Theory of Graph Spectra, North-Holland, Amsterdam, 1988. D.M. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Theory, Cambridge University Press, 2010. D.C. Fisher, J. Ryan, Bounds on the largest roots of the matching polynomial, Discrete Math. 1–3 (110) (1992) 275–278. I. Gutman, A note on analogies between the characteristic and the matching polynomial of a graph, Publ. Inst. Math. Nouv. Sér. tome 31 (45) (1982) 27–31. I. Gutman, O.E. Polansky, Topological Concepts in Organic Chemistry, Springer, Berlin, 1986. O.J. Heilmann, E.H. Leib, Theory of monomer-dimer systems, Comm. Math. Phys. 25 (1972) 190–232. H. Hosoya, Topological index,a newly proposed quantity characterizing the topological nature of structural isomers of satuated hydrocarbons, Bull. Chem. Soc. Jpn. 44 (1971) 2332–2339. W. Liu, Q. Guo, Y. Zhang, Lihua Feng*, I. Gutman, Further results on the largest matching root of unicyclic graphs, Discrete Appl. Math. 221 (2017) 82–88. H.L. Zhang, R.F. Lin, The Hosoya index order of three type of special graphs, J. Combin. Math. Combin. Comput. 8 (2012) 225–232. H.L. Zhang, R.F. Lin, G.L. Yu, The Largest matching root of unicyclic graphs, Inform. Process. Lett. 113 (2013) 804–806.

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