South Asian Journal of Mathematics 2014, Vol. 4 ( 2 ) : 107 118 www.sajm-online.com ISSN 2251-1512 RESEARCH ARTICLE Sharp lower bound for the total number of matchings of graphs with given number of cut edges Hongzhuan Wang 12, Rongrong Gu 2 1 Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai an, Jiangsu 223003, P.R. China 2 Faculty of Foreign Languages, Huaiyin Institute of Technology, Huai an, Jiangsu 223003, P.R. China E-mail: hongzhuanwang@gmail.com Received: March-18-2014; Accepted: April-10-2014 *Corresponding author This research was partially supported Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (No. 12KJB110001). Abstract The total number of matchings of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent. In this paper we determine a sharp lower bound for the total number of matchings among the set of graphs with k cut edges for all possible values of k, and characterize the corresponding extremal graphs as well. Key Words MSC 2010 Total number of matchings; Cut edge; 2-Edge-connected graph; Extremal graph 05C90, 05C50 1 Introduction In the chemistry literature total number of matchings is called the Hosoya index of a molecular graph. If we denote by m(g, k) the number of matchings with k edges in G, then its Hosoya index Z(G) can be expressed as Z(G) = n 2 k=0 m(g, k), where n is the number of vertices of G and n 2 is the greatest integer n 2. As a chemical descriptor of molecular structures, the Hosoya index has received much attention in the literature since it was introduced by Hosoya [4]. An important direction is to determine the graphs with maximal or minimal Hosoya indices in a given class of graphs. In [7] Gutman showed that the linear hexagonal chain is the unique chain with minimal Hosoya index among all hexagonal chains. In [11] Zhang showed that the Zig-zag hexagonal chain is the unique chain with maximal Hosoya index among all hexagonal chains. In [12] Zhang and Tian gave another proof of Gutman s and Zhang s results above mentioned. In [13] Zhang and Tian determined the graphs with minimal and second minimal Hosoya indices among catacondensed Citation: Hongzhuan Wang, Rongrong Gu, Sharp lower bound for the total number of matchings of graphs with given number of cut edges, South Asian J Math, 2014, 4(2), 107-118.
H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges systems. In [8] the path and star have been shown to have the maximal and minimal Hosoya indices, respectively, among all trees on n vertices. Hou [16] characterized the trees having minimal and second minimal Hosoya indices among all trees with a given size of matching. Yu et al. [1] investigated the graphs having minimal Hosoya index among all graphs with given edge independence number and cyclomatic number. In [2] Yu et al. investigated the trees having minimal Hosoya index among all trees with k- pendent vertices. In [14] Li, Li and Zhu determined the n-vertex unicyclic graphs with the minimal, second-, third-, fourth-, fifth- and sixth-minimal Hosoya indices. In [3] Heuberger and Wagner gave a characterization of the trees with given maximum degree which maximize the number of independent subsets, and showed that these trees also minimize the number of independent edge subsets. Zhu and one of the present authors [15] characterized the unique unicyclic graphs of a given diameter with the maximum number of independent sets. In order to state our results, we introduce some notation and terminology. For other undefined notation we may refer to Bondy and Murty [9]. We only consider finite, undirected and simple graphs. For a vertex v of a graph G, we denote N(v) = {u uv E(G)} and N[v] = N G (v) {v}. We denote d(v) be the cardinality of N(v). A pendent vertex is a vertex of degree one of G. If e is an edge in G incident with one pendent vertex, we call e a pendent edge. The graph that arises from G by deleting the vertex u V (G) or the edge uv E(G) will be denoted by G u or G uv. As usual, P n, C n and S n denote a path, a cycle and a star on n vertices, respectively. A cut vertex of a connected graph G is a vertex whose deletion results in a disconnected graph. A cut edge is defined similarly. A connected graph without cut edges is also called 2-edge-connected graph. We denote by K m,n the complete bipartite graph whose partition sets are of size m, n respectivly. If G is a connected graph with k cut edges, then clearly 1 k n 1 and k n 2. Let C n k (1 k ) be the graph obtained from C n k by attaching k pendent edges to one of its vertices. We denote G 0 (n, k) be the graph obtained from K 2,n k 2 by attaching to one of its maximum-degree vertices k pendent edges. The path P t = v 1 v 2 v t is called a pendent path of G if d(v 1 ) 3, d(v 2 ) = = d(v t 1 ) = 2 and d(v t ) = 1 in G. Let F n denote the nth Fibonacci number. Then we have F n + F n+1 = F n+2, with initial conditions F 1 = F 2 = 1. Let tg stand for the disjiont union of t copies of G. To our best knowledge, the total number of matchings of graphs with given cut edges was, so far, not considered. In this paper we investigate the total number of matchings for the set of graphs with k cut edges. In next section, we give some preliminary results. In Section 3, we first present a sharp lower bound on the total number of matchings among the set of the graphs without cut edges and characterize the extremal graphs achieved the bound. Secondly, we characterize the graphs with the smallest value of total number of matchings among the set of graphs with k ( 1) cut edges for different values of k. In Section 4, we give a conclusion of this paper. 2 Some preliminary results In this section, we list some necessary results which are needed in this paper. Lemma 2.1 ([8]). Let G = (V, E) be a graph. Then 108
South Asian J. Math. Vol. 4 No. 2 (i) If uv E(G), we have Z(G) = Z(G uv) + Z(G {u, v}); (ii) If v V (G), we have Z(G) = Z(G v) + Z(G {u, v}); u N(v) (iii) If G 1, G 2,, G t, (t 1) are the components of graph G, we have Z(G) = t Z(G j ). For a graph G, according to the definition of Z(G), by Lemma 2.1 (ii), if v is a vertex of G, then Z(G) Z(G v). In particular, when v is a pendent vertex of G and u is the unique vertex adjacent to v, we have Z(G) = Z(G v) + Z(G {u, v}). Corollary 2.2. Let G be a graph with e E(G), then we have Z(G e) < Z(G). Lemma 2.3 ([16]). Let T be a tree on n vertices. Then Z(S n ) Z(T ) Z(P n ), and Z(T ) = n if and only if T = S n and Z(T ) = F n+1 if and only if T = P n. Lemma 2.4 ([5]). If G 1 is a proper subgraph of G 2, then Z(G 2 ) > Z(G 1 ). Lemma 2.5 ([10]). If G is a connected unicyclic graph on n vertices, then Z(G) 2n 2, with equality if and only if G = Sn, 3 where Sn 3 denoted by the graph obtained by adding one edge between two pendent edges of star with n vertices. Lemma 2.6 ([15]). Let X, Y and Z be three connected graphs with disjoint vertex sets and let u, v V (X), v 0 V (Z), u 0 V (Y ). Let G be the graph obtained from X, Y and Z by identifying v with v 0 and u with u 0, respectively. Let H be the graph obtained from X, Y and Z by identifying three vertices v, v 0 and u 0, and let J be the graph obtained from X, Y and Z by identifying three vertices u, v 0 and u 0 (see Fig. 1). Then Z(G) > Z(H) or Z(G) > Z(J), where G, J, H are all connected graphs having at least four vertices. Y Y u X v u v H X Z Z G Y u J Z X v Fig 1: The graphs in Lemma 2.6 From Lemma 2.6, the following two results are obvious, we omit their proofs. Lemma 2.7. Let uv E(G) be a cut edge of a graph G and G uv = G 1 G 2, where u V (G 1 ) and v V (G 2 ). Let G be the graph obtained from G by identifying vertex u with v (the new vertex is labeled as w) and attaching at w a pendent vertex w 0. Then Z(G) > Z(G ). 109
H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges Lemma 2.8. Let G be a connected graph with u, v V (G). Denote by G(s, t) the graph obtained by attaching s 0 pendent vertices to vertex u of G and t 0 pendent vertices to vertex v of G. Then Z(G(s, t)) > Z(G(0, s + t)) or Z(G(s, t)) > Z(G(s + t, 0). 3 Main results In the following subsection, we first consider the 2-edge-connected graphs, namely, graphs without cut edges. 3.1 The smallest values of total number of matchings in 2-edge-connected graphs Let Span(G) denote the spanning tree of a graph G. Let F(n) denote the set of 2-edge-connected graphs on n vertices. If n = 3, then F(3) contains a single element C 3, so we always assume that n 4 in the remainder part of this section. Lemma 3.1. Let G be a connected graph. If G is not a tree, then Z(G) 2n 2 and this equality holds if and only if G = C 3 (1 n 3 ). Proof. Let T be a spanning tree of G and e an edge in E(G\T ). Then G = T + e is a unicycle spanning subgraph of G. Hence, by Lemmas 2.4 and 2.5, we have that Z(G) Z(G ) 2n 2, and equality holds if and only if G = C 3 (1 n 3 ). Lemma 3.2. If t 2 and n j 2 (j = 1, 2,, t), then Z( t S nj ) Z(S t ), and equality holds if n j and only if n j = 2 and t = 2. Proof. We shall proceed by induction on t. First, consider the case of t = 2. In this case, we have Z(S n1+n 2 ) Z(S n1 S n2 ) = (n 1 + n 2 ) n 1 n 2 = 1 (n 1 1)(n 2 1) 0. Now, suppose that t 3 and the statement is true for smaller values of t. Thus, by Lemma 2.1 (iii) and the induction hypothesis, we have Z( t S nj ) Z( t 1 S n j )Z(S nj ) Z(S t 1 )Z(S nt ) Z(S t ), n j n j as desired. The equality holds if and only if n j = 2, t = 2. We first choose G min F(n) such that the Hosoya index of G min is as small as possible. Now we will show some properties of graph G min. when 4 n 7, by direct calculation, G min = Cn. When n = 8 and n = 9, by direct calculation G min = K2,6 and G min = K2,7 respectively. In the following we assume that n 10. Lemma 3.3. G min C n for n 10. 110
South Asian J. Math. Vol. 4 No. 2 Proof. Suppose that G min = Cn for n 10. A simple computation gives that for n 10. Z(G min ) = Z(C n ) > Z(K 2,n 2 ), contradiction to the choice of G min. Lemma 3.4. For any edge e in G min, G min e contains at least one cut edge. Proof. Suppose that G min e has no cut edge. Then by Corollary 2.2, Z(G min e) < Z(G min ), contradicting the choice of G min. Lemma 3.5. Suppose that G C n, for any edge e E(G min ), If G min e contains at least a non-pendent cut edge, then Z(G min ) > n 2 3n + 3. Proof. Let e = uv be an edge in G min such that G min e contains a non-pendent cut edge. We first know that G min {u, v} is a connected graph, otherwise, if G min {u, v} is not connected, then Z(G min {u, v}) 1, the equality holds if all vertices in G min {u, v} are isolated vertices. Therefore, we deduce that G min = K2,n 4 + e, then Z(G min ) > Z(K 2,n 4 ), contradiction to the choice of G min. Next, we consider the following two cases. Case 1. G min {u, v} is a tree. In this case, by Lemma 2.3, Z(G min {u, v}) Z(S n 2 ) = n 2. For G C n, then G min has at least two cycles, then e must be one edge of a cycle. If G min e contains non-pendent cut edges, that is to say, there exist a pendent path of length at least 2 in G min e. When G min {u, v} = S n 2, firstly, all the pendent vertices of S n 2 must be adjacent to u or v, otherwise, there exist cut edge in G min, contradiction to the choice of G min. Secondly, u, v are both adjacent to the pendent vertices of S n 2, otherwise, then is no pendent path of length at least 2 in G min e, contradiction to the assumption. Thirdly, we conclude that one vertex of u and v is adjacent to one pendent vertices of S n 2 and another is adjacent to all the remainder pendent vertices of S n 2, if not, G min e has no cut edges, contradiction to Lemma 3.4, then Z(G min e) Z(G ) = 2n 2 13n + 23, where G is the graph obtained from K 2,n 4 by attaching to one of its maximum degree vertices a pendent path of length 2, (see Fig. 2). By Lemma 2.1, we have Z(G min ) = Z(G min e) + Z(G min {u, v}) 2n 2 13n + 23 + n 2 > n 2 3n + 3 (n 10). Case 2. G min {u, v} is not a tree. In this case, by Lemma 3.1, Z(G min {u, v}) Z(C 3 (1 n 3 )) = 2n 2, if G min e contains a non-pendent cut edge, similar as argument in Case 1, we have Z(G min e) Z(G ) = 4n 2 38n + 94, where G is the graph obtained by identifying the pendent vertex of P 3, one vertex of C 3 and one maximum degree vertex of K 2,n 6, (see Fig. 2). By Lemma 2.1, Z(G min ) = Z(G min e) + Z(G min {u, v}) 4n 2 38n + 94 + 2n 2 > n 2 3n + 3 (n 10). 111
H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges u v G u u G Fig 2: The graphs in Lemma 3.5 This proves Lemma 3.5. By an elementary calculation, we obtain that Z(K 2,n 2 ) = n 2 3n + 3, thus, by Lemma 3.5, all cut edges in G min e must be pendent edges. If not, Z(G min ) > Z(K 2,n 2 ), a contradiction to our choice of G min. Furthermore, we claim that all cut edges in G min e are pendent edges attached to one common vertex of a 2-edge-connected graph. Since G min is a 2-edge-connected graph, G min e has at most two pendent edges. If there is only one pendent edges in G min e, the claims holds immediately. If there are two pendent edges, by contradiction assume that the pendent edges in G min e attached to different vertices of a 2-edge-connected graph, then move the pendent edges to the common vertex, we obtain a new graph G such that G + e is a 2-edge-connected graph, by Lemma 2.1 (i) and Lemma 2.8, we have that Z(G + e) < Z(G min ), then contradiction to the choice of G min. From above argument, the following results holds immediately. Corollary 3.6. For any e E(G min ), all cut edges in G min e are pendent edges and attached to one common vertex of a 2-edge-connected graph. Lemma 3.7. For any edge e in G min, G min e contains only one cut edge. Proof. Suppose to the contrary that G min e contains two cut edges. In what follows we shall show that Z(G min ) > n 2 3n + 3 = Z(K 2,n 2 ), while K 2,n 2 is a 2-edge-connected graph, which contradicting the choice of G min. If G min e contains two cut edges, then G min must be the graph as depicted in G 1 (see Fig. 3). Let e = uv be any edge of G min, we denote by H = G min {u, v} the 2-edge-connected subgraph of G min, we have that N H (u) = N H (v) = x. Note that Z(G min ) = Z(G min uv) + Z(G min {u, v}) = Z(G min e) + Z(H). (1) In Equation (1), if Z(G min e) and Z(H) attain the least at the same time, the inequality Z(G min ) > n 2 3n + 3 holds, then any other cases ensure the inequality holds clearly. Therefore, in the following, we assume that the value of Hosoya index of Z(G min e) and Z(H) are as small as possible. Next, we distinguish two steps to obtain our result. First, we consider the Hosoya index of G min e in Equation (1). In the following, we shall prove that Z(G min e) n 2 5n + 7. 112
South Asian J. Math. Vol. 4 No. 2 By Lemma 2.1 (ii) and G 1 (see Fig. 3), we have Z(G min e) = Z(G min e x) + Z(G min e {x, y}) y N(x) = Z(H x) + Z(H {x, y}) + 2Z(H x) y N H (x) = 3Z(H x) + Z(H {x, y}). (2) y N H (x) Since H is 2-edge-connected graph, H x has no isolated vertex. For the above chosen vertex x, we consider the following cases: Case 1. Suppose that H x and H {x, y} is connected. Then Z(H x) Z(Span(H x)) n 3. We know that H {x, y} attains the smallest Hosoya index when H {x, y} = S n 4, and d H (x) = n 4, where y N H (x). By above statement and Equation (2), we have Z(G min e) 3(n 3) + (n 4)(n 4) = n 2 5n + 7. Case 2. Suppose that H x has t (t 2) components, say G 1, G 2,..., G t, and for any vertex y N H (x), H {x, y} has s (s 2) components and has no isolated vertices. Let V (G j = n j for 1 j t. Then, from Lemmas 2.3 and 2.4, it follows that Z(G j ) Z(Span(G j )) Z(S nj ). Again by Lemmas 2.1 (iii), 2.3, 2.4 and 3.2, we have t t Z(H x) = Z( G j ) = Z(G j ) t Z(Span(G j )) t t Z(S nj ) = Z( S nj ) Z(S t ) = Z(S n 3 ) = n 3. n j Similarly, we can prove that Z(H {x, y}) n 4. By above statement and Equation (2), we obtain Z(G min e) 3(n 3) + (n 4)(n 4) = n 2 5n + 7. Case 3. Suppose that H x is connected and for any vertex y N H (x), H {x, y} has s (s 2) components without isolated vertices. By using the same method as above, we can obtain that Z(H x) Z(Span(H x)) n 3, and Z(H {x, y}) n 4. Hence, by Equation (2), we have Z(G min e) 3(n 3) + (n 4)(n 4) = n 2 5n + 7. Case 4. Suppose that there exists a vertex y N H (x) such that Z(H {x, y}) has at least an isolated vertex and let w be an isolated vertex in Z(H {x, y}). Then d H (w) = 2, that is, N H (w) = {x, y}, and 113
H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges so w N H (x). By the definition of Z(G), it is easy to see that the more vertices as w, the smaller Hosoya index of G min e. Then, if all the vertices of H {x, y} are isolated, the Hosoya index of G min e attains minimum. Hence G min e must be the graph as depicted in G 2 (see Fig. 3). So, by Equation (2), we have Z(G min e) 3(n 3) + (n 4)(n 4) + 1 = n 2 5n + 7 + 1 > n 2 5n + 7, thus the proof of the first step is completed. Secondly, we consider the Hosoya index of H in Equation (1). In the following, we shall prove that Z(H) > 2n 4. In order to prove the inequality, we first distinguish the following assumptions. Assumption 1. For any e = u v E(H), the graph H e is connected and contains at least a cycle. Indeed, if H e is a tree, in view of H is a 2-edge-connected graph, then H = C n 2. Hence, the result follows from a direct calculation for n 10, Z(H) = Z(C n 2 ) > Z(K 2,n 4 ), which contradicting the previous assume of Z(H). Thus Assumption 1 follows. By Assumption 1 and Lemma 3.1, we obtain that Z(H e) 2(n 2) 2, but then H e C 3 (1 n 5 ), contradicting the choice of H. Consequently, Z(H e) > 2(n 2) 2 = 2n 6. Assumption 2. For any e = u v E(H), we have that Z(H {u, v }) 2. Otherwise, if Z(H {u, v }) = 1, then all the vertices of H {u, v } are isolated. Hence H = K 2,n 4 + e. Note that K 2,n 4 + e contains K 2,n 4 as its proper subgraph. So we obtain that Z(H) > Z(K 2,n 4 ), which also contradicting the previous assume of Z(H). By combining Assumptions 1 and 2, we have Z(H) = Z(H e) + Z(H {u, v }) > 2(n 2) 2 + 2 = 2n 4, as desired. From the combination of two steps above, Lemma 2.1 (i) and Equation (1), it follows that Z(G min ) = Z(G min e) + Z(H) > n 2 5n + 7 + 2n 4 = n 2 3n + 3. But this contradicting the choice of G min. This completes the proof of Lemma 3.7. e u x u x H y v v H v u e v G 1 G 2 G 3 Fig 3: The graphs in Lemma 3.7 and Theorem 3.8 Now we prove the main result of this section. 114
South Asian J. Math. Vol. 4 No. 2 Theorem 3.8. If G is any graph in F(n) with n 8, then Z(G) n 2 3n + 3, and equality holds if and only if G = K 2,n 2. Proof. In view of Lemmas 3.4, 3.7, and Corollary 3.6, we obtain that G min e must has exactly one pendent edge. Let e = uv be any edge in G min such that G min e contains only one pendent edge. Then one of u and v must have degree 2 in G min. We may assume that d Gmin (v) = 2. Without loss of generality, suppose vv E(G min ) and let H = G min v, where H is a 2-edge-connected graph. It is evident that u H. In this case, G min is the graph shown in G 3 (see Fig. 3). We proceed by induction on n. First, consider the validity of the above statement for the case of n = 8. By the properties of G min and direct calculation, we have that Z(G min ) > Z(K 2,6 ), the assertion is true. Now let n 9 and suppose that the above statement is true for smaller values of n. By Lemma 2.1 (ii) and G 3 (see Fig. 3), we obtain Z(G min ) = Z(G min v) + Z(G min {u, v}) u N(v) = Z(G min v) + Z(G min {u, v}) + Z(G min {v, v }) = Z(H ) + Z(H u) + Z(H v ). (3) Thus by Equation (3), we distinguish three cases to obtain our result. For H is 2-edge-connected graph, then H u and H v have no isolated vertices. Case 1. Both H u and H v are connected. By Lemmas 2.3 and 2.4, we have Z(H u) Z(Span(H u)) Z(S n 2 ) n 2. Similarly, by Lemmas 2.3 and 2.4, we can show that Z(H v ) n 2. Again by induction hypothesis, we have that Z(H ) (n 1) 2 3(n 1) + 3. In view of above statement and Equation (3), we obtain Z(G min ) (n 1) 2 3(n 1) + 3 + 2(n 2) = n 2 3n + 3, equality holds if and only if H u = S n 2, H v = Sn 2, H = K2,n 3. Case 2. Suppose that H u is connected, while H v has r (r 1) components. By Lemmas 3.2, 2.3 and 2.4, we have that Z(H v ) n 2 and, by Lemmas 2.3 and 2.4, we further obtain that Z(H u) Z(Span(H u)) Z(S n 2 ) n 2. Therefore, again by induction hypothesis, above statement and Equation (3), we have that Z(G min ) n 2 3n + 3, and this equality holds if and only if Z(H u) = n 2, Z(H ) = (n 1) 2 3(n 1) + 3, Z(H v ) = n 2, that is, H u = S n 2, H v = S2 S2, H = K2,n 3. 115
H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges If H v = S 2 S2, then G min must be the graph as depicted in G 4 (see Fig. 4), but then, H K 2,n 3, H u S n 2. Hence, Z(H ) > (n 1) 2 3(n 1) + 3 and Z(H u) > n 2, and thus Z(G min ) > n 2 3n + 3. That is, the equality doesn t holds in Case 2. Case 3. Suppose that H u is not connected, while H v is connected. we can use the same method as above to show that Z(G min ) > n 2 3n + 3. Consequently, by Cases 1-3, the equality holds if and only if H u = S n 2, H v = Sn 2, H = K2,n 3, which is equivalent to G min = K2,n 2. Therefore, the proof of Theorem 3.8 is completed. In Section 3.2, we shall consider the graph with at least one cut edge. 3.2 The smallest values of total number of matchings in graphs with k ( 1) cut edges Let H(n, k) denote the set of connected graphs with n vertices and k cut edges. Also, we use (kp 2 )vh to denote the graph arisen from H by pasting k paths P 2 to the vertex v of H (see (kp 2 )vh in Fig. 4 for instance). u H v v v G 4 G 5 (kp 2 )vh Fig 4: The graphs in Theorem 3.9 Theorem 3.9. Let any G H(n, k) with 1 k n 4 and n 8. Then Z(G) (n 1)+(n 2)(n k 2). Equality holds if and only if G = G 0 (n, k), where G 0 (n, k) is the graph obtained from K 2,n k 2 by attaching to one of its maximum-degree vertices k pendent edges. Proof. Let G min be chosen from H(n, k), such that Z(G) Z(G min ) for any G in H(n, k). Next, we shall prove that G min = G 0 (n, k). Suppose to the contrary assume that G min G 0 (n, k). From Lemma 2.6, Lemma 2.7 and Lemma 2.8, it follows that all cut edges in G min are pendent edges attached to one common vertex of a 2- edge-connected graph H with n k vertices. Moreover, G min has exactly one cut vertex, say v. Thus G min = (kp2 )vh, as depicted in Fig. 4. In the following, we shall prove that Z(G min ) = Z((kP 2 )vh) > Z(G 0 (n, k)) by contradiction. 116
South Asian J. Math. Vol. 4 No. 2 If k = 1, then we have Z(G 0 (n, 1)) = Z(K 2,n 3 ) + Z(S n 2 ). (4) Z(P 2 vh) = Z(H) + Z(H v). (5) Since H is a 2-edge-connected graph with n 1 vertices, Z(H) Z(K 2,n 3 ) with equality if and only if H = K 2,n 3 by Theorem 3.8. If H v is connected, then, by Lemmas 2.3 and 2.4, Z(H v) Z(Span(H v)) Z(S n 2 ), with equality if and only if H v = S n 2. By our assumption that G min G 0 (n, k) and Equations (4-5), we have that Z(G min ) = Z(P 2 vh) > Z(G 0 (n, 1), which contradicting the choice of G min. Assume that H v has components Q 1, Q 2,, Q t (t 2). Let n j denote the order of Q j for j = 1, 2,, t. Similar to above, we have that Z(Q j ) Z(Span(Q j )) Z(S nj ). By Lemmas 2.3 and 2.4, we obtain t t Z(H v) Z(S nj ) = Z( S nj ) Z(S n 2 ), with equality if and only if H v = 2S 2, namely, G min is isomorphic to the graph G 5 (see Fig. 4). But then Z(G 5 ) > Z(G 0 (6, 1)), a contradiction. Hence Z(H v) > Z(S n 2 ). By Equations (4-5), we have that Z(G min ) = Z(P 2 vh) > Z(G 0 (n, 1)), again a contradiction. If k 2, then by Lemma 2.1 (ii), we obtain the following recursion relations: Z(G 0 (n, k)) = Z(G 0 (n 1, k 1)) + Z(S n k 1 ). (6) Z(G min ) = Z((kP 2 )vh) = Z(((k 1)P 2 )vh) + Z(H v). (7) Combining Equations (6-7) and the initial condition Z(P 2 vh) > Z(G 0 (n 1, 1)), we have that Z(G min ) = Z((kP 2 )vh) > G 0 (n, k) for all 2 k n 4, since G min G 0 (n, k). So Z(G min ) = Z((kP 2 )vh) > G 0 (n, k) for all 1 k n 4, which contradicting the choice of G min. This contradiction gives G min = G 0 (n, k) for all 1 k n 4. It is not difficult to verify that Z(G 0 (n, k)) = n 1+(n 2)(n k 2). Therefore, the proof of Theorem 3.9 is completed. By a similar discussion as in the proof of Theorem 3.9, we may also show the following result is true. We omit the procedure here. Theorem 3.10. For any G H(n, k) with 1 k n 4 and 4 n 7. Then Z(G) (k + 1)F n k + 2F n k 1. Equality holds if and only if G = C n k (1 k ). 4 Conclusion In this paper we characterized the graphs with the smallest values of total number of matchings among the set of graphs with k cut edges for different values of k. It is surprised to see that the graph on n vertices graphs with given cut edges which attains the smallest values of total number of matchings is not unique. When 4 n 7 there exist one case and when n 8, there is another case. Let any G H(n, k) with 0 k n 1, n 4 and k n 2. Then the following statements hold. If k = n 1, then Z(G) n equality holds if and only if G = S n ; 117
H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges If k = n 3, then Z(G) 2n 2 equality holds if and only if G = C 3 (1 n 3 ); If k = 0 and 4 n 7, then Z(G) F n+1 + 2F n equality holds if and only if G = C n ; If k = 0 and n 8, then Z(G) n 2 3n + 3 equality holds if and only if G = K 2,n 2. If 1 k n 4 and 4 n 7, then Z(G) (k + 1)F n k + 2F n k 1 equality holds if and only if G = C n k (1 k ); If 1 k n 4 and n 8, then Z(G) (n 1) + (n 2)(n k 2) equality holds if and only if G = G 0 (n, k). On the other hand, it is natural to consider the following problem which may be much more difficulty. Problem 4.1 How can we determine a sharp upper bound on the total number of matchings of graphs with given cut edges. References 1 A. M. Yu, F. Tian, A kind of graphs with minimal Hosoya indices and maximal Merrield-Simmons indices, MATCH Commun. Math. Comput. Chem. 55(1) (2006) 103 108. 2 A. M. Yu, X. Z. Lv, The Merrield-Simmons indices and Hosoya indices of trees with k pendent vertices, J. Math. Chem. 41 (2007) 33 43. 3 C. Heuberger, S. G. Wagner, Maximizing the number of independent subsets over trees with bounded degree, J. Graph Theory 58(1) (2008) 49 68. 4 H. Hosoya, Topological index, Bull. Chem. Soc. Jpn. 44 (1971) 2332. 5 H. B. Hua, Hosoya index of unicyclic graphs with prescribed pendent vertices, J. Math. Chem. 43(2) (2008) 831 844. 6 H. Liu, M. Liu, A unified approach to extremal cati for different indices, MATCH Commun. Math. Comput. Chem. 58 (2007) 193 204. 7 I. Gutman, Extremal hexagonal chains, J. Math. Chem. 12 (1993) 197 210. 8 I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986. 9 J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Macmillan, New York, 1976. 10 J. P. Ou, On extremal unicyclic molecular graphs with prescribed girth and minimual Hosoya index, J. Math. Chem. 42(3) (2007) 423 432. 11 L. Z. Zhang, The proof of Gutman s conjectures concerning extremal hexagonal chains, J. Sys. Sci. Math. Sci. 18 (1998) 460 465. 12 L. Z. Zhang, F. Tian, Extremal hexagonal chains concerning largest eigenvalue, Sci. China A 44 (2001) 1089 1097. 13 L. Z. Zhang, F.Tian, Extremal catacondensed benzenoids, J. Math. Chem. 34 (2003) 111 122. 14 S. Li., X. Li, Z. Zhu, On minimal energy and Hosoya index of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 61 (2009) 325 339. 15 S. Lin, C. Lin, Trees and forests with large and small independent indices, Chinese J. Math. 23 (1995) 199 210. 16 Y. Hou, On acyclic systems with minimal Hosoya index, Discr. Appl. Math. 119 (2002) 251 257. 118