In this note we give an explanation for two phenomena mentioned in the concluding remarks of “The matching polynomial of a polygraph” by Babić et al. The following results are obtained: \ud
1.\ud
Although three matrices for given polygraphs defined in the above article in general have different orders, they determine the same recurrence relations for the matching polynomial of these polygraghs.\ud
2.\ud
Under certain symmetry conditions, the order of the recurrence relations can be reduced by almost a half

Broersma, H.J.

Li, Xueliang

English

In this note we give an explanation for two phenomena mentioned in the concluding remarks of “The matching polynomial of a polygraph” by Babić et al. The following results are obtained: 1. Although three matrices for given polygraphs defined in the above article in general have different orders, they determine the same recurrence relations for the matching polynomial of these polygraghs. 2. Under certain symmetry conditions, the order of the recurrence relations can be reduced by almost a half

Broersma, Haitze J.

University of Twente Research Information

Discrete Applied Mathematics 46 (1993) 79-86 North-Holland Note 79 On “The matching polynomial of a polygraph” H.J. Broersma and Li Xueliang* Department of Applied Muikematics, Unirersit.v of Twente, Enschede, Nrther1md.s Received 11 December 1990 Revised 25 January 1993 Abstract In this note we give an explanation for two phenomena mentioned in the concluding remarks of “The matching polynomial of a polygraph” by Babic et al. The following results are ob- tained: (1) Although three matrices for given polygraphs defined in the above article in general have different orders, they determine the same recurrence relations for the matching polynomial of these polygraphs. (2) Under certain symmetry conditions, the order of the recurrence relations can be reduced by almost a half. Keywords. Polygraph, matching polynomial, recurrence relation. 1. Terminology For notation and terminology not defined here we refer to [l] and [4,11]. Let G = (V, E) be a graph on IZ vertices, that is, with 1 VI = n. Then the matching polynomial of G, denoted by cr(G;x), is defined by I n/2 I cx(G;x)=-I-(- l)kp(G,k)~“-2k, k=O Correspondence to: Professor H.J. Broersma, Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands. *On leave from Department of Mathematics, Xinjiang University, Urumchi, Xinjiang, P.R. China. 0166-218X/93/$06.00 c 1993-Elsevier Science Publishers B.V. All rights reserved where p(G, k) denotes the number of k-matchings of G, i.e., the number of ways we can choose k independent edges in G, and p(G, 0) := 1. If E = 8, then cc(G; x) = xn. The sum of the absolute values of all coefficients of cc(G;x) is the so-called Hosoya index of G. For a subset F of E, M(F) denotes the set of all matchings in F, i.e., the set of all independent subsets of F. Let G1,Gz,..., G, be a set of mutually disjoint graphs, and let Xi, X2,. ,X, be a set of binary relations such that Xi E: V(Gi) x V(Gi+ i) (1 I i I m - 1) and X, E V(G,) x V(G,). Then the polygraph Cl,,, is defined as follows: V(s2,) = V(G,) u V(G,) u ... u V(G,), E(Q,) = E(G,) u X, u E(G,) u X2 u ... u E(G,) u X,. Denote by Ti the subgraph of Q, with v(ri) = V(G,) u ... u V(G,) and E(~,)=E(G,)uX,u~~~uE(Gi)(i~{1,2,...,m}).1fG1=G2= ... =G,=Gand Xi = X2 = ... = X, = X, we denote R, by w, and Ti by yi; o, and Y,,, are called a rotagraph and a ,fasciagraph, respectively. 2. Introduction Hosoya’s index has many applications, e.g., in physical chemistry, statistical physics and thermodynamics. The computation of this index is a very difficult problem (# P-complete). Recently, many authors (see [6-IO]) gave explicit expressions or recurrence rela- tions for matching polynomials of some polygraphs. Furthermore, Babii- et al. [l] obtained a systematic method to compute matching polynomials for polygraphs. Many of the known results can be regarded as special cases of their results. In [l], they introduced three types of matrices to obtain recurrence relations for the matching polynomials of w, and yrn by using the Cayley-Hamilton Theorem from linear algebra (see [4,11]). We repeat some of the definitions in [l]. By hil(X) we denote the set of all matchings in X, with a matching viewed as a subset of indepen- dent edges, including the empty set. To distinguish between the elements of M(X) we use a one-to-one mapping of the set M(X) onto the set of indices {1,2, , IM(X)I ) and denote the elements of M(X) with the assigned index j by Wj. The matching Wj can also be viewed as a binary relation with domain D( Wj) c V(Gi) and range R(Wj) ~ V(Gi+l) for i = 1,2, ,m - 1, V(Gi 1 for i = m. The three types of matrices are defined as follows (see [l, p. 181): [Ti]jk:= (- l)‘@“a(G - R(Wj) ~ D(Wk);x), where Wj and W, are elements of M(X) with range R( Wj) and domain D( W,), respectively. [TJjk:= ( - I)‘“\’ 1 a(G - R(W) - D( Wk.; x), WlD(W)=D(W,) where j runs through only those values with different D( Wj). [TJjk:= ( - 1)‘“; c cc(G - R( Wj) - D(W); x), WR(W=R(W,) where k runs through only those values with different R( W,). In the above formulas, x(G - A - B;x) = 0 if A n B # 8. Let T be one of the matrices T, , T, or TX. Then, by [ 1, Corollary lo], i cc(o,;x) = tr(T) and @(Yfn - R( Wj) - D( W,);X) = ( - l)l”” [T”]jk. Let the characteristic polynomial of T be 4(T;2) := det(j,1, - T) = f LZ~(.~)~.~-~, i=o where N is the order of T and so(x) := 1. Then, by [l, Corollary 111, the recurrence relations for the matching polynomials of w, and ;I~ are given by 1 i% ai(x)cc(om-i;X) = OS i~o Ui(X)~(~,~i - R( Wj) - D( W,);x) = 0 ) (*) where m 2 N, z(o,,;x) := N and r(g, - R( Wj) - D( Wk); X) := ( - I)““‘dj, (Sj, is Kronecker’s symbol). It is obvious from the above results that the order of the recurrence relations is closely related to the rank of T: the higher the rank of T, the higher the order of (*). In the concluding remarks of [ 11, BabiC et al. proposed two problems with regard to the order of (*). Problem 1. In [l, Example 141, three matrices with different orders give rise to three recurrence relations with the same order and coefficients. Is this a general phenom- enon? In the next section we prove that the answer is affirmative. Problem 2. In [I, Example 131, it is possible to obtain another matrix T’ with lower order than T for the special case when y,,, is constructed from a graph G and a 1-l binary relation X that are both symmetric (in a sense to be defined in Section 4). What is the role of the symmetry of G and X in a possible reduction of the order? 82 H.J. Broersma, Li Xueliang We answer this question in Section 4 and show that the order of the recurrence relation of cc(y,;x) in this special case can be reduced from 22p + 1 to 22p-’ + 2P-’ + 1,where p = (X1/2. 3. Answer to Problem 1 We claim that, although in general T,, T2 and T, have different orders (especially when X is not l-l), they determine the same recurrence relations (*). Before we prove this, we introduce the following matrix transformation. Let A be an n x n matrix with entries from a commutative ring. Denote by C(~ the ith row of A and by Aj thejth column of A. Two columns are said to be in the same class if and only if they are equal. Assume that the columns of A are classified into s classes. Let A* be the IZ x s matrix obtained from A by deleting all columns except for those with minimum index in every class. Now, two rows a: and G$ of A * are said to be in the same class if and only if Ai and Aj are in the same class. Obviously, the rows of A* also fall into s classes. Let B be the s x s matrix obtained from A * by replacing the rows with minimum index in every class by the sum of all rows in that class and by deleting the other rows. Lemma 3.1. det (21, - A) = j.“-“det (,?I, - B). Proof. From linear algebra (see [4,1 l]), we know that the coefficient ai of xnmi (0 < i I n) in the characteristic polynomial det ().I,, - A) = c,!=, aixnSi of A is equal to ( - 1)’ times the sum of all main minors of A with order i. It is clear that all minors with order > s are zero. Hence, a, = a,_ 1 = ... = a,, 1 = 0. It remains to prove that the sum of all main minors of A with order i (0 I i 5 s) is equal to that of B. This is an easy consequence of the following property 1 Al .. . Ai + Ai ... A, 1 = 1 Al ... Ai ... A, 1 + IA, . . . A; . . . A,1 and other elementary properties of determinants. 0 Theorem 3.2. The recurrence relations (*) for the matching polynomials of w, and y,,, determined by T,, T2 and T3, respectively, have the same order and coefficients. Proof. It is not difficult to check that T, is obtained from T1 by applying the above transformation, and that T, is obtained by applying this transformation to the transpose T: of T1. Let the orders of T1, T2 and T3 be n, s and t, respectively. Then, by Lemma 3.1, we have and det (/II, - Ti) = j_“-“det (1.1, - T2) det (AI,, - T,) = det (j-1, - TT) = l.“-‘det (i1, - T3). As a consequence, T1, T2 and T, yield three recurrence relations (*) with the same order and coefficients. 0 On “The matching polynomial qf a polygraph” 83 4. Answer to Problem 2 Throughout this section we assume X is a l-l binary relation, implying that T1 = T2 = T3 = T. Definition 4.1. Let G be a graph with vertex set V(G) = { 1,2, . . . ,m} u { 1’,2’, . ,m’j and X a relation on V(G). Then G and X are said to be symmetric if (i) the mapping cp defined by q(i) = i’ and q(i’) = i (i = 1,2, . . . , m) is an auto- morphism of G, and (ii) X = {(i,,j,),..., (i,, j,)} u {(i; ,j;), . ,(iz, jb)}, where i: = i if i, = i’ and j: = j if j, = j’. The following facts are obvious and stated without proofs. Proposition 4.2. Let G and X be as in Dejinition 4.1. Then: (1) R(Wj) isan r-subset(r=0,1,...,2p)of{j,,j;,...,j,,jb} and thus thereare 22p difSerent R( Wj). Similarly, there are 22p diflerent D( W,); (2) T is a matrix of order 22p, and the recurrence relation (*) for yrn determined by T has 22p + 1 terms; (3) let S = {u1,u2, . . . . u,} u {w;,w;, . . . . wi} G V(G). Then S’ := {II;,& . . . . u:> u (w1,w2,...,wt) c V(G) and G[V(G) - S] z G[V(G) - S’], where G[U] denotes the subgraph of G induced by a subset U of V(G); (4) for every D( W,) there exists a Wj such that cp(D(W,)) = D(y) and q(R( Wk)) = R( Wj) and such that D( WJ = D( Wj) if and only ifR( Wk) = R( Wj). By arranging T in a proper way it is possible to reduce the order of (*) for Y,,, from 22p + 1 to 22p-’ + 2p-’ + 1. B e ore f we prove this, we need some more ter- minology. Definition 4.3. Let Y = (yr, y,, . . . ,y,,) and Z = (z1,z2, . . . ,z,) be two n-dimensional vectors. We call Y and Z a pair of associated vectors if, for all i E { 1,2, . . , n}, either yi = Zi Or yi = Zif 1 and yi+ 1 = zi; an index i for which yi = zi is called an E-index, an index i for which yi = Zi+ 1 and yi+ 1 = Zi is called an F-index. Definition 4.4. Let P be an n x n matrix. Denote by pi the ith row of P and by Pj the jth column of P. If P satisfies the following conditions: (i) Pi, and Pir, + I are associated vectors, where k = 1,2, . . . , r, and / i, - i, 1 2 2 for all S,tE {1,2 ,..., r}, s # t; (ii) i, , i2, . . . , ‘* z are precisely all the F-indices of pir, and pi,+ 1 (for all k = 1,2, . . . , r), then we define the associate (n - r) x (n - r) matrix P” of P as follows: First, we construct an n x (n - r) matrix P* from P by replacing Pi, by Pi, + Pi,+ 1 and deleting Pi, + 1 for all k E { 1,2, . . , r}. Now P” is obtained from P* by deleting the (ik + 1)th row for all k E {1,2, . . . . rI. 1 We call ii, iz, . , i, the associate indices of P and P”. If the y1 x y1 matrix P satisfies (i) and (ii) above, then pairs of associated row vectors of P share no vectors, and for all associated vectors Pir = (yr, . . . , y,,) and pik+i = (zi ,..., z,) (kE{1,2 ,..., r}) we have: i L’i = Zi for i # iI, . . ..i., yi = zifl, yicl = zi for i = iI, . . . . i,. This implies that, replacing column Pi, by Pi, + Pik+ 1 and deleting Pik+l for all kE(1,2,..., r}, yields an y1 x (n - r) matrix P* in which every i,th and (ik + 1)th row are identical (k = 1, . . , r). Lemma 4.5. Let A, B and C he three n x n matrices such that A = BC. Suppose C has an associate matrix C” with associate indices iI, i2, . . , i,. Construct A” and B” from A and B as ifiI,i2,..., r i were also associate indices of A and A” and B and B”, respectively. (Note that, in general, A” and B” are not associate matrices of A and B.) Then A” = B”C”. Proof. The result can be verified immediately by comparing the corresponding entries of (BC)a and B”C”. 0 Corollary 4.6. Let G and X be as in Definition 4.1 and let T = T1. Then T has an associate matrix T” such that (T”)” = (T”)” for any positive integer m. Proof. By Proposition 4.2(4), we can arrange the elements of M(X) in such a way that all pairs Wj, W, with j # k and cp(D( W,)) = D( Wj) (and cp(R( Wk)) = R(Wj)) are consecutive, i.e., 1 k - ji = 1. Except for such pairs, all elements W of M(X) satisfy O(W)) = D(W) (and q@(W)) = R(W)). Now consider a pair Wj, Wj+l with cp(R( W,)) = R( Wj+l). If Wk is such that cp(D( W,)) = D( W,), then, by Proposition 4.2(3), [T]jk = ( - l)‘“~‘r(G - R(Wj) - D(W,);X) =(-l)“C”~(G-R(Wj+l)-D(W~);~)=[T],j+l)k If W,, W,, 1 is a pair with cp(D( W,)) = D( W,+ l)r then [T]jk = ( - l)l”“‘r(G - R(Wj) - D(W,);x) = ( _ 1 ) I Lt; / I 1 r(G-R(Wj+,)-D(W,+,);x)=CTl,j+,,,,+,,, and CCl(j+l)k = ( - l) ‘““‘cz(G-R(Wj+l)-D(Wk);x) = (- l)““tl’~(G - R(Wj) - D(Wk+,);x) = [T]j(k+l). Clearly, by the above arrangement, the associated rows of Tare consecutive. Let T” be the associate matrix of T. Using Lemma 4.5 it is easy to prove, by induction on m, that ( Tm)’ = ( Ta)m for any positive integer m. 0 Theorem 4.7. Let G, X and T he as hcfore. Then: (i) T has 2”-’ - 2p-1 associated pairs of rows; (ii) the order of T” is 22p-1 + 2p- ‘; (iii) the order of the recurrence relation (*) for the matching polynomial of 7, can be reduced jiom 22p + 1 to 22p-1 + 2p-’ + 1, where 1x1 = 2p. Proof. Statements (i) and (ii) follow immediately from Proposition 4.2 (l), (2) and (4). For (iii), we use the characteristic polynomial of T” and Cayley-Hamilton’s Theorem to obtain igo al(x)(T”)N’-i = 0, where ~~~, a!(x)AN’mi is the characteristic polynomial of T” and N’ is the order of T”. If m 2 N’, multiplying by (Ta)m-N’ and applying Corollary 4.6 yields i& af(x)(Tm-i)a = 0 (m 2 N’). From this formula and the definition of T, we can obtain the recurrence relation for the matching polynomial of yrn (and of many subgraphs of y,). Since N’ = 22P-’ + 2P-1, the order of the recurrence relation can be reduced to N’ + 1 = 22p-’ + 2p-’ + 1,where p = 1X1/2. 0 References ct1 121 131 M I51 [61 171 PI D. Babit, A. Graovac, B. Mohar and T. Pisanski, The matching polynomial of a polygraph, Discrete Appl. Math. 15 (1986) 11-24. E.J. Farrell and S.A. Wahid, Matchings in benzene chains, Discrete Appl. Math. 7 (1984) 45-54. E.J. Farrell and S.A. Wahid, Matchings in long benzene chains, Match 18 (1985) 37-47. J.N. Franklin, Matrix Theory (Prentice-Hall, Englewood Cliffs, NJ 1968). A. Graovac and D. Babik, On the matching spectrum of rotagraphs, Z. Naturforsch. A 40 (1985) 66-72. A. Graovac and D. BabiC, Comment on “Matchings in long benzene chains” by E.J. Farrell and S.A. Wahid, Match 18 (1985) 49-53. A. Graovac, O.E. Polansky and N.N. Tyutyulkov, Acyclic and characteristic polynomial of regular polymers and their derivatives, Croat. Chem. Acta 56 (1983) 325-356. I. Gutman, E.J. Farrell and S.A. Wahid, On the matching polynomials of graphs containing benzenoid chains, J. Combin. Inform. System Sci. 8 (1983) 159- 168. 86 H.J. Broersmu, Li Xuelimg [9] H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Phys. 26 (1985) 1577167. [lo] H. Hosoya and N. Okhami, Operator technique for obtaining the recursion formulas of characteristic and matching polynomials as applied to polyhex graphs, J. Comput. Chem. 4 (1983) 5855593. [l l] Zhang Yuanda, Linear Algebra (Shanghai Educational Publishing House, Shanghai, 1978). 

On "The matching polynomial of a polygraph"

AbstractIn this note we give an explanation for two phenomena mentioned in the concluding remarks of “The matching polynomial of a polygraph” by Babić et al. The following results are obtained: 1.Although three matrices for given polygraphs defined in the above article in general have different orders, they determine the same recurrence relations for the matching polynomial of these polygraghs.2.Under certain symmetry conditions, the order of the recurrence relations can be reduced by almost a half

Xueliang, Li

Elsevier - Publisher Connector 

Discrete Applied Mathematics 46 (1993) 79-86 
North-Holland 
Note 
79 
On “The matching polynomial 
of a polygraph” 
H.J. Broersma and Li Xueliang* 
Department of Applied Muikematics, Unirersit.v of Twente, Enschede, Nrther1md.s 
Received 11 December 1990 
Revised 25 January 1993 
Abstract 
In this note we give an explanation for two phenomena mentioned in the concluding remarks 
of “The matching polynomial of a polygraph” by Babic et al. The following results are ob- 
tained: 
(1) Although three matrices for given polygraphs defined in the above article in general have 
different orders, they determine the same recurrence relations for the matching polynomial of these 
polygraphs. 
(2) Under certain symmetry conditions, the order of the recurrence relations can be reduced by 
almost a half. 
Keywords. Polygraph, matching polynomial, recurrence relation. 
1. Terminology 
For notation and terminology not defined here we refer to [l] and [4,11]. 
Let G = (V, E) be a graph on IZ vertices, that is, with 1 VI = n. Then the matching 
polynomial of G, denoted by cr(G;x), is defined by 
I n/2 I 
cx(G;x)=-I-(- l)kp(G,k)~“-2k, 
k=O 
Correspondence to: Professor H.J. Broersma, Faculty of Applied Mathematics, University of Twente, 
P.O. Box 217, 7500 AE Enschede, Netherlands. 
*On leave from Department of Mathematics, Xinjiang University, Urumchi, Xinjiang, P.R. China. 
0166-218X/93/$06.00 c 1993-Elsevier Science Publishers B.V. All rights reserved 
where p(G, k) denotes the number of k-matchings of G, i.e., the number of ways we can 
choose k independent edges in G, and p(G, 0) := 1. If E = 8, then cc(G; x) = xn. The sum 
of the absolute values of all coefficients of cc(G;x) is the so-called Hosoya index 
of G. 
For a subset F of E, M(F) denotes the set of all matchings in F, i.e., the set of all 
independent subsets of F. 
Let G1,Gz,..., G, be a set of mutually disjoint graphs, and let Xi, X2,. ,X, be 
a set of binary relations such that Xi E: V(Gi) x V(Gi+ i) (1 I i I m - 1) and 
X, E V(G,) x V(G,). Then the polygraph Cl,,, is defined as follows: 
V(s2,) = V(G,) u V(G,) u ... u V(G,), 
E(Q,) = E(G,) u X, u E(G,) u X2 u ... u E(G,) u X,. 
Denote by Ti the subgraph of Q, with v(ri) = V(G,) u ... u V(G,) and 
E(~,)=E(G,)uX,u~~~uE(Gi)(i~{1,2,...,m}).1fG1=G2= ... =G,=Gand 
Xi = X2 = ... = X, = X, we denote R, by w, and Ti by yi; o, and Y,,, are called 
a rotagraph and a ,fasciagraph, respectively. 
2. Introduction 
Hosoya’s index has many applications, e.g., in physical chemistry, statistical physics 
and thermodynamics. The computation of this index is a very difficult problem 
(# P-complete). 
Recently, many authors (see [6-IO]) gave explicit expressions or recurrence rela- 
tions for matching polynomials of some polygraphs. Furthermore, Babii- et al. [l] 
obtained a systematic method to compute matching polynomials for polygraphs. 
Many of the known results can be regarded as special cases of their results. 
In [l], they introduced three types of matrices to obtain recurrence relations for 
the matching polynomials of w, and yrn by using the Cayley-Hamilton Theorem from 
linear algebra (see [4,11]). We repeat some of the definitions in [l]. By hil(X) we 
denote the set of all matchings in X, with a matching viewed as a subset of indepen- 
dent edges, including the empty set. To distinguish between the elements of M(X) we 
use a one-to-one mapping of the set M(X) onto the set of indices {1,2, , IM(X)I ) 
and denote the elements of M(X) with the assigned index j by Wj. The matching 
Wj can also be viewed as a binary relation with domain D( Wj) c V(Gi) and 
range 
R(Wj) ~ 
V(Gi+l) for i = 1,2, ,m - 1, 
V(Gi 1 for i = m. 
The three types of matrices are defined as follows (see [l, p. 181): 
[Ti]jk:= (- l)‘@“a(G - R(Wj) ~ D(Wk);x), 
where Wj and W, are elements of M(X) with range R( Wj) and 
domain D( W,), respectively. 
[TJjk:= ( - I)‘“\’ 1 a(G - R(W) - D( Wk.; x), 
WlD(W)=D(W,) 
where j runs through only those values with different D( Wj). 
[TJjk:= ( - 1)‘“; c cc(G - R( Wj) - D(W); x), 
WR(W=R(W,) 
where k runs through only those values with different R( W,). 
In the above formulas, x(G - A - B;x) = 0 if A n B # 8. 
Let T be one of the matrices T, , T, or TX. Then, by [ 1, Corollary lo], 
i 
cc(o,;x) = tr(T) and 
@(Yfn - R( Wj) - D( W,);X) = ( - l)l”” [T”]jk. 
Let the characteristic polynomial of T be 
4(T;2) := det(j,1, - T) = f LZ~(.~)~.~-~, 
i=o 
where N is the order of T and so(x) := 1. Then, by [l, Corollary 111, the recurrence 
relations for the matching polynomials of w, and ;I~ are given by 
1 
i% ai(x)cc(om-i;X) = OS 
i~o Ui(X)~(~,~i - R( Wj) - D( W,);x) = 0 ) 
(*) 
where m 2 N, z(o,,;x) := N and r(g, - R( Wj) - D( Wk); X) := ( - I)““‘dj, (Sj, is 
Kronecker’s symbol). 
It is obvious from the above results that the order of the recurrence relations 
is closely related to the rank of T: the higher the rank of T, the higher the order 
of (*). 
In the concluding remarks of [ 11, BabiC et al. proposed two problems with regard to 
the order of (*). 
Problem 1. In [l, Example 141, three matrices with different orders give rise to three 
recurrence relations with the same order and coefficients. Is this a general phenom- 
enon? 
In the next section we prove that the answer is affirmative. 
Problem 2. In [I, Example 131, it is possible to obtain another matrix T’ with lower 
order than T for the special case when y,,, is constructed from a graph G and a 1-l 
binary relation X that are both symmetric (in a sense to be defined in Section 4). What 
is the role of the symmetry of G and X in a possible reduction of the order? 
82 H.J. Broersma, Li Xueliang 
We answer this question in Section 4 and show that the order of the recurrence 
relation of cc(y,;x) in this special case can be reduced from 22p + 1 to 
22p-’ + 2P-’ + 1,where p = (X1/2. 
3. Answer to Problem 1 
We claim that, although in general T,, T2 and T, have different orders (especially 
when X is not l-l), they determine the same recurrence relations (*). 
Before we prove this, we introduce the following matrix transformation. Let A be an 
n x n matrix with entries from a commutative ring. Denote by C(~ the ith row of A and 
by Aj thejth column of A. Two columns are said to be in the same class if and only if 
they are equal. Assume that the columns of A are classified into s classes. Let A* be the 
IZ x s matrix obtained from A by deleting all columns except for those with minimum 
index in every class. Now, two rows a: and G$ of A * are said to be in the same class if 
and only if Ai and Aj are in the same class. Obviously, the rows of A* also fall into 
s classes. Let B be the s x s matrix obtained from A * by replacing the rows with minimum 
index in every class by the sum of all rows in that class and by deleting the other rows. 
Lemma 3.1. det (21, - A) = j.“-“det (,?I, - B). 
Proof. From linear algebra (see [4,1 l]), we know that the coefficient ai of xnmi 
(0 < i I n) in the characteristic polynomial det ().I,, - A) = c,!=, aixnSi of A is equal 
to ( - 1)’ times the sum of all main minors of A with order i. It is clear that all minors 
with order > s are zero. Hence, a, = a,_ 1 = ... = a,, 1 = 0. It remains to prove that 
the sum of all main minors of A with order i (0 I i 5 s) is equal to that of B. This is an 
easy consequence of the following property 1 Al .. . Ai + Ai ... A, 1 = 1 Al ... Ai ... A, 1 
+ IA, . . . A; . . . A,1 and other elementary properties of determinants. 0 
Theorem 3.2. The recurrence relations (*) for the matching polynomials of w, and y,,, 
determined by T,, T2 and T3, respectively, have the same order and coefficients. 
Proof. It is not difficult to check that T, is obtained from T1 by applying the above 
transformation, and that T, is obtained by applying this transformation to the 
transpose T: of T1. Let the orders of T1, T2 and T3 be n, s and t, respectively. Then, by 
Lemma 3.1, we have 
and 
det (/II, - Ti) = j_“-“det (1.1, - T2) 
det (AI,, - T,) = det (j-1, - TT) = l.“-‘det (i1, - T3). 
As a consequence, T1, T2 and T, yield three recurrence relations (*) with the same 
order and coefficients. 0 
On “The matching polynomial qf a polygraph” 83 
4. Answer to Problem 2 
Throughout this section we assume X is a l-l binary relation, implying that 
T1 = T2 = T3 = T. 
Definition 4.1. Let G be a graph with vertex set V(G) = { 1,2, . . . ,m} u { 1’,2’, . ,m’j 
and X a relation on V(G). Then G and X are said to be symmetric if 
(i) the mapping cp defined by q(i) = i’ and q(i’) = i (i = 1,2, . . . , m) is an auto- 
morphism of G, and 
(ii) X = {(i,,j,),..., (i,, j,)} u {(i; ,j;), . ,(iz, jb)}, where i: = i if i, = i’ and j: = j if 
j, = j’. 
The following facts are obvious and stated without proofs. 
Proposition 4.2. Let G and X be as in Dejinition 4.1. Then: 
(1) R(Wj) isan r-subset(r=0,1,...,2p)of{j,,j;,...,j,,jb} and thus thereare 22p 
difSerent R( Wj). Similarly, there are 22p diflerent D( W,); 
(2) T is a matrix of order 22p, and the recurrence relation (*) for yrn determined by 
T has 22p + 1 terms; 
(3) let S = {u1,u2, . . . . u,} u {w;,w;, . . . . wi} G V(G). Then S’ := {II;,& . . . . u:> 
u (w1,w2,...,wt) c V(G) and G[V(G) - S] z G[V(G) - S’], where G[U] denotes 
the subgraph of G induced by a subset U of V(G); 
(4) for every D( W,) there exists a Wj such that cp(D(W,)) = D(y) and 
q(R( Wk)) = R( Wj) and such that D( WJ = D( Wj) if and only ifR( Wk) = R( Wj). 
By arranging T in a proper way it is possible to reduce the order of (*) for Y,,, 
from 22p + 1 to 22p-’ + 2p-’ + 1. B e ore f we prove this, we need some more ter- 
minology. 
Definition 4.3. Let Y = (yr, y,, . . . ,y,,) and Z = (z1,z2, . . . ,z,) be two n-dimensional 
vectors. We call Y and Z a pair of associated vectors if, for all i E { 1,2, . . , n}, either 
yi = Zi Or yi = Zif 1 and yi+ 1 = zi; an index i for which yi = zi is called an E-index, an 
index i for which yi = Zi+ 1 and yi+ 1 = Zi is called an F-index. 
Definition 4.4. Let P be an n x n matrix. Denote by pi the ith row of P and by Pj the jth 
column of P. If P satisfies the following conditions: 
(i) Pi, and Pir, + I are associated vectors, where k = 1,2, . . . , r, and / i, - i, 1 2 2 for all 
S,tE {1,2 ,..., r}, s # t; 
(ii) i, , i2, . . . , ‘* z are precisely all the F-indices of pir, and pi,+ 1 (for all k = 1,2, . . . , r), 
then we define the associate (n - r) x (n - r) matrix P” of P as follows: 
First, we construct an n x (n - r) matrix P* from P by replacing Pi, by Pi, + Pi,+ 1 
and deleting Pi, + 1 for all k E { 1,2, . . , r}. Now P” is obtained from P* by deleting the 
(ik + 1)th row for all k E {1,2, . . . . rI. 1 We call ii, iz, . , i, the associate indices of 
P and P”. 
If the y1 x y1 matrix P satisfies (i) and (ii) above, then pairs of associated row 
vectors of P share no vectors, and for all associated vectors Pir = (yr, . . . , y,,) and 
pik+i = (zi ,..., z,) (kE{1,2 ,..., r}) we have: 
i 
L’i = Zi for i # iI, . . ..i., 
yi = zifl, yicl = zi for i = iI, . . . . i,. 
This implies that, replacing column Pi, by Pi, + Pik+ 1 and deleting Pik+l for all 
kE(1,2,..., r}, yields an y1 x (n - r) matrix P* in which every i,th and (ik + 1)th row 
are identical (k = 1, . . , r). 
Lemma 4.5. Let A, B and C he three n x n matrices such that A = BC. Suppose C 
has an associate matrix C” with associate indices iI, i2, . . , i,. Construct A” and B” from 
A and B as ifiI,i2,..., r i were also associate indices of A and A” and B and B”, 
respectively. (Note that, in general, A” and B” are not associate matrices of A and B.) 
Then A” = B”C”. 
Proof. The result can be verified immediately by comparing the corresponding entries 
of (BC)a and B”C”. 0 
Corollary 4.6. Let G and X be as in Definition 4.1 and let T = T1. Then T has an 
associate matrix T” such that (T”)” = (T”)” for any positive integer m. 
Proof. By Proposition 4.2(4), we can arrange the elements of M(X) in such a way that 
all pairs Wj, W, with j # k and cp(D( W,)) = D( Wj) (and cp(R( Wk)) = R(Wj)) are 
consecutive, i.e., 1 k - ji = 1. Except for such pairs, all elements W of M(X) satisfy 
O(W)) = D(W) (and q@(W)) = R(W)). 
Now consider a pair Wj, Wj+l with cp(R( W,)) = R( Wj+l). If Wk is such that 
cp(D( W,)) = D( W,), then, by Proposition 4.2(3), 
[T]jk = ( - l)‘“~‘r(G - R(Wj) - D(W,);X) 
=(-l)“C”~(G-R(Wj+l)-D(W~);~)=[T],j+l)k 
If W,, W,, 1 is a pair with cp(D( W,)) = D( W,+ l)r then 
[T]jk = ( - l)l”“‘r(G - R(Wj) - D(W,);x) 
= ( _ 1 ) I Lt; / I 1 r(G-R(Wj+,)-D(W,+,);x)=CTl,j+,,,,+,,, 
and 
CCl(j+l)k = ( - l) ‘““‘cz(G-R(Wj+l)-D(Wk);x) 
= (- l)““tl’~(G - R(Wj) - D(Wk+,);x) = [T]j(k+l). 
Clearly, by the above arrangement, the associated rows of Tare consecutive. Let T” 
be the associate matrix of T. Using Lemma 4.5 it is easy to prove, by induction on m, 
that ( Tm)’ = ( Ta)m for any positive integer m. 0 
Theorem 4.7. Let G, X and T he as hcfore. Then: 
(i) T has 2”-’ - 2p-1 associated pairs of rows; 
(ii) the order of T” is 22p-1 + 2p- ‘; 
(iii) the order of the recurrence relation (*) for the matching polynomial of 7, can be 
reduced jiom 22p + 1 to 22p-1 + 2p-’ + 1, where 1x1 = 2p. 
Proof. Statements (i) and (ii) follow immediately from Proposition 4.2 (l), (2) and (4). 
For (iii), we use the characteristic polynomial of T” and Cayley-Hamilton’s Theorem 
to obtain 
igo al(x)(T”)N’-i = 0, 
where ~~~, a!(x)AN’mi is the characteristic polynomial of T” and N’ is the order 
of T”. 
If m 2 N’, multiplying by (Ta)m-N’ and applying Corollary 4.6 yields 
i& af(x)(Tm-i)a = 0 (m 2 N’). 
From this formula and the definition of T, we can obtain the recurrence relation 
for the matching polynomial of yrn (and of many subgraphs of y,). Since 
N’ = 22P-’ + 2P-1, the order of the recurrence relation can be reduced to 
N’ + 1 = 22p-’ + 2p-’ + 1,where p = 1X1/2. 0 
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On “the matching polynomial of a polygraph” 

NARCIS 

[l l] Zhang Yuanda, Linear Algebra

Acyclic and characteristic polynomial of regular polymers and their derivatives,

Comment on “Matchings in long benzene chains” by E.J.

Matchings in benzene chains,

Matchings in long benzene chains,

Matrix Theory (Prentice-Hall,

On the matching polynomials of graphs containing benzenoid chains,

On the matching spectrum of rotagraphs,

Operator technique for obtaining the recursion formulas of characteristic and matching polynomials as applied to polyhex graphs,

The matching polynomial of a polygraph,

https://ris.utwente.nl/ws/files/6505231/Broersma93on.pdf

On "the matching polynomial of a polygraph"

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