Matchings in starlike trees

1. I N T R O D U C T I O N Ordering of graphs with respect to the number of matchings, and finding the graphs extremal with regard to this property, has been the topic of several earlier works [1-4]. These results have chemical applications, in connection with the so-called total 1r-electron energy [5-7]. Let G be a graph without loops and multiple edges. For k being a positive integer, m ( G , k) denotes the number of k-matchings in G, tha t is, the number of k-element sets of independent edges of G. In addition to this, it is consistent to define re(G, 0) = 1 for all graphs G, as well as m ( G , k ) = 0 for k < 0. I f for two graphs G1 and G2, the relation re(G1, k) < m(G2, k) is obeyed for all values of k, k >_ 1, then we say tha t G2 is m-greater than G1 or tha t G1 is m-smaller than G2 and write G1 -~ G2 or G2 __ G1. If G1 _ G2, but not G1 ~ G2, then G1 is strictly m-smaller than G2 and G2 is strictly m-greater than G1, which we denote by G1 -~ G2 or G2 ~G1. If both relations G1 _ G2 and G1 ~ G2 are valid, then G1 and G2 are said to be m-equivalent (which does not mean tha t they need to be isomorphic). If neither G1 _ G2 nor G1 _ G2, then G1 and G2 axe said to be m-incomparable. The r e l a t i o n _ induces a quasiordering in any set of graphs. In what follows, we shall need a few elementary results on the number of matchings and on the quasiordering ~ [4,8]. LEMMA 1.1. L e t G be a graph and e i ts edge connec t ing the vert ices u and v. Then re(G, k) = m ( G e, k) + m ( G u v, k 1). (1) /£ the degree of the vertex u is unity, then m ( G , k) = m ( G u, k) + m ( G u v, k 1). (2) 0893-9659/01/$ see front matter (~) 2001 Elsevier Science Ltd. All rights reserved. Typeset by JtA/~S-TEX PII: S0893-9659(01)00054-4 844 I. GUTMAN et al. A graph consisting of disconnected components H1 and/-/2 will be denoted by H1 U/-/2. /(,~ denotes the n-vertex graph without edges. LEMMA 1.2. I f G = Go U K n , then G and Go are m-equivalent. I f H is a subgraph of G, having fewer edges than G, then H -~ G. LEMMA 1.3. Let G1, G2, and G be graphs with disjoint vertex sets. Then G1 "4 G2 i f and only i f G 1 U G ~ G 2 U G . A tree is a connected graph without cycles. A tree in which exactly one vertex has degree (= number of first neighbors) greater than two is said to be starlike. The n-vertex tree in which no vertex has degree greater than two is the path P,~. The set of all starlike trees on n vertices, in which the maximal vertex degree is d, will be n--1 denoted by S(n, d). The set of all n-vertex starlike trees, Ud=3 S(n, d), will be denoted by S(n). It is both convenient and consistent to define S(n, 2) = {pn}. A starlike tree belonging to S(n ,d) has d branches (attached to the vertex of degree d). If these branches possess bl, b2, . . . , bd vertices, bl + b2 4. . . + bd 41 ---n, then the respective tree will be denoted by T(bl, b2 . . . . , bd). The vertex of degree d is called the branching vertex. 2. C O M P A R I N G S T A R L I K E T R E E S W I T H D I F F E R E N T N U M B E R O F B R A N C H E S THEOREM 2.1. I f T1 C S(n, dl) and T2 c S(n, d2) and if dl > d2, then either ( a ) T1 -~ T2 or (b) T1 and T2 are m-incomparable. PROOF. It is sufficient to observe that m(T1, 1) = re(T2, 1) --n 1 and that for any starlike tree T ~ S(n, d), m(T, 2 ) = 1-~(n . . . . 1)(n 4) ~d(d 3), implying re(T1, 2) < re(T2, 2) whenever dl > d2. | Both Cases (a) and (b), specified in Theorem 2.1, may occur. For instance, the tree T(2, 2, 2, 1) c S(8, 4) is strictly m-smaller than T(4, 2, 1) E S(8, 3), whereas the same tree is m-incomparable with T(5, 1, 1) e S(8, 3). THEOREM 2.2. I f T1 C S(n ,d) , d >_ 3, then there exists a tree T2 E S ( n , d 1 ) , such that T2 ~T1. Before proving Theorem 2.2, we introduce a few abbreviations in order to make the formulas tha t follow more compact. Let T(bl, b2, . . . , bd) be a starlike tree, such that the lengths of two of its branches are x and y, and the lengths of all other branches are fixed. Without loss of generality, we may choose bl = x and b2 = y. The ordered (d 2)-tuple b3, b4 , . . . , bd will then be denoted by b, so that instead of T(x, y, b3, b4, . . . , bd) we write T(x , y, b). Further, we denote the tree T(0, 0, b) by R and the forest Pb3 U Pb4 U .. . U Pb,L (which is obtained by deleting the branching vertex from T(0, 0, b)) by R'. PROOF. We show that T(x , y, b) c S(n, d) is strictly m-smaller than T(x + y, b) E S(n, d 1). Applying equation (1) of Lemma 1.1 to the edge connecting the first branch of T(x , y, b) with the branching vertex, we get m ( T ( x , y, b), k) = m (Px U T(y, b), k) + m (Px-1 U P~ U R', k 1). Applying equation (1) of Lemma 1.1 to the (y + 1) th edge of the first branch of T ( x + y, b), we get m ( T ( x + y, b), k) = m (Pz U T(y, b), k) + m(P~i u T(y 1, b), k 1). Therefrom, m ( T ( x + y, b), k) m ( T ( x , y, b), k) = m (Px-1 U T(y 1, b), k 1) m (P~-I U Py U R', k 1). Starlike Trees 845 By Lemmas 1.2 and 1.3, the right-hand side of this difference is positive-valued because Py U R ~ is obtained by deleting from T ( y 1, b) the edges connecting the branching vertex with the branches 3, 4 , . . . , d. Therefore, m ( T ( x + y, b), k) > m ( T ( x , y, b), k) for all values of k >_ 2, i.e., T ( x + y, b) ~T ( x , y, b). | 3. C O M P A R I N G S T A R L I K E T R E E S W I T H E Q U A L N U M B E R O F B R A N C H E S We first deduce two auxiliary results. LEMMA 3.1. Let x and y be integers, such that 1 < x < y 2. A s s u m e that the graphs Px, P~, and G have disjoint vertex sets. Then for all graphs G and for all vMues o f k, m(Px U Py U C, k) rn(Px+] U Py-l U C, k) = (-1)Zm(Py_x_2 U C, k x 1). PROOF. Denote m(P~ U P~ U G, k) m(Px+l U Pv-1 u G, k) by D1. Using equation (2) of Lemma 1.1, we have and m(Px U Py U a , k) = m(P~ U Pv-1 U a,k) + m(Px U Py-2 U C , k 1) m(P:~+l u Py-1 U G, k) = m(Px U Py-1 U G, k) +m(Px-1 u Py-1 U G , k 1), from which D i = [ r n ( P x 1 U Pyi U G , k 1) rn(P~ U Py-2 UG, k 1)]. Continuing the same reasoning, we arrive at D1 = ( 1 ) t [,~ (Px t U Py_t U G, k t) ,~ (Px+lt U P y l t U G, k t)], which in the special case t = x yields D1 = ( -1) x [m (Py -x U G, k x) m (Py -x -1 U G, k x)]. Lemma 3.1 follows now from another application of equation (2): m(Py_xUG, k x ) = rn (Py_x_ lUC , k x ) + m ( P y _ z _ 2 U C , k x 1 ) . | LEMMA 3.2. Le t 2 < x < y 4 and everything else as in Lemma 3.1. Then for all graphs G and for all values of k, m(P~ U Py U a ,k ) m (Px+2 U Py-2 U a,k) = ( -1)Xm (Py_:~_3 U a , k x 1). PROOF. Denote m(P~ U Py U G, k) m(P~+2 U Py-2 U G, k) by D2. Noting that D2 = [m (P~ U Py U G, k) rn (Px+l U Py-1 U G, k)] + [m (Px+l U By-1 U G, k) m (P~+2 U By-2 U G, k)] , and applying Lemma 3.1, we obtain D2 = ( -1) z [m (Py-x-2 U G, k x 1) m (Py -~ -4 U G, k x 2)]. By equation (2), m ( P ~ x _ 2 U G , k x 1 ) = m ( P y x 3 U G , k x 1 ) + m ( P y x 4 U G , k x 2 ) . | 846 I. GUTMAN et al. We are now r eady to r e tu rn to starl ike trees. Bear ing in mind the no ta t ion in t roduced before the p roof of T h e o r e m 2.2, let T ( x , y, b) E S (n , d) be such a t ree and let the p a r a m e t e r s n and d be fixed. Apply ing two t imes L e m m a 1.1 to T ( x , y, b), we readily arrive a t m ( T ( x , y, b), k) = m (Px U Py U R, k) + m ( p x uP -i u R ' , k 1) +m(Px_l u R ' , k 1). Deno te the difference m ( T ( x , y, b), k) m ( T ( x + 1, y 1, b), k) by A1. T h e n A1 = [ m (Px U Py U R, k) m (Px+l U Py-1 U R, k)] + [ m (P~ U Py-1 U R' , k 1) m (Px+l U Py-2 U R' , k 1)] + [ m ( P x 1 UP y U R ' , k 1) m ( P x U P y 1 U R ' , k 1)].