On Some Distance-Based Indices of Trees With a Given Matching Number

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1 Applied Mathematical Sciences, Vol. 8, 204, no. 22, HIKARI Ltd, On Some Distance-Based Indices of Trees With a Gien Matching Number Shu Wen Faculty of Mathematics Physics Huaiyin Institute of Technology Huai an, Jiangsu , P. R. China Copyright c 204 Shu Wen. This is an open access article distributed under the Creatie Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, proided the original work is properly cited. Abstract Let G be a connected simple graph. For any edge e = u in G, we use n u (e, G) to denote the number of ertices in G lying closer to u than to m u (e, G) to denote the number of edges in G lying closer to u than to, respectiely. The second third geometric-arithmetic indices GA 2 (G) GA 3 (G) of G are defined as mu(e,g)m (e,g) nu(e,g)n(e,g) 2 [nu(e,g)+n(e,g)] [mu(e,g)+m(e,g)], respectiely. The Szeged edge Szeged 2 indices of G are defined, respectiely, as Sz(G) = n u (e, G) n (e, G) Sz e (G) = m u (e, G) m (e, G). In this paper, we shall proide a unified approach to characterize the trees with the minimum GA 2, GA 3, Szeged edge Szeged indices among all trees with a gien matching number. As applications, we deduce a result of Fath- Tabar et al. concerning tree with the minimum GA 2 index a result of Zhou et al. concerning tree with the minimum GA 3 index. Mathematics Subject Classification: 05C05, 05C2, 05C35, 05C75 Keywords: GA 2 index; GA 3 index; Szeged index; Edge Szeged index; tree; matching number

2 6094 Shu Wen Introduction Let G be a connected graph with ertex set V (G) edge set E(G). Recently, a class of geometric-arithmetic topological indices were proposed in [2] as GA = GA general (G) = Qu Q (Q 2 u + Q ), where Q u is some quantity that in a unique manner can be associated with the ertex u of the graph G. The reason why this class of topological indices is named geometric-arithmetic index is that Q u Q Q u + Q are the geometric arithmetic means, respectiely, of the numbers Q u Q. The first member of geometric-arithmetic topological indices was conceied [] by setting Q u to be the degree d u of the ertex u of the graph G, namely, GA = GA(G) = du d (d 2 u + d ). The second member of geometric-arithmetic topological indices, called GA 2 index, was recently studied [2-3, 7-9] defined by setting Q u to be n u (e, G), the number of ertices in G lying closer to u than to in the graph G, namely, GA 2 = GA 2 (G) = nu (e, G)n (e, G) [n 2 u(e, G) + n (e, G)], () where e = u is an edge of G. The third member of geometric-arithmetic topological indices, called GA 3 index, was recently studied [3, 9] defined by setting Q u to be m u (e, G), the number of edges in G lying closer to u than to in the graph G, namely, GA 3 = GA 3 (G) = mu (e, G)m (e, G) [m 2 u(e, G) + m (e, G)], (2) where e = u is an edge of G. The other two preiously established molecular structure descriptors are, respectiely, the Szeged index [4, 5]), defined as Sz(G) = the edge Szeged index [6], defined as Sz e (G) = n u (e, G) n (e, G) (3) m u (e, G) m (e, G). (4)

3 Distance-based indices of trees 6095 More recently, Fath-Tabar et al. [2] obtained arious lower upper bounds of GA 2 index for a connected graph in terms of Sz(G) Zhou et al. [3] obtained arious lower upper bounds of GA 3 index for a connected graph in terms of Sz e (G). In particular, they proed [2, 3] that the n ertex path is the unique tree with the maximum GA 2 GA 3 indices the n ertex star is the unique tree with the minimum GA 2 GA 3 indices, respectiely. Other papers concerning GA indices can be found in [7 9]. In this paper, we shall proide a unified approach to characterize the tree with the minimum GA 2, GA 3, Sz Sz e indices among all trees with a gien matching number. As applications, we deduce a result of [2] concerning tree with the minimum GA 2 index a result of [3] concerning tree with the minimum GA 3 index, respectiely. 2 Main results For any edge e = u in a tree T of n ertices, we always hae n u (e, T ) + n (e, T ) = n, m (e, T ) = n (e, T ), m u (e, T ) = n u (e, T ) m u (e, T ) + m (e, T ) = n 2. In particular, if e = u is a pendent edge with pendent ertex u, then m u (e, T ) = 0. So, for a n ertex tree T, Eq. s () (2) are simplified as GA 2 = GA 2 (T ) = GA 3 = GA 3 (G) = 2 nu (e, T )n (e, T ) (5) n 2 mu (e, G)m (e, G), (6) n 2 respectiely. Actually, the aboe Eq.s (3) (6) proide us a unified way of comparing the GA 2, GA 3, Sz Sz e indices of two trees of the same order. Let F = {GA 2, GA 3, Sz, Sz e }. Gien two trees T, T 2 of n ertices. Let f be a one to one map from E(T ) to E(T 2 ) such that for any e i = u i i in T, there exists a unique edge e i = u i i in T 2 corresponding to it. Under the map f, e i its image e i constitute an edge pair {e i, e i}. Then {{e, e }, {e 2, e 2},, {e n, e n }} is called to be an edge partition of E(T )

4 6096 Shu Wen E(T 2 ). By the definition of edge partition, there exists (n )! edge partition of E(T ) E(T 2 ). If there exists an edge partition of E(T ) E(T 2 ) such that for each i =,, n, n ui (e i, T ) n i (e i, T ) n u (e i, T 2 ) n i (e i, T 2 ), i m ui (e i, T ) m i (e i, T ) m u (e i, T 2 ) m i (e i, T 2 ) i there exists an edge pair {e j, e j} such that n uj (e j, T ) n j (e j, T ) > n u (e j, T 2 ) n j (e j, T 2 ), j m uj (e j, T ) m j (e j, T ) > m u (e j, T 2 ) m j (e j, T 2 ), j then f(t ) > f(t 2 ) for any f F. A matching M of a graph G is a subset of E(G) with the property that no two edges in M share a common ertex. A matching M of G is said to be maximum, if for any other matching M of G, M M. The matching number of G is the number of edges of a maximum matching in G. Let T n, m denote the set of trees of n ertices a gien matching number m. Clearly, n 2m. If n = 2m, then each tree T in T n, m has a perfect matching, T is said to be a conjugated tree. An edge is said to be a pendent edge in a tree T if it is incident to a ertex of degree. We begin with an elementary result without proof, which is helpful in proing our following lemmas. Lemma. Let x i, y i be positie integers satisfying x i +y i = n. If x k y k > x j y j, then x k y k < x j y j. Next, we shall gie some graph transformations that decrease the GA 2, GA 3, Sz Sz e indices of trees under consideration. In the following, if not stated, we will always use n(t ) to denote the number of ertices in a tree T. T 0 T 00 T 0 u T 00 u T T 2 Fig.. The grafting transformation I: T T 2.

5 Distance-based indices of trees 6097 Lemma 2. Let T T 2 be trees on n ertices, as shown in Fig.. Then for any f F, we hae f(t ) > f(t 2 ), where n(t 0 ), n(t 00 ) 2. Proof. Let f be any element in F. Since n(t 0 ), n(t 00 ) 2, we hae T T 2. From Fig. we conclude that if e = xy E(T i ) \ {u}(i =, 2), then we hae n x (e, T )n y (e, T ) = n x (e, T 2 )n y (e, T 2 ) if e = u, then we hae m x (e, T )m y (e, T ) = m x (e, T 2 )m y (e, T 2 ), n u (e, T )n (e, T ) = n(t 0 )n(t 00 ) > (n ) = n u (e, T 2 )n (e, T 2 ) m u (e, T )m (e, T ) = (n(t 0 ) )(n(t 00 ) ) > 0 = m u (e, T 2 )m (e, T 2 ) by Lemma. So f(t ) > f(t 2 ). This completes the proof. } } T 0. s T 0 u. s u T 3 T 4 Fig. 2. The grafting transformation II: T 3 T 4. Lemma 3. Let T 3 T 4 be trees on n ertices, as shown in Fig. 2. Then for any f F, we hae f(t 3 ) > f(t 4 ), where n(t 0 ) 2, s 2 n 6. Proof. Let f be any element in F. It can be concluded from Fig. 2 that if e = xy E(T i ) \ {u}(i = 3, 4), then we hae n x (e, T 3 )n y (e, T 3 ) = n x (e, T 4 )n y (e, T 4 ),

6 6098 Shu Wen m x (e, T 3 )m y (e, T 3 ) = m x (e, T 4 )m y (e, T 4 ), if e = u, then we hae n u (e, T 3 )n (e, T 3 ) = (s + )(n s ) > 2 (n 2) = n u (e, T 4 )n (e, T 4 ), m u (e, T 3 )m (e, T 3 ) = s(n s 2) > (n 3) by Lemma. So f(t 3 ) > f(t 4 ), completing the proof. = m u (e, T 4 )m (e, T 4 ) T 0 w }. } t + u t T 0 u w }{{}}{{} s s T 5 T 6 Fig. 3. The grafting transformation III: T 5 T 6. Lemma 4. Let T 5 T 6 be trees on n ertices, as shown in Fig. 3. Then for any f F, we hae f(t 5 ) > f(t 6 ), where n(t 0 ) 3. Proof. Let f be any element in F. From Fig. 3 we know that if e = xy E(T i ) \ {u}(i = 5, 6), then we hae n x (e, T 5 )n y (e, T 5 ) = n x (e, T 6 )n y (e, T 6 ) m x (e, T 5 )m y (e, T 5 ) = m x (e, T 6 )m y (e, T 6 ),

7 Distance-based indices of trees 6099 if e = u, then we hae n u (e, T 5 )n (e, T 5 ) = (s + 2t + )(n s 2t ) > 2 (n 2) = n u (e, T 6 )n (e, T 6 ) m u (e, T 5 )m (e, T 5 ) = (s + 2t)(n s 2t 2) by Lemma the fact n(t 0 ) 3. So f(t 5 ) > f(t 6 ) we are done. > (n 3) = m u (e, T 6 )m (e, T 6 ) Remark. If n(t 0 ) =, then the size of maximum matching in T 5 is not equal to that of maximum matching in T 6. If n(t 0 ) = 2, then T 5 = T6. Thus, if n(t 0 ) = or 2, the grafting transformation III will play no role in helping us proe our main result of this paper. So we assume that n(t 0 ) 3 in this lemma. Remark 2. Let T be a tree in T n,m T 2 be the tree obtained by using grafting transformation I on T. Let M be a maximum matching in T. If u M, then M is also a maximum matching in T 2. If u is not saturated by M in T, then must be saturated by M in T, for otherwise, M {u} is a matching in T, a contradiction. Thus u must be saturated by M in T 2 is not saturated by M in T 2. So T 2 T n,m. Similarly, after grafting transformations II III, the matching number of graphs under consideration remains unchanged..... m Fig. 4. The graph ˆT n,m. Lemma 5. Let T be a tree in T n, m. Then T has at most n m pendent edges. Proof. Let M = {u,, u i i,, u m m } be a maximum matching in T. So T has n 2m ertices different from u i, i (i =,, m). Connecting each of these n 2m ertices to u, also connecting u to each i. Now,

8 600 Shu Wen T has the largest possible number of pendent edges, which is equal to (n 2m) + m = n m. This completes the proof. Theorem. Among all trees in T n, m, n 2m, f F, ˆTn,m is the unique tree with the minimum f alue. Proof. Let T be a tree in T n, m such that T has the minimum f alue. We shall proe that T = ˆT n,m. By contradiction, suppose that T ˆT n,m. By Lemma 5, T has at most n m pendent edges. If T has fewer than n m pendent edges, in other words, it contains greater than m non-pendent cut edges. Thus, we can use grafting transformation I on T get a new tree, say T, in T n, m. Obiously, P ( T ) = P (T ) +, where P (G) denote the number of pendent edges in G. After a series of grafting transformation I on T, we finally get a new tree, say T, in T n, m with exactly n m pendent edges. By Lemma 2, f(t ) > f( T ), a contradiction. So T has exactly n m pendent edges. If T ˆT n,m, then by using grafting transformations II III on T, we will arrie at ˆT n,m in the end, since grafting transformations II III do not change the number of pendent edges in T. Now, by Lemmas 3 4, we know that f(t ) > f( ˆT n,m ), a contradiction once again. This completes the proof. By setting n = 2m in Theorem, we obtain the conjugated tree with the minimum f alue. Corollary. Among all trees in T 2m, m, f F, ˆT2m,m is the unique tree with minimum f alue. Lemma 6. For n 2m, we hae f( ˆT n,m ) > f( ˆT n,m ). Proof. Let u be an edge in ˆT n,m with d(u) = n m d(u) = 2. We contract the u edge in ˆT n,m attach one additional pendent edge u to the ertex u. Then we obtain ˆT n,m. Obiously, we hae n u (e, ˆT n,m )n (e, ˆT n,m ) = 2(n 2) > (n ) m u (e, ˆT n,m )m (e, ˆT n,m ) = (n 3) = n u (e, ˆT n,m )n (e, ˆT n,m ) > 0 = m u (e, ˆT n,m )m (e, ˆT n,m ). Also, for each edge e = xy different from u in ˆT n,m, we hae n x (e, ˆT n,m )n y (e, ˆT n,m ) = n x (e, ˆT n,m )n y (e, ˆT n,m ) m x (e, ˆT n,m )m y (e, ˆT n,m ) = m x (e, ˆT n,m )m y (e, ˆT n,m ).

9 Distance-based indices of trees 60 Thus, f( ˆT n,m ) > f( ˆT n,m ). By repeatedly using of Lemma 6, we obtain f( ˆT n,m ) > f( ˆT n,m ) > f( ˆT n,m 2 ) > > f( ˆT n, ). Thus, for any tree T in T n, m with a gien m, we hae f(t ) > f( ˆT n,m ) > f( ˆT n,m ) > f( ˆT n,m 2 ) > > f( ˆT n, ) = f(s n ). So, we arrie at Corollary 2. Among all trees of n ertices, f F, S n is the unique tree with the minimum f alue. Remark 3. In [2], Fath-Tabar et al. proed that the star S n is the unique tree with minimum GA 2 index in [3], Zhou et al. proed that the star S n has the minimum GA 3 index within all trees of n ertices. So, by setting f = GA 2 or GA 3 in Corollary 2, we obtain the results in [2] [3], respectiely. References [] D. Vuki ceić, B. Furtula, Topological index based on the ratios of geometrical arithmetical means of end-ertex degrees of edges, J. Math. Chem., 46 (2009) [2] G. Fath-Tabar, B. Furtula, I. Gutman, A new geometric-arithmetic index, J. Math. Chem., 47 (200) [3] B. Zhou, I. Gutman, B. Furtula, Z. Du, On two types of geometricarithmetic index, Chem. Phys. Lett., 482 (2009) [4] M.V. Diudea, M.S. Florescu, P.V. Khadikar, Molecular Topology its Applications, EfiCon Press, Bucharest, [5] I. Gutman, A formula for the Wiener number of trees its extension to graphs containing cycles, Graph Theory Notes of New York, 27 (994) 9-5. [6] I. Gutman, A.R. Ashrafi, The edge ersion of the Szeged index, Croat. Chem. Acta, 8 (2008) [7] B. Furtula, I. Gutman, Geometric-arithmetic indices, in: I. Gutman, B. Furtula (Eds.), Noel Molecular Structure Descriptors Theory Applications, Uni. Kragujeac, Kragujeac, 200, pp

10 602 Shu Wen [8] H. Hua, Trees with gien diameter minimum second geometricarithmetic index, MATCH Commun. Math. Comput. Chem., 64 (200) [9] H. Hua, S. Zhang, A unified approach to extremal trees with respect to geometric-arithmetic, Szeged edge Szeged indices, MATCH Commun. Math. Comput. Chem., 65 (20) Receied: August 29, 204