Further results on the largest matching root of unicyclic graphs

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Further results on the largest matching root of unicyclic graphs✩ Weijun Liu a,b , Qiang Guo a , Yanbo Zhang b , Lihua Feng b, *, Ivan Gutman c,d a

School of Science, Nantong University, Nantong, Jiangsu, 226019, China School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, China Faculty of Science, University of Kragujevac, Kragujevac, Serbia d State University of Novi Pazar, Novi Pazar, Serbia b c

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Article history: Received 29 January 2016 Received in revised form 9 November 2016 Accepted 29 December 2016 Available online xxxx

a b s t r a c t Let G be a simple connected with vertex set V (G). The matching polynomial of G is ∑n/2 graph k n−2k defined as MG (x) = , where m(G, k) denotes the number of ways k=0 (−1) m(G, k) x in which k independent edges can be selected in G. Let λ1 (G) be the largest root of MG (x). We determine the unicyclic graphs with the four largest and the two smallest λ1 (G)-values. © 2017 Elsevier B.V. All rights reserved.

Keywords: Matching polynomial Unicyclic graphs The largest matching root

1. Introduction All graphs we consider in this paper are finite and simple. Let G = (V (G), E(G)) be a graph with n vertices and m edges. Let Γ (u) denote the neighbor set of the vertex u of G. The degree of u in G is denoted by dG (u), which is equal to |Γ (u)|. A k-matching in G is a set of k pairwise non-incident edges ∑ and the number of k-matchings in G is denoted by m(G, k). The original definition of the matching polynomial is [11] k m(G, k) xk . However, it is nowadays customary [7,14,15,17] to define this polynomial as MG (x) =

n/2 ∑

(−1)k m(G, k) xn−2k .

(1)

k=0

For convenience, we set m(G, 0) = 1. Clearly, m(G, k) = 0 if k > 2n . The roots of MG (x) are called the matching roots of G. It was proven in [19] (independently in [22]) that all roots of the matching polynomial of any graph are real numbers. The largest root of MG (x), denoted by λ1 (G), is called the largest matching root of G. It has been proven [12] that, except in the case of the edgeless graphs Kn , λ1 (G) is always positive. The reason for defining the matching polynomial via Eq. (1) is that in this case it is related to the characteristic polynomial of the adjacency matrix of G. In particular the matching and characteristic polynomials coincide if and only if G is a forest [15]. Moreover, the matching polynomial of any connected graph is a factor of the characteristic polynomial of some tree, see ✩ This research was supported by NSFC (Nos. 11671402, 11371207, 11301302), Hunan Provincial Natural Science Foundation ( 2016JJ2138), Mathematics and Interdisciplinary Sciences Project of CSU. Corresponding author. E-mail addresses: [email protected] (W. Liu), [email protected] (Q. Guo), [email protected] (Y. Zhang), [email protected] (L. Feng), [email protected] (I. Gutman).

*

http://dx.doi.org/10.1016/j.dam.2016.12.022 0166-218X/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: W. Liu, et al., Further results on the largest matching root of unicyclic graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2016.12.022.

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Fig. 1. The Kelmans Transformation.

[14, Theorem 6.1.1]. This provides another way to explain that the roots of matching polynomial are real numbers since the adjacency matrix of any graph is a symmetric matrix and so the roots of its characteristic polynomial are necessarily real-valued. Graph polynomials and their roots have been much studied in algebraic graph theory (see the recent works [8,10] and the references cited therein). In particular, the matching polynomial, as well as the problems related with its roots, have been studied in due detail [7,14,15,17,21,27,28]. In analogy with the traditional graph–energy concept [16,23], defined to be the sum of the absolute values of the eigenvalues of the adjacency matrix, the matching energy of a graph has been conceived as the sum of the absolute values of the roots of the matching polynomial [18]. This graph invariant has recently attracted much attention; see [1,3–5,24,25] and the references cited therein. In [12], Fisher and Ryan obtained several bounds for λ1 (G). Ghorbani [13] determined the graphs with at most five distinct matching roots. Zhang et al. [27,28] studied λ1 (G) of unicyclic graphs and graphs with six distinct matching roots. Other related works can be found in [7,17]. Motivated by the previous research, in this paper, we focus on the ordering unicyclic graphs with respect to the largest matching root. We determine the extremal graphs with the four largest and the two smallest λ1 (G). 2. Preliminaries If u ∈ V (G), then G − u is the graph obtained from G by deleting the vertex u and the edges of G incident to u. Similarly if e ∈ E(G), then G − e is the graph obtained from G by deleting the edge e. The following lemmas are well known. Lemma 2.1 ([7]). Let G1 + G2 be the direct sum (disjoint union) of the graphs G1 and G2 . Then MG1 +G2 (x) = MG1 (x) MG2 (x). Lemma 2.2 ([7]). ∑ Let G be a graph and u ∈ V (G). Suppose the neighborhood of u is Γ (u) = {v1 , v2 , . . . , vd }. Then MG (x) = x MG−u (x) − v ∈Γ (u) MG−uvi (x). i

Corollary 2.3 ([7]). If Pn is the path on n vertices, then MPn (x) = x MPn−1 (x) − MPn−2 (x). Lemma 2.4 ([7]). Let u, v ∈ V (G) and uv ∈ E(G). Then m(G, k) = m(G − uv, k) + m(G − {u, v}, k), and therefore MG (x) = MG−uv (x) − MG−u−v (x). Lemma 2.5 ([7]). Let G∗ be a spanning subgraph of G, λ1 (G) be the largest matching root of G. If x ≥ λ1 (G) then MG∗ (x) ≥ MG (x). If G∗ is a proper subgraph of G and x > λ1 (G), then MG∗ (x) > MG (x). The following transformation, called Kelmans Transformation, plays an prominent role in the sequel. This transformation is widely used in many other problems [6]. Definition 2.6. Let u, v be two vertices of the graph G. The Kelmans Transformation of G is as follows (cf. Fig. 1): erase all edges between v and N(v ) \ (N(u) ∪ u) and add all edges between u and N(v ) \ (N(u) ∪ u). Let us call u and v the beneficiary and the co-beneficiary of the transformation, respectively. The obtained graph has same number of edges as G; in general we will denote it by G′ without referring to the vertices u and v . Lemma 2.7 ([6,20]). Assume that G′ is a graph obtained from G by some Kelmans transformation. Then λ1 (G′ ) ≥ λ1 (G). Lemma 2.8 ([28]). For two disjoint graphs G1 and G2 with u ∈ V (G1 ) and v ∈ V (G2 ), let G be the graph obtained from G1 and G2 by adding an edge uv . Let G′ be the graph obtained from G1 and G2 by identifying u and v (to a new vertex say w ), and then adding a pendent edge to w . Then λ1 (G) < λ1 (G′ ). Lemma 2.9 ([27]). Let Gn,k be the graph of order n obtained from a cycle Ck by attaching n − k pendent edges at one vertex of Ck . Then λ1 (Gn,k ) < λ1 (Gn,k−1 ). Please cite this article in press as: W. Liu, et al., Further results on the largest matching root of unicyclic graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2016.12.022.

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Fig. 2. Graphs in Theorem 3.1.

3. Main results Let G1 be the graph obtained from a graph H and an isolated vertex v ̸ ∈ V (H), by connecting v to p vertices of H. For two disjoint graphs H1 and H2 , let G2 be the graph obtained by connecting v ∈ H2 and p vertices of H2 , as shown in Fig. 2. By a direct application of Lemma 2.2, we get: Theorem 3.1 ([7]). Let G1 and G2 be simple graphs shown in Fig. 2. Then

∑p

(1) MG1 (x) = xMH (x) − i=1 M ∑Hp−ui (x); (2) MG2 (x) = MH1 (x)MH2 (x) − i=1 MH1 −ui (x)MH2 −v (x). For a graph G with u ∈ V (G), and two disjoint paths Pa = u1 · · · ua and Pb = w1 · · · wb , let G1 be the graph obtained from G by adding two edges uu1 and uw1 . Let G2 = G1 − uw1 + ua w1 . Theorem 3.2. Let G1 and G2 be the graphs described above. Then λ1 (G1 ) > λ1 (G2 ). Proof. By Lemmas 2.1, 2.2 and 2.4, we have MG1 (x) = MG (x)MPa (x)MPb (x) − MG−u (x)MPa (x)MPb−1 (x)

− MG−u (x)MPa−1 (x)MPb (x), MG2 (x) = MG (x)MPa (x)MPb (x) − MG (x)MPa−1 (x)MPb−1 (x)

− MG−u (x)MPa (x)MPb−1 (x) + MG−u (x)MPa−1 (x)MPb−2 (x). Thus it follows that MG1 (x) − MG2 (x)

= MG (x)MPa−1 (x)MPb−1 (x) − MG−u (x)MPa−1 (x)MPb (x) − MG−u (x)MPa−1 (x)MPb−2 (x) = MG (x)MPa−1 (x)MPb−1 (x) − MG−u (x)MPa−1 (x)(MPb (x) + MPb−2 (x)) = MG (x)MPa−1 (x)MPb−1 (x) − xMG−u (x)MPa−1 (x)MPb−1 (x) = MPa−1 (x)MPb−1 (x)(MG (x) − xMG−u (x)) ⎛ ⎞ ∑ = MPa−1 (x)MPb−1 (x) ⎝− MG−uwi (x)⎠ . wi ∈Γ (u)

From Lemma 2.5 and putting x = λ1 (G1 ), we have MG1 (x) − MG2 (x) < 0. Hence we obtain λ1 (G1 ) > λ1 (G2 ). ■ Let G be a connected graph with u, v ∈ V (G) and uv ∈ E(G). For a path Pa of order a with two end vertices y, z, let G1 be the graph obtained identifying u and y. Let G2 = G1 − uv + z v . Theorem 3.3. Let G1 and G2 be the graphs described above. Then λ1 (G1 ) > λ1 (G2 ). Proof. By Lemmas 2.1, 2.4 and Theorem 3.1, we have, MG1 (x) = MG1 −uv (x) − MG−u−v (x)MPa−1 (x), MG2 (x) = MG2 −v z (x) − MG2 −v−z (x)

⎞

⎛ = MG1 −uv (x) − ⎝MPa−1 (x)MG−u−v (x) − MPa−2 (x)

∑

MG−v−uui (x)⎠ .

ui ∈ΓG (u)

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Fig. 3. Graph Transformation in Theorem 3.4.

Fig. 4. Five graphs in Theorem 3.5.

Therefore MG1 (x) − MG2 (x) = −MPa−2 (x)

∑

MG−v−uui (x).

ui ∈Γ (u)

By Lemma 2.5 and putting x = λ1 (G1 ), we have MG1 (x) − MG2 (x) < 0. Hence λ1 (G1 ) > λ1 (G2 ). ■ Theorem 3.4. Let G1 and G2 be unicyclic graphs with n vertices as shown in Fig. 3. If 2 ≤ k ≤ equality holding if and only if k = 2n − 1.

n 2

− 1, then λ1 (G1 ) ≤ λ1 (G2 ), with

Proof. We first easily get m(G1 , 1) = n, m(G1 , 2) = kn − k2 − 3k + n − 3 and m(G2 , 1) = n m(G2 , 2) = kn − k2 − k − 1. From the definition of matching polynomial, we have MG1 (x) = xn − nxn−2 + (kn − k2 − 3k + n − 3)xn−4 , MG2 (x) = xn − nxn−2 + (kn − k2 − k − 1)xn−4 . So we have MG1 (x) − MG2 (x) = (n − 2k − 2)xn−4 . By Lemma 2.5, if k ≤

n 2

− 1, putting x = λ1 (G1 ), one has MG1 (x) − MG2 (x) ≥ 0. Hence λ1 (G1 ) ≤ λ1 (G2 ). ■

Theorem 3.5. Let Sn2 , Sn3 , Sn4 , Sn5 and Sn6 be unicyclic graphs with n(≥8) vertices as shown in Fig. 4. Then λ1 (Sn2 ) > λ1 (Sn3 ) >

λ1 (Sn4 ) > λ1 (Sn5 ) > λ1 (Sn6 ).

Proof. We can easily get that m(Sni , 1) = n (for i = 2, 3, 4, 5, 6), m(Sn2 , 2) = 2n − 7, m(Sn3 , 2) = 2n − 6, m(Sn3 , 3) = n − 5, m(Sn4 , 2) = 2n − 6 and m(Sn5 , 2) = 3n − 13, m(Sn6 , 2) = 3n − 11. Therefore MS 2 (x) = xn − nxn−2 + (2n − 7)xn−4 , n

MS 3 (x) = xn − nxn−2 + (2n − 6)xn−4 − (n − 5)xn−6 , n

MS 4 (x) = xn − nxn−2 + (2n − 6)xn−4 , n

MS 5 (x) = xn − nxn−2 + (3n − 13)xn−4 , n

MS 6 (x) = xn − nxn−2 + (3n − 11)xn−4 . n

√

√

√

So we have MS 2 (x) − MS 3 (x) = xn−6 [−x2 + (n − 5)]. Since MS 2 ( n − 4) < 0, thus λ1 (Sn2 ) > n − 4. For any x > n − 4, n n n we have MS 2 (x) − MS 3 (x) < 0. By Lemma 2.5 and putting x = λ1 (Sn2 ), we have MS 2 (x) − MS 3 (x) < 0. Hence λ1 (Sn2 ) > λ1 (Sn3 ). n

n

n

n

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Similarly, MS 3 (x) − MS 4 (x) = −(n − 5)xn−6 . By Lemma 2.5 and putting x = λ1 (Sn3 ), we have MS 3 (x) − MS 4 (x) < 0. Hence n

λ1 (Sn3 ) > λ1 (Sn4 ).

n

n

n

Since MS 4 (x) − MS 5 (x) = (−n + 7)xn−4 , by Lemma 2.5 and putting x = λ1 (Sn4 ), we have MS 4 (x) − MS 5 (x) < 0. Hence n n n n λ1 (Sn4 ) > λ1 (Sn5 ). Since MS 5 (x) − MS 6 (x) = −2xn−4 , by Lemma 2.5 and putting x = λ1 (Sn5 ), we have MS 5 (x) − MS 6 (x) < 0. Hence n n n n λ1 (Sn5 ) > λ1 (Sn6 ). The proof is now complete. ■ For a cycle Cn−k with an edge uv , and a path Pk = u1 u2 · · · uk , let G1 be the unicyclic graph obtained from Cn−k and Pk by adding an edge uu1 . Let G2 = G1 − uv + v u1 . Theorem 3.6. Let G1 and G2 be unicyclic graphs with n vertices as described above. If k ≤ n − 1 < k ≤ n − 3, then λ1 (G1 ) < λ1 (G2 ). 2

n 2

− 1, then λ1 (G1 ) > λ1 (G2 ). If

Proof. By Lemmas 2.1, 2.2 and 2.4, we have MG1 (x) = MPn (x) − MPk (x)MPn−k−2 (x), MG2 (x) = MPn (x) − MPk−1 (x)MPn−k−1 (x). Applying Corollary 2.3, if k ≤

n 2

− 1, we have

MG1 (x) − MG2 (x) = MPk−1 (x)MPn−k−1 (x) − MPk (x)MPn−k−2 (x)

= MPk−1 (x)(xMPn−k−2 (x) − MPn−k−3 (x)) − MPn−k−2 (x)(xMPk−1 (x) − MPk−2 (x)) = MPk−2 (x)MPn−k−2 (x) − MPk−1 (x)MPn−k−3 (x) ··· = MP1 (x)MPn−2k+1 (x) − MP2 (x)MPn−2k (x) = xMPn−2k+1 (x) − (x2 − 1)MPn−2k (x) = x(xMPn−2k (x) − MPn−2k−1 (x)) − (x2 − 1)MPn−2k (x) = MPn−2k (x) − xMPn−2k−1 (x) = −MPn−2k−2 (x). By Lemma 2.5 and putting x = λ1 (G1 ), we have MG1 (x) − MG2 (x) < 0. Hence λ1 (G1 ) > λ1 (G2 ). If 2n − 1 < k ≤ n − 3 we have MG1 (x) − MG2 (x) = MPk−1 (x)MPn−k−1 (x) − MPk (x)MPn−k−2 (x)

= MPk−1 (x)(xMPn−k−2 (x) − MPn−k−3 (x)) − MPn−k−2 (x)(xMPk−1 (x) − MPk−2 (x)) = MPk−2 (x)MPn−k−2 (x) − MPk−1 (x)MPn−k−3 (x) ··· = MP2 (x)MP2k−n+2 (x) − MP1 (x)MP2k−n+3 (x) = (x2 − 1)MP2k−n+2 (x) − xMP2k−n+3 (x) = (x2 − 1)MP2k−n+2 − x(xMP2k−n+2 (x) − MP2k−n+1 (x)) = xMP2k−n+1 (x) − MP2k−n+2 (x) = MP2k−n (x). From Lemma 2.5 and putting x = λ1 (G1 ), we have MG1 (x) − MG2 (x) > 0. Hence λ1 (G1 ) < λ1 (G2 ). The proof is now complete. ■ Let Sn+ be the unicyclic graph obtained by introducing a new edge to the star Sn of order n. We denote by Un,t the graph of order n with t vertices on the cycle of degree at least 3. We denote by C (m1 , m2 , . . . , mg ) the unicyclic graph with girth g, such that there are mi pendent edges attached to the vertex vi of the cycle, 1 ≤ i ≤ t. This graph is sometimes referred to as the sun graph. For u ∈ V (Sk ), the center of Sk , v ∈ V (Sn+−k+1 ) with degree 2, let Cn,k be the graph obtained by identifying u and v . For u ∈ V (Sk ), the center of Sk , v ∈ V (Sn+−k+1 ) with degree 1, let Dn,k be the graph obtained by identifying u and v . Let Cn2 be the graph obtained by adding a new pendent edge to the cycle Cn−1 . Let D2n be the graph obtained by identifying one end vertex of Pn−2 and one vertex of C3 . Theorem 3.7. Among all unicyclic graphs of order n(≥8), the first four graphs with the largest matching root are Sn+ , Sn2 , Sn3 , and Sn4 . Among unicyclic graphs of order n(≥8), the last three graphs with the largest matching root are Cn2 or D2n and Cn . Please cite this article in press as: W. Liu, et al., Further results on the largest matching root of unicyclic graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2016.12.022.

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Proof. Assume that Un,t is a unicyclic graph of order n with girth g, such that on its cycle there are t vertices of degree at least 3. We have to separately consider the following three cases. Case 1: g ≥ 3 and t ≥ 2. First, by Lemma 2.8, we contract the cut edges of the trees rooted at the cycle, and get the graph C (m1 , m2 , . . . , mg ), with t of g vertices on the cycle of degree at least 3. Second we apply the Kelmans Transformation of Lemma 2.7 on two vertices on the cycle of C (m1 , m2 , . . . , mg ), one of degree at least 3, the other being its neighbor. Then we get a sun graph with smaller girth. Finally we arrive at the graph Cn,k . Each step of the above procedure increases the largest matching root. Case 2: g > 3 and t = 1. First we apply Lemma 2.8 to contract the cut edges of the trees rooted at the cycle, resulting in the graph C (n − k, 0, . . . , 0). We then apply Lemma 2.9 to shrink the cycle Cg and add more vertices on the root of the tree. Each step increases the largest matching root. At last we arrive at Sn4 , having the maximal λ1 in this case. Case 3: g = 3 and t = 1. If outside the cycle, there is no vertex of degree at least 2, then the graph is Sn+ . If there exists at least one such vertex, then by applying Lemma 2.8 in a similar manner as above, we obtain Dn,k . Now, by Lemma 2.9, λ1 (Sn4 ) < λ1 (Sn+ ). By Lemma 2.8, λ1 (Dn,k ) < λ1 (Sn+ ). By Lemma 2.7, λ1 (Cn,k ) < λ1 (Sn+ ). Thus, Sn+ has the largest λ1 among all unicyclic graphs. For any Dn,k1 , bearing in mind Lemma 2.7, by comparing the vertex u and one vertex of degree 2 in Dn,k1 , we find that there exists one Cn,k2 such that λ1 (Dn,k1 ) < λ1 (Cn,k2 ). From Theorem 3.4, Cn,2 ∼ = Sn2 has the maximal λ1 among all graphs of the form Cn,k . From Theorem 3.5, λ1 (Sn4 ) < λ1 (Sn2 ). Thus Sn2 has the second largest λ1 . By Theorem 3.4, Cn,3 ∼ = Sn5 has the second largest λ1 among all graphs of the form Cn,k . By Theorem 3.5, λ1 (Sn4 ) > λ1 (Sn5 ). For any Dn,k1 except for the two graphs Dn,2 ∼ = Sn3 and Dn,n−3 ∼ = Sn6 , by Lemma 2.7 by comparing the vertex u and one vertex of degree 2 in Dn,k1 , we conclude that there exists one Cn,k2 such that λ1 (Dn,k1 ) < λ1 (Cn,k2 ). Thus by Theorem 3.5, Sn3 has the third largest λ1 . In order to determine the candidate with the fourth largest λ1 , we need only consider the graphs Cn,3 ∼ = Sn5 , Sn4 , and Sn6 . By 3 Theorem 3.5, Sn is the fourth largest one. We now consider Un,k with smaller λ1 . First, for each rooted tree on the cycle, by Theorem 3.2, when the tree becomes a path, λ1 decreases. Next, by Theorem 3.3, we can obtain a unicyclic graph with exactly one vertex on the cycle having degree 3, like the graph G1 in Theorem 3.6. This transformation also decreases the largest matching root. Finally, by Theorem 3.6, if k ≤ 2n − 1, then we conclude that Cn2 has the second smallest λ1 . If 2n < k ≤ n − 4, then D2n has the second smallest λ1 . We can easily check that MC 2 (x) ≡ MD2 (x) (see, for example [26]) and therefore λ1 (Cn2 ) = λ1 (D2n ). n n This completes the proof. ■ 4. Further remarks In the present paper, the largest and smallest greatest roots of the matching polynomial (or bounds for them) could be determined explicitly. In the general case, these roots would be the solutions of equations of the type aν xν + aν−1 xν−1 +· · ·+ a0 = 0, where ν is some integer smaller than (or in the worst case, equal to) the number of vertices. It would be interesting to tackle such problems in the future. Similar results, but for different polynomials, have been proven in [8,9]. As already mentioned, the matching energy attracted recently much attention [1–5,18,24,25]. The graphs with largest and smallest λ1 -values are evident candidates for being the graphs with smallest and largest matching energies. Thus, any result for λ1 sheds some light on the structure–dependency of matching energy, or –at least –provides a guideline for its investigation. Several recent studies of matching energy pertain to unicyclic graphs [2–4]. For these, the results of the present paper would be directly applicable. Acknowledgments The authors are grateful to the anonymous referees for their remarks and suggestions which improve the manuscript a lot. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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Please cite this article in press as: W. Liu, et al., Further results on the largest matching root of unicyclic graphs, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2016.12.022.