THE MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS WITH GIVEN GIRTH

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 46, Number 4, 2016 THE MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS WITH GIVEN GIRTH HONG-HAI LI AND LI ZOU ABSTRACT. The matching energy of a graph was introduced by Gutman and Wagner in 2012 and defined as the sum of the absolute alues of zeros of its matching polynomial. Let θ(r, s, t) be the graph obtained by fusing two triples of pendant ertices of three paths P r+2, P s+2 and P t+2 to two ertices. The graph obtained by identifying the center of the star S n g with the degree 3 ertex u of θ(1, g 3, 1) is denoted by S n g (u)θ(1, g 3, 1). In this paper, we show that, S n g (u)θ(1, g 3, 1) has minimum matching energy among all bicyclic graphs with order n and girth g. 1. Introduction. All graphs in this paper are finite, simple and nondirected. Let G = (V, E) be such a graph with order V = n and size E = m. In a graph a matching is a set of pairwise nonadjacent edges, and we denote the number of k-matchings of G by m k (G). Note that m 1 (G) = m and m k (G) = 0 for k > n/2. It is both consistent and conenient to define m 0 (G) = 1. The matching polynomial of G is defined as α(g, x) = k 0( 1) k m k (G)x n 2k. All the zeros of α(g, x) are real-alued and the theory of matching polynomials is well elaborated in [3, 5]. Recently, Gutman and Wagner [7] introduced the matching energy of a graph G, denoted by ME(G) and defined as 2010 AMS Mathematics subject classification. Primary 05C35, 05C50. Keywords and phrases. Bicyclic graph, matching energy, girth. The first author is supported by National Natural Science Foundation of China, (grant Nos , ), the Scientific Funds of the Education Department of Jiangxi Proince (grant No. GJJ150345), and the Sponsored Program for Cultiating Youths of Outstanding Ability in Jiangxi Normal Uniersity. The first author is the corresponding author. Receied by the editors on September 20, DOI: /RMJ Copyright c 2016 Rocky Mountain Mathematics Consortium 1275

2 1276 HONG-HAI LI AND LI ZOU (1.1) ME(G) = 2 π 0 [ 1 ] x 2 ln m k (G)x 2k dx, k 0 which coincides with the Coulson-type integral formula for the energy which has been studied extensiely (see an excellent monograph [13] and [14] and the references therein for recent adances), when the graph under consideration is a tree. As mentioned in [7], matching energy can also be defined in another form as follows. Let G be a simple graph, and let µ 1, µ 2,..., µ n be the zeros of its matching polynomial. Then n ME(G) = µ i. The integral on the right side of equation (1.1) is increasing in all the coefficients m k (G). This means that if two graphs G and G satisfy m k (G) m k (G ) for all k 1, then ME(G) ME(G ). If, in addition, m k (G) < m k (G ) for at least one k, then ME(G) < ME(G ). This then motiates the introduction of a quasi-order, defined by i=1 G H m k (G) m k (H), for all nonnegatie integers k. If G H and there exists some k such that m k (G) > m k (H), then we write G H. It is said that G is m-greater than H if G H and strictly m-greater than H if G H. It is easy to see that and G H = ME(G) ME(H) G H = ME(G) > ME(H). Initial work on matching energy of graphs is attributed to [7] and then followed by Li and Yan [16] who characterized the maximal connected graphs with gien connectiity and chromatic numbers. The extremal graphs in connected bicyclic graphs were determined by Ji, Li and Shi [10] and further by Chen, Liu and Shi [1, 2] for unicyclic, bicyclic and tricyclic graphs. More generally, the minimal matching

3 MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS 1277 energy of (m, n)-graphs with a gien matching number was obtained by Xu, Das and Zheng [19]. Huang, Kuang and Deng [9] characterized the extremal graph for a random polyphenyl chain. For more results, see [11, 12, 17] and there may be other results which are unknown to the authors. 1 w 1 C g n g q +2 {}}{ C q 2 3 u 1 u 2 w 2 w 3 r 1 u s w t 1 (i) n (g, q) r u (ii) θ(r, s, t) w t Figure 1. Two types of braces: n(g, q) and θ(r, s, t). Denote the set of all connected bicyclic graphs with order n and girth g by B n,g. We now define two special classes of bicyclic graphs. Let n (g, q) denote the graph obtained by the coalescence of two end ertices of a path P n g q+2 with one ertex of two cycles C g and C q, respectiely, and θ(r, s, t) the graph obtained by fusing two triples of pendant ertices of three paths P r+2, P s+2 and P t+2 to two ertices, as gien in Figure 1. The distance of two cycles C g and C q in G is defined as d G (C g, C q ) = min{d G (x, y) x V (C g ), y V (C q )}, sometimes written as d G for short. Note that d G (C g, C q ) = 0 if C g and C q hae a common ertex, e.g., for G = n (g, q) such that q = n g+1, and in this case, n (g, q) with q = n g+1 is simply written as (g, q) for conenience. Clearly, any bicyclic graph must contain either graph (i) or (ii) in Figure 1 as an induced subgraph, called its brace. Then the set B n,g can be partitioned into two subsets Bn,g 1 and Bn,g, 2 where Bn,g 1 is the set of all bicyclic graphs which contain a brace of the form n (g, q), and Bn,g 2 is the set of all bicyclic graphs which contain a brace of the form θ(r, s, t).

4 1278 HONG-HAI LI AND LI ZOU There hae been some papers on characterizing the minimal Hosoya index of graphs, see [4, 18]. In this paper, minimal graphs in B i n,g for i = 1, 2, are determined respectiely and, by comparing them, we show that S n g (u)θ(1, g 3, 1), obtained by identifying the center of the star S n g with a ertex u of degree 3 in θ(1, g 3, 1), has minimum matching energy among all bicyclic graphs in B n,g. 2. Preliminaries. In this section, we shall present some basic results which will be used in the proof of our main results. Gien a graph G and an edge u of G, we denote by G u (respectiely G ) the graph obtained from G by deleting the edge u (respectiely the ertex and edges incident to it). Lemma 2.1 ([10]). If u, are adjacent ertices of G, then m k (G) = m k (G u) + m k 1 (G u ), for all nonnegatie integers k. Let G()S t+1 (or S t+1 ()G) denote the graph obtained by identifying the ertex of a graph G with the center of the star S t+1, as gien in Figure 2. G {}}{ t Figure 2. G()S t+1. Note. Consider the graph in Figure 2. m k (G) is the number of k-matchings that do not contain any of the edges of S t+1. tm k 1 (G ) is the number of k-matchings that contain one of these edges, as there are t choices for these edges in S t+1. Consequently, we hae what follows without giing a formal proof.

5 MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS 1279 Lemma 2.2. Let G be a graph, and let be a ertex of G. m k (G()S t+1 ) = m k (G) + tm k 1 (G ). Then, Recall a result in [13], which establishes the order of the union of two paths with a gien number of ertices according to the quasi-order as stated in the introduction. Lemma 2.3 ([13]). Let n = 4k, 4k + 1, 4k + 2 or 4k + 3. Then, P n P 2 P n 2 P 4 P n 4 P 2k P n 2k P 2k+1 P n 2k 1 P 2k 1 P n 2k+1 P 3 P n 3 P 1 P n 1. Lemma 2.4 ([6]). If G 1 G 2, then G 1 H G 2 H, where H is an arbitrary graph. Applying Lemma 2.4, we can generalize Lemma 2.3 to the following form, the union of three paths. Lemma 2.5. Let r, s, t be nonnegatie integers with r s 2. If r is een, then P r 2 P s+2 P t P r P s P t P r+1 P s 1 P t P r 1 P s+1 P t. Lemma 2.6 ([7]). Suppose that G is a connected graph and T an induced subgraph of G such that T is a tree and is connected to the rest of G only by a cut ertex. If T is replaced by a star of the same order and centered at, then the matching energy decreases (unless T is already such a star). If T is replaced by a path with one end at, then the matching energy increases (unless T is already such a path). Recall the definition of a generalized π-transform in [15]. We say Q is a branch of a connected graph G with root u if Q is a connected induced subgraph of G for which u is the only ertex in Q that has a neighbor not in Q. Let P and Q be branches of a component of a graph G with a common root u 0, which is also their only common ertex. Assume that P is a path and u 0 has at least one neighbor in G that does not lie on P or Q. Form a graph from G by relocating

6 1280 HONG-HAI LI AND LI ZOU the branch Q from u 0 to where is the other end ertex of the path P (by deleting edges u 0 w and adding new edges w for eery ertex w in Q adjacent to u 0 ). We refer to the resulting graph as a generalized π-transform of G. Theorem 2.7 ([15]). If G is a generalized π-transform of G, then G G and so ME(G ) > ME(G). G C g C q 1 = r n (i) P n ( r,)g (ii) S n (g, q) Figure 3. Let P n ( r, )G denote the graph obtained by identifying the ertex r of P n with the ertex of G (see Figure 3 (i)). For conenience, we use P n ()G (or G()P n ) to stand for P n ( 1, )G. Note that P n ( r, )G and P r ()G()P n r+1 are isomorphic. Theorem 2.8 ([8]). If is an arbitrary ertex of the graph G, then for n = 4k + i, i { 1, 0, 1, 2}, k 1, P n ( 1, )G P n ( 3, )G P n ( 2k+1, )G P n ( 2k, )G P n ( 2k 2, )G P n ( 2, )G. Let G be an arbitrary graph with a specified ertex. The graph obtained from G is denoted by Ĝi for i = 1, 2,..., n 1 (as gien in Figure 4), by adding n 1 new ertices to G in the following way. Attach i 1 pendant edges and a pendant path of length n i at. By Theorem 2.8, we easily obtain the following. Lemma 2.9. Ĝ 1 Ĝ2 Ĝn 1. Proof. Ĝ 1 Ĝ2 follows immediately from Theorem 2.8, as Ĝ1 = P n ( 1, )G and Ĝ2 = P n ( 2, )G. In fact, other cases can be erified

7 MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS 1281 G 1 G 2 n 1 2 G 1 G 2 n 1 2 G G n 3 1 n 1 n n 2 2 G 3 G n 1 Figure 4. in the same way. Note that if we denote the graph G()S i by H, then Ĝ i = Pn i+1 ( 1, )H and Ĝi+1 = P n i+1 ( 2, )H. Let S n (g, q) be the graph in B 1 n,g with n + 1 (g + q) pendant edges attached at the common ertex of C g and C q (see Figure 3 (ii)). Theorem S n (g, q) S n (g, g) with equality if and only if g = q. Proof. Let u (u, respectiely) be the common ertex of C g and C q (C g, respectiely) in S n (g, q) (S n (g, g), respectiely), and u 1 u 2 (u 1u 2, respectiely) an edge of C q (C g, respectiely) such that u 1 (u 1, respectiely) is adjacent to u (u, respectiely). By Lemma 2.1, we hae m k (S n (g, q)) = m k (S n (g, q) u 1 u 2 ) + m k 1 (S n (g, q) u 1 u 2 ) and m k (S n (g, g)) = m k (S n (g, g) u 1u 2) + m k 1 (S n (g, g) u 1 u 2).

8 1282 HONG-HAI LI AND LI ZOU Note that S n (g, q) u 1 u 2 = Pq 1 (u)c g (u)s n+3 g q S n (g, g) u 1u 2 = P g 1 (u)c g (u)s n+3 2g. If we denote the graph P i 1 (u)c g (u)s n+3 g i by G i, then S n (g, q) u 1 u 2 = Gq and S n (g, g) u 1u 2 = G g. By Lemma 2.9, we hae G q G q 1 G g. Thus, S n (g, q) u 1 u 2 S n (g, g) u 1u 2 with equality if and only if g = q. In the same way, we also get S n (g, g) u 1 u 2 S n (g, g) u 1 u 2 with equality if and only if g = q. Therefore, S n (g, q) S n (g, g) with equality if and only if g = q. Theorem If t 2 and r is een, then (2.1) θ(r 2, s + 2, t) θ(r, s, t) θ(r + 1, s 1, t) where r s 2 and r + s + 2 = g. θ(r 1, s + 1, t), Proof. Let G = θ(r, s, t), which can be obtained by merging two triples of pendant ertices u 0, 0, w 0 and u r+1, s+1, w t+1 of three paths and P r+2 = u 0 u 1 u r u r+1, P s+2 = 0 1 s s+1, P t+2 = w 0 w 1 w t w t+1, to two ertices, say u and, respectiely. By Lemma 2.1, we hae m k (G) = m k (G uw 1 ) + m k 1 (G u w 1 ) = m k (G uw 1 w t ) + m k 1 (G uw 1 w t ) + m k 1 (G u w 1 w t ) + m k 2 (G u w 1 w t ) = m k (P t C g ) + 2m k 1 (P g 1 P t 1 ) + m k 2 (P r P s P t 2 ).

9 MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS 1283 For conenience, let G 1 = θ(r 1, s + 1, t), G 2 = θ(r 2, s + 2, t) and G 3 = θ(r + 1, s 1, t). Applying the same method to the graphs G 1, G 2 and G 3, we get Thus, m k (G 1 ) = m k (P t C g ) + 2m k 1 (P g 1 P t 1 ) + m k 2 (P r 1 P s+1 P t 2 ), m k (G 2 ) = m k (P t C g ) + 2m k 1 (P g 1 P t 1 ) + m k 2 (P r 2 P s+2 P t 2 ), m k (G 3 ) = m k (P t C g ) + 2m k 1 (P g 1 P t 1 ) + m k 2 (P r+1 P s 1 P t 2 ). m k (G 1 ) m k (G) = m k 2 (P r 1 P s+1 P t 2 ) m k 2 (P r P s P t 2 ), m k (G 2 ) m k (G) = m k 2 (P r 2 P s+2 P t 2 ) m k 2 (P r P s P t 2 ), m k (G 3 ) m k (G 1 ) = m k 2 (P r+1 P s 1 P t 2 ) m k 2 (P r 1 P s+1 P t 2 ). By Lemma 2.5, it follows directly that if r s 2 and r is een, then P r 2 P s+2 P t P r P s P t P r+1 P s 1 P t P r 1 P s+1 P t, and so assertion (2.1) holds. As an immediate consequence, we hae the following result. Corollary If t 2, then θ(r, s, t) θ(1, r+s 1, t) with equality if and only if r = 1 or s = 1. Proof. Without loss of generality, assume that r s. If r = 0, by Theorem 2.11, we hae either θ(0, s, t) = θ(r, s, t) θ(r + 1, s 1, t) = θ(1, s 1, t)

10 1284 HONG-HAI LI AND LI ZOU when s 2 or s = 1 and, in this case, θ(0, s, t) is already of the form θ(1, r + s 1, t). Now assume that r 1. If r is een, then θ(r, s, t) θ(r 1, s + 1, t) θ(r 3, s + 3, t) θ(1, s + r 1, t). If r is odd, then θ(r, s, t) θ(r 2, s+2, t) θ(r 4, s+4, t) θ(1, s+r 1, t). 3. Main results. In this section, we first show that S n (g, g) has minimum matching energy in Bn,g 1 and S n g (u)θ(1, g 3, 1) is the minimal graph in Bn,g. 2 Further, by comparing these two graphs, we conclude that S n g (u)θ(1, g 3, 1) is the unique graph with minimum matching energy in B n,g. Theorem 3.1. For any graph G B 1 n,g, we hae G S n (g, g) with equality if and only if G = S n (g, g). Proof. For any graph G B 1 n,g, its brace must be of the form n (g, q) for some q g. By Lemma 2.6, if any tree branch is replaced by a star of equal order centered at the root, its matching energy strictly decreases unless the branch is already such a star. To show that G S n (g, g) for any G B 1 n,g, it suffices to show it holds for such a graph G, all of whose tree branches are stars. We distinguish two cases according to the alue of d G, the distance between C g and C q in its brace. Case 1. d G = 0. As aboe, if G has l tree branches, we can assume that these l branches are stars. Without loss of generality, for i = 1,..., l, we will assume G is the coalescence of the ertex u i in its brace (g, q) and the center of S ri +1. For conenience, we use the notation G i to denote such graphs recursiely defined as follows. Let G 0 = (g, q), and if G i 1 is already defined, then G i is defined to be G i 1 (u i )S ri +1. Note that G l = G. Applying Lemma 2.2 to S n (g, q), we hae (3.1) m k (S n (g, q)) = m k ( (g, q))+(n+1 g q)m k 1 (P g 1 P q 1 ).

11 MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS 1285 Similarly applying Lemma 2.2 to G, we hae m k (G) = m k (G l 1 ) + r l m k 1 (G l 1 u l ) (3.2) where = m k (G l 2 ) + r l 1 m k 1 (G l 2 u l 1 ) + r l m k 1 (G l 1 u l ) = = m k (G 0 ) + l r i m k 1 (G i 1 u i ), i=1 l r i = n + 1 g q. i=1 First, G 0 = (g, q). Second, G i 1 u i (g, q) u i P g 1 P q 1 for i = 1,..., l, because each graph is a subgraph of the former. If G S n (g, q), then for some i, G i 1 u i has P g 1 P q 1 as a proper subgraph. Therefore, G S n (g, q) with equality if and only if G = S n (g, q). By Theorem 2.10, the assertion holds. Case 2. d G 1. By Lemma 2.6, we can assume that all tree branches of G are stars. Suppose P = w 1 w 2 w t is the unique path connecting two cycles C g and C q in G. We proceed by induction on the number of pendant ertices along the path P. If there are no pendant ertices along the path, by applying the inerse generalized π-transform to G, i.e., deleting all edges w t u, where u N(w t ) \ {w t 1 } and adding new edges w 1 u, it becomes a graph in B 1 n,g with d G (C g, C q ) = 0 and so, by Case 1, G S n (g, g). Now assume that there is at least a pendant edge u on the path P. By Lemma 2.1, we hae m k (G) = m k (G u) + m k 1 (G u ). By the induction hypothesis, G u S n 1 (g, q). Also, it is easy to see that G u has P g 1 P q 1 as its subgraph. Thus, m k (G) = m k (G u) + m k 1 (G u ) m k (S n 1 (g, q)) + m k 1 (P g 1 P q 1 ) = m k (S n (g, q)), and strict inequality holds for at least one k. Therefore, G S n (g, q) and then G S n (g, g) by Theorem 2.10.

12 1286 HONG-HAI LI AND LI ZOU Theorem 3.2. For any positie integers r, s, t with r + s + 2 = g, let G 1, G 1, G 1 Bn,g 2 be defined as G 1 = G 0 (u)s n r s t 1, G 1 = G 0(u)S n g t+1 and G 1 = G 0(u)S n g, where G 0 = θ(r, s, t), G 0 = θ(1, g 3, t), G 0 = θ(1, g 3, 1), as gien in Figure 5. Then: (i) G 0 with a ertex of degree 2 deleted is strictly m-greater than G 0 with a ertex of degree 3 deleted; (ii) for any G B 2 n,g with θ(r, s, t) as its brace, G G 1 with equality if and only if G = G 1 ; (iii) G 1 G 1 with equality if and only if G 1 = G 1 ; (i) G 1 G 1 with equality if and only if G 1 = G 1. Consequently, for any G B 2 n,g, G G 1 with equality if and only if G = G 1. Proof. (i) Choose two ertices of degrees 2 and 3 respectiely in G 0, say w m and u. Consider G 0 w m, which can be iewed as a cycle C g together with two pendant paths, namely, P = uw 1 w m 1 at u and Q = w t w m+1 at, possibly of length 0. If one of the two paths P or Q is of length 0, then we can choose an appropriate edge e such that G 0 w m e = P n 1. Since P n 1 is m-greater than any tree T of order n 1, we hae G 0 w m P n 1 G 0 u as G 0 u is a tree. Now assume that both P and Q are not of length 0. By Lemma 2.1, we hae m k (G 0 w m ) m k (G 0 u) = m k (G 0 w m uw 1 ) Note that + m k 1 (G 0 w m u w 1 ) m k (G 0 u w m 1 w m ) m k 1 (G 0 u w m 1 w m ). G 0 w m u w 1 and G 0 u w m 1 w m are the union of the same graph T g+t m w m (see Figure 5) and the path P m 2 and so are isomorphic. Thus, m k 1 (G 0 w m u w 1 ) = m k 1 (G 0 u w m 1 w m ) for any k. It is clear that G 0 w m uw 1 is the union of a graph P t m+1 ()C g and a path P m 1, and G 0 u w m 1 w m is the union of a graph T g+t m and a path

13 MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS 1287 u 1 u 2 u 1 w 2 w 1 u 1 w 2 w 1 1 u u 3 C g 2 C q w 3 u 1 2 C g C q w 3 u 1 C g 2 C g w 1 u r 1 s w t 1 g 3 w t 1 g 3 u r G 0 w t G 0 w t G 1 u 1 u 2 u 3 C g 1 2 u C q w 2 w 3 w 1 1 u 1 2 u C g C q w 2 w 3 w 1 u 1 u r w t w m+1 w m u s u r 1 s w t 1 g 3 w t 1 u r G 1 w t G 1 w t T g+t m Figure 5. P m 1. Due to P t m+1 ()C g P r+s+t m+2 T g+t m and Lemma 2.4, we hae G 0 w m uw 1 P r+s+t m+2 P m 1 G 0 u w m 1 w m. Therefore, G 0 w m G 0 u. (ii) As in the proof of Theorem 3.1, by Lemma 2.6, we can assume that all tree branches at the cycles of G are stars. Without loss of generality, suppose that G is the coalescence of the ertex t i (here we use new notation for these ertices) in G 0 and the center of S ri +1 for i = 1,..., l, and l i=1 r i = a. For conenience, we use H i to denote graphs defined recursiely as follows. Let H 0 = G 0, and if H i 1 is already defined, then H i is defined to be H i 1 (t i )S ri+1. Note that H l = G. Applying Lemma 2.2, we hae (3.3) m k (G 1 ) = m k (G 0 ) + am k 1 (G 0 u) and

14 1288 HONG-HAI LI AND LI ZOU m k (G) = m k (H l 1 ) + r l m k 1 (H l 1 t l ) = m k (H l 2 ) + r l 1 m k 1 (H l 2 t l 1 ) + r l m k 1 (H l 1 t l ) + = m k (G 0 ) + l r i m k 1 (H i 1 t i ). i=1 First note that G 0 t i is a (proper) subgraph of H i t i, and so H i t i G 0 t i. Further, by (i) aboe, we hae G 0 t i G 0 u. Therefore, H i t i G 0 t i G 0 u and then G G 1 unless G is already such a graph G 1. (iii) By Corollary 2.12, we hae G 0 G 0. Note that G 0 u = P r+s+1 (u r+1, )P t and G 0 u = P r+s+1 (u 2, )P t. By Lemma 2.9, we get G 0 u G 0 u. So we hae m k (G 1 ) = m k (G 0 ) + am k 1 (G 0 u) m k (G 0) + am k 1 (G 0 u) = m k (G 1). Therefore, G 1 G 1. From the process aboe, equality holds only if G 1 = G 1. and (i) By Lemma 2.1, m k (G 1) = m k (G 1 w t ) + m k 1 (G 1 w t ) m k (G 1 ) = m k (G 1 w 1 ) + m k 1 (G 1 w 1 ). By Lemma 2.9, we get G 1 w t G 1 w 1 and G 1 w t G 1 w 1. Hence, G 1 G 1 with equality if and only if G 1 = G 1. Next, we shall compare the minimal graphs from B 1 n,g and B 2 n,g. Theorem 3.3. (see Figure 5). S n (g, g) S n g (u)θ(1, g 3, 1) (= G 1 )

15 MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS 1289 Proof. Let u 0 be the common ertex of the two copies of C g in S n (g, g), where G 1 = S n g (u)θ(1, g 3, 1), and let u 1 u 2 be an edge with u 1 adjacent to u 0. Note that S n (g, g) u 1 u 2 is the coalescence of a ertex in C g with the center of a star and an end ertex of a path, i.e., S n (g, g) u 1 u 2 = Pg 1 (u 0 )C g (u 0 )S n+3 2g. Similarly, S n (g, g) u 1 u 2 = Pg 2 (u 0 )C g (u 0 )S n+2 2g, and let u 1u 0 be an edge of C g in S n (g, g) u 1 u 2, S n (g, g) u 1 u 2 u 1u 0 = Pg (u 0 )S n+2 2g (u 0 )P g 2. It is obious that S n (g, g) u 1 u 2 S n (g, g) u 1 u 2 u 1u 0. In the same way, we hae G 1 w 1 = Cg (u)s n g+1 and G 1 w 1 = Pg 2 (u)s n g+1. and By Lemma 2.1, we hae m k (G 1 ) = m k (G 1 w 1 ) + m k 1 (G 1 w 1 ) = m k (C g (u)s n g+1 ) + m k 1 (P g 2 (u)s n g+1 ), m k (S n (g, g)) = m k (S n (g, g) u 1 u 2 ) + m k 1 (S n (g, g) u 1 u 2 ) By Lemma 2.9, we get = m k (P g 1 (u 0 )C g (u 0 )S n+3 2g ) + m k 1 (P g 2 (u 0 )C g (u 0 )S n+2 2g ). P g 1 (u 0 )C g (u 0 )S n+3 2g C g (u)s n g+1. Choosing an appropriate edge e, we hae P g 2 (u 0 )C g (u 0 )S n+2 2g e = P g (u 0 )S n+2 2g (u 0 )P g 2, and again, by Lemma 2.9, P g (u 0 )S n+2 2g (u 0 )P g 2 P g 2 (u)s n g+1. Thus, S n (g, g) G 1. From Theorems 3.1, 3.2 and 3.3, we obtain the following main result.

16 1290 HONG-HAI LI AND LI ZOU Theorem 3.4. For any graph G B n,g, we hae G S n g (u)θ(1, g 3, 1), and therefore, ME(G) ME(S n g (u)θ(1, g 3, 1)), where equality holds if and only if G = S n g (u)θ(1, g 3, 1). REFERENCES 1. L. Chen, J. Liu and Y. Shi, Matching energy of unicyclic and bicyclic graphs with a gien diameter, Complexity 21 (2015), L. Chen and Y. Shi, The maximal matching energy of tricyclic graphs, MATCH Comm. Math. Comp. Chem. 73 (2015), D.M. Cetkoić, M. Doob, I. Gutman and A. Torgaše, Recent results in the theory of graph spectra, North-Holland, Amsterdam, H. Deng, The smallest Hosoya index in (n, n + 1)-graphs, J. Math. Chem. 43 (2008), I. Gutman, The matching polynomial, MATCH Comm. Math. Comp. Chem. 6 (1979), I. Gutman and D.M. Cetkoić, Finding tricyclic graphs with a maximal number of matchings Another example of computer aided research in graph theory, Publ. Inst. Math. Nou. 35 (1984), I. Gutman and S. Wagner, The matching energy of a graph, Discr. Appl. Math. 160 (2012), I. Gutman and F. Zhang, On the ordering of graphs with respect to their matching numbers, Discr. Appl. Math. 15 (1986), G. Huang, M. Kuang and H. Deng, Extremal graph with respect to matching energy for a random polyphenyl chain, MATCH Comm. Math. Comp. Chem. 73 (2015), S. Ji, X. Li and Y. Shi, Extremal matching energy of bicyclic graphs, MATCH Comm. Math. Comp. Chem. 70 (2013), S. Ji and H. Ma, The extremal matching energy of graphs, Ars Combin. 115 (2014), S. Ji, H. Ma and G. Ma, The matching energy of graphs with gien edge connectiity, J. Inequal. Appl. 1 (2015), X. Li, Y. Shi and I. Gutman, Graph energy, Springer, New York, X. Li, Y. Shi, M. Wei and J. Li, On a conjecture about tricyclic graphs with maximal energy, MATCH Comm. Math. Comp. Chem. 72 (2014), H. Li, B. Tan and L. Su, On the signless Laplacian coefficients of unicyclic graphs, Linear Alg. Appl. 439 (2013),

17 MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS S. Li and W. Yan, The matching energy of graphs with gien parameters, Discr. Appl. Math. 162 (2014), H. Li, Y. Zhou and L. Su, Graphs with extremal matching energies and prescribed parameters, MATCH Comm. Math. Comp. Chem. 72 (2014), R. Sun, Z. Zhu and L. Tan, On the Merrifield-Simmons index and Hosoya index of bicyclic graphs with a gien girth, Ars Combin. 103 (2012), K. Xu, K.C. Das and Z. Zheng, The minimal matching energy of (n, m)- graphs with a gien matching number, MATCH Comm. Math. Comp. Chem. 73 (2015), College of Mathematics and Information Science, Jiangxi Normal Uniersity, Nanchang, , China address: lhh@jxnu.edu.cn College of Mathematics and Information Science, Jiangxi Normal Uniersity, Nanchang, , China address: @qq.com