The Higher-Order Matching Polynomial of a Graph

THE HIGHER-ORDER MATCHING POLYNOMIAL OF A GRAPH

OSWALDO ARAUJO, MARIO ESTRADA, DANIEL A. MORALES, AND JUAN RADA

Received 27 April 2004 and in revised form 1 March 2005

G n p G j j Given a graph with vertices, let ( , ) denote the number of ways mutually non- G M x = [n/2] − j p G j xn−2j incident edges can be selected in .Thepolynomial ( ) j=0 ( 1) ( , ) , called the matching polynomial of G, is closely related to the Hosoya index introduced in applications in physics and chemistry. In this work we generalize this polynomial by introducing the number of disjoint paths of length t, denoted by pt(G, j). We compare this higher-order matching polynomial with the usual one, establishing similarities and diﬀerences. Some interesting examples are given. Finally, connections between our generalized matching polynomial and hypergeometric functions are found.

1. Introduction Let G be a graph with n vertices and let p(G, j) be equal to the number of ways in which j mutually nonincident edges can be selected in G.Bydeﬁnition,p(G,0)= 1 and clearly p(G,1) is equal to the number of edges. The Hosoya topological index Z(G)isdeﬁnedas follows [13]:

[n/2] Z(G) = p(G, j). (1.1) j=0

This index was used by Hosoya and coworkers to correlate boiling points and other physicochemical properties of alkanes (hydrocarbons of the general formula CnH2n+2) with their structure [14, 17]. It was shown by Hosoya that for the path Pn and the cycle Cn graphs, his index leads to the Fibonacci and Lucas numbers, respectively. The numbers p(G, j) in the deﬁnition of the Hosoya index can be used to construct the polynomial

[n/2] M(G) = M(G,x) = (−1)j p(G, j)xn−2j (1.2) j=0 called the matching polynomial of the graph G [4]. As it turned out, this polynomial had

Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:10 (2005) 1565–1576 DOI: 10.1155/IJMMS.2005.1565 1566 The higher-order matching polynomial of a graph been independently introduced in applications to physics and chemistry [1, 10, 11, 12, 15]. Many properties of the matching polynomial have been established. For instance, the matching polynomial and the characteristic polynomial of forests are identical, its roots real, and for several classes of graphs their matching polynomials are identical to well- known orthogonal polynomials, such as Chebyshev, Hermite, and Laguerre polynomials [3, 8]. On the other hand, the Hosoya index was generalized to higher-order Hosoya numbers by Randicetal.[´ 18]. In this generalization, instead of using the numbers p(G, j) which count the disjoint edges in G, one uses numbers pt(G, j) that count the number of ways of selecting j mutually nonincident paths of length t (t-paths, for short) in G (with pt(G,0):= 1). This new generalized index leads to higher-order Fibonacci and Lu- cas numbers in the cases of linear chains and cycles, respectively. Those numbers were introduced by Staklov [20] and, independently, by Randicetal.[´ 18, 19]. The main goal of this work is to use the numbers pt(G, j) to construct a polynomial, called the higher-order matching polynomial, which generalizes the standard matching polynomial, and to extend many of the results which appeared in [3, 8, 9] to this new polynomial. For example, we establish connections between the higher-order matching polynomial and hypergeometric functions, generalizing in this way, the well-known relations with orthogonal polynomials. The higher-order matching polynomial is a natural generalization of the usual matching or acyclic polynomial, in which the paths of length t>1 play the role of the edges. It is deﬁned as

[n/(t+1)] j n− t j Mt(G) = Mt(G,x) = (−1) pt(G, j)x ( +1) . (1.3) j=0

Actually, as we will see in Section 2, Mt(G)isaspecialcaseofanF-polynomial of the graph G, introduced and developed by Farrell [5, 6, 7]. In particular, the results in Farrell’s papers can be used to obtain explicit formulas of Mt(G) for signiﬁcant classes of graphs. This is done in Section 2.InSection 3 we establish connections between the higher-order matching polynomial and hypergeometric functions, generalizing in this way the well- known relations between the matching polynomial and orthogonal polynomials.

2. The path polynomial of a graph We follow the basic definitions and notations given by Farrell [5, 7]. Let Ᏺ be a family of connected graphs. Let w := Ᏺ → W be a function from Ᏺ to a Ᏺ G G set of weights. An -cover of a graph is a spanning subgraph of in which every component belongs to Ᏺ.ForeachᏲ-cover C of G we define π(C) = α w(α), where this product is taken over all components of the cover. The F-polynomial of G is defined as F(G,w) = π(C), (2.1) C where the sum is taken over all Ᏺ-covers of G. Oswaldo Araujo et al. 1567

In this paper, Ᏺ :={π0,π1,...},withπi a generic path of length i. The corresponding F(G,w) will be called the path polynomial, written as P(G,w). Further, the tth order matching polynomial Mt(G,x)willbeP(G,w) for the special weights w1 = x, wt+1 =−1 and wi = 0 otherwise, where wj is the weight of any path in Ᏺ with j vertices. Next we give explicit formulas for the higher-order matching polynomials for some important classes of graphs. Theorem 2.1. (1) Let G be a graph with at least one t-path Q,ande an edge of G,then: Mt(G) = Mt(G − e) − Mt(G − Q), (2.2) Q

where the summation is taken over all paths Q in G containing the edge e. (2) If v is a vertex of G,then: Mt(G) = Mt(G − v) − Mt(G − Q), (2.3) Q

where the summation is taken over all paths Q of length t containing the vertex v. Proof. (1) is a particular case of [7, Theorem 1], and (2) follows easily by applying (2.2) to all edges of G incident with v. P G w = r P H w G r From [7,Theorem2], ( , ) i=1 ( i, ), where consists of components H H ... H M G = r M H 1, 2, , r , one gets the expression t( ) i=1 ( i).

Proposition 2.2. Let v1,...,vn be the vertices of G. Then

d n M G = M G − v . dx t( ) t i (2.4) i=1

Proof. It is not diﬃcult to prove that

d n P G w = P G − v w . dw ( , ) i, (2.5) 1 i=1

Making the already mentioned substitution w1 = x, wt+1 =−1 one gets (2.4)from (2.5).

2.1. Chains Pn. In [7] Theorem 3 asserts that

n P Pn,w = wiP Pn−i,w , (2.6) i=1 which in our case reduces to Mt Pn = xMt Pn−1 − Mt Pn−(t+1) . (2.7) 1568 The higher-order matching polynomial of a graph

2.2. Circuits Cn. Theorem 11 of [7]statesthat n P Cn,w = rwr P Pn−r ,w . (2.8) r=1

Again, making the substitution w1 = x, wt+1 =−1produces Mt Cn = xMt Pn−1 − (t +1)Mt Pn−(t+1) , (2.9) Mt Cn = Mt Pn − tMt Pn−(t+1) .

FromtheaboverelationsitisalsoeasytoobtaintherecursiveformulaforCn, Mt Cn = xMt Cn−1 − Mt Cn−(t+1) , (2.10) and the following explicit formulas for Mt(Cn)andMt(Pn).

Proposition 2.3. The matching polynomials of order t of Cn and Pn are (1)

n/ t [ ( +1)] n − jt M P = − j xn−(t+1)j t n ( 1) j , (2.11) j=1 (2)

n/ t [ ( +1)] n n − jt M C = − j xn−(t+1)j. t n ( 1) n − jt j (2.12) j=1

Before we consider the cases of the complete graph Kn and the complete bipartite graph Kn,m, we include some formulas which generalize an analogous one when t = 1 (see [16]).

Proposition 2.4. Let V(G) ={v1,v2,...,vn} and E(G) ={e1,e2,...,em} be the vertices and the edges of the graph G, then m n (t +1) Mt G − ei,x = tx Mt G − vj,x + (t +1)m − tn Mt(G,x), (2.13) i=1 j=1 and n − t k p G k = n p G − v k (1) ( ( +1) ) t( , ) j=1 t( j, ), m − tk p G k = m p G − e k (2) ( ) t( , ) i=1 t( i, ) for all 1 ≤ k ≤ [(n − 1)/(t +1)].

Proposition 2.5. Let G beagraph,v avertexofG,ande1,e2,...,ed the edges incident with v. Then for each integer t ≥ 1 and l such that 1 ≤ l ≤ d − 1, Mt(G) = Mt G − e1 − e2 −···−el + Mt G − el+1 − el+2 −···−ed (2.14) − Mt G − e1 − e2 −···−ed − Mt(G − Q), Q Oswaldo Araujo et al. 1569 where the last sum is taken over all such paths containing an edge ei, 1 ≤ i ≤ l,andanedge ej, l +1≤ j ≤ d.

2.3. Complete graphs Kn. In [5,Section5]theF-polynomials of complete graphs are considered. Given a partition of the set V(G)ofverticesofthegraphG into ji sets with i vertices, formula (2.2) of that section gives the contribution of each partition to F(Kn,w)

n ji 1 φiwi Bj = n! , (2.15) ji i i=1 ! ! where φi denotes the number of spanning subgraphs of Ki,foreach1≤ i ≤ n. In the case of path polynomials one has φ1 = 1andφi = i!/2for2≤ i ≤ n,henceone gets

n n! 1 1 Bj = j , (2.16) j ji i 1! i=2 ! 2 and the corresponding term to this partition in P(Kn,w) is, of course,

j j B · w 1 ·····w n . j 1 n (2.17)

In the case of the higher-order matching polynomial, setting j1 = n − (t +1)j, jt+1 = j, the above formula turns out to be j n (t +1)j ! 1 Bj = (−1) (2.18) (t +1)j j! 2j

n− t j and the corresponding term will be Bj x ( +1) . As far as the recursive relations for F(Kn,w) are concerned, the following is proved in [5]: n+1 n F K w = φ w F K w . n+1, i − i i n−i+1, (2.19) i=1 1

Making the substitution Mt(Kn)forF(Kn,w), w1 = x, wt+1 =−1, φ1 = 1, φt+1 = (t + 1)!/2, results in n (t +1)! Mt Kn = xMt Kn − Mt Kn−t . (2.20) +1 t 2

The explicit expression for Mt(Kn)is [n/(t+1)] n t j M K = − j ( +1) ! 1 xn−(t+1)j. t n+1 ( 1) t j j j (2.21) j=0 ( +1) ! 2 1570 The higher-order matching polynomial of a graph

2.4. Complete bipartite graphs Kn,m, n ≥ m. Here we give the general procedure to obtain Mt(Kn,m)wheren is greater than or equal to m. It is convenient to consider two cases: (a) t is even and (b) t is odd, the latter being the easier one.

Proposition 2.6. The number pt(Kn,j,k) ofthecompletebipartitegraphKn,j of n + j vertices, where j ≤ n and t = 2l is an even number, is given by

1 n! j! p l Kn j,k = . (2.22) 2 , k! (n − kl)! j − k(l +1) !

Proof. We call Vn and Vj the two disjoint sets whose union is the vertex set of the graph Kn,j.Recallthatwedenotel = t/2. Now, if we represent by [(l,l + 1)] a path with l vertices in Vn and (l +1)verticesinVj, there are the following possibilities for k different nonincident t-paths in Kn,j:(k − s)different [(l,l + 1)] paths and s different [(l +1,l)] paths, which explains the combinatorial numbers and the division by 1/(k − s)!s!inorderto avoid repetitions, in the expression k k k −1 n − il j − i l p K k = 1 l l 1 ( +1) . 2l n,j, ( +1)!! k − s s l l (2.23) 2 s=0 ( )! ! i=0 +1

Now, any (2l + 1) vertices determine (l!)2((l +1)/2) diﬀerent 2l-paths. This explains the factor at the beginning of the above formula for p2l(Kn,j,k). We note that k runs from 1 to [(n + j)/(2l + 1)]. Since k −1 n − il n = ! l l k n − kl , i=0 ( !) ( )! (2.24) k k 1 = 2 k − s s k , s=0 ( )! ! ! we obtain (2.22). Proposition 2.7. Let t = 2l be an even number. Then the following recurrence relations hold for Mt(Kn,m), n ≥ m,

l 2 +1 M2l Kn,m = xM2l Kn−1,m − (l!) 2 n − 1 m n − 1 m × Mt Kn− l m−l + M l Kn−l m− l , l l ( +1), l − 1 l +1 2 , ( +1) (2.25)

l 2 +1 M2l Kn,m = xM2l Kn,m−1 − (l!) 2 n m − 1 n m × Mt Kn−l m− l + M l Kn− l m−l . l l , ( +1) l +1 l − 1 2 ( +1), (2.26) Oswaldo Araujo et al. 1571

Proof. Let Vn and Vm denote the two sets of vertices making up the vertex set of the graph Kn,m.Only(2.25)isprovenhere,since(2.26) follows by symmetry. We apply, again, Theorem 2.1,takingoff one vertex, say v of Vn. Then according to whether the different t- paths having v as end vertex have l additional points in Vn andl inVm or (l − 1) additional n−1 m n−1 m Vn l Vm ff points in and ( +1)in ,thereare l l or l−1 l+1 , respectively, di erent possibilities of choosing the other 2l vertices that, together with v,formthe2l-paths. The number of different 2l-paths joining the (2l + 1)-vertices is (l!)2((l +1)/2). Similarly, taking v in Vm yields (2.26). Proposition 2.8. Let t = 2l be an even number, then the matching polynomial of order t of thecompletebipartitegraphKn,n is given by the following expression:

n/ t [2 ( +1)] n 2 n − kl − M K = 2 ( !) (2 2 1)! x2n−(t+1)k. t n,n k k n − kl n − kl − n − k l (2.27) k=0 ! 2 ( )!( 1)! 2 (2 +1) ! Proof. Making the substitution j = n in the formula (2.22) and making the necessary simpliﬁcations, one gets (2.27). The general case of the complete bipartite graph is more complicated and we omit it. Proposition 2.9. Let t = 2l − 1 be an odd number. Then for n ≥ m,

m/l [] − j n m M K = ( 1) jl 2 xn−2lj. 2l−1 n,m j ( )! jl jl (2.28) j=0 !

Proof. We note, using the same argument as for the case t even (Proposition 2.6), that the possible selections of j mutually nonincident (2l − 1)-paths in Kn,m are

j− 1 n − il m − il n m = ! ! . l l l j n − jl l j m − jl (2.29) i=0 ( !) ( )! ( !) ( )!

2j Multiplying (2.29)by(l!) , one gets p2l−1(Kn,m, j). Summing up such p2l−1(Kn,m, j)and simplifying the combinatorial expression, we obtain (2.28). Proposition 2.10. Let t = 2l − 1 be an odd number. Then the following recurrence relations hold for M2l−1(Kn,m), n ≥ m: n − m 2 1 M l− Kn m = xM l− Kn− m − (l!) M l− Kn−l m−l , (2.30) 2 1 , 2 1 1, l − 1 l 2 1 , n m − 2 1 M l− Kn m = xM l− Kn m− − (l!) M l− Kn−l m−l . (2.31) 2 1 , 2 1 , 1 l l − 1 2 1 ,

Proof. To obtain (2.30) we apply the same criterion as inProposition 2.7.Takingv in n−1 m Vn Vn Vm ,wehave l−1 possible choices of points in and l in . As before, the factor (l!)2 is the number of diﬀerent (2l − 1)-paths joining 2l-vertices. By symmetry we obtain (2.31). 1572 The higher-order matching polynomial of a graph

3. Higher-order matching polynomial and orthogonal polynomials It is known that matching polynomials have connections to orthogonal polynomials [2, 8]. It is also known that those orthogonal polynomials can be expressed in terms of hypergeometric functions [2]. Here we want to explore the connections of the higher-order matching polynomial with hypergeometric functions for some simple graphs and to show that in the particular case of t = 1 the hypergeometric functions corresponding to our higher-order matching polynomial reduce to the well-known relations. Let Γ(z) represent the Euler gamma function. The Pochhammer symbol, deﬁned as (a)k = a(a +1)···(a + k − 1) = Γ(a + k)/Γ(a), is useful in the common expression of the generalized hypergeometric function ∞ a1 k a2 k ···(ap)k 1 k pFq(a;b;x) = · x , (3.1) b b ··· bq k k=0 1 k 2 k k ! where a = (a1,a2,...,ap)andb = (b1,b2,...,bq). In order to transform polynomial expressions into hypergeometric ones, it is very convenient to know the basic properties of the Pochhammer symbol, such as the following:

(−1)i (a)−i = , (3.2) − a i (1 ) a a +1 a +2 a + k − 1 kn (a)kn = ··· k . (3.3) k n k n k n k n

3.1. Paths Pn. Making use of the deﬁnition of (a)k and formula (3.1), we can write (2.11) as

[n/(t+1)] −n n ( )j(t+1) − t j Mt Pn = x x ( +1) . (3.4) j −n jt j=0 !( )

From (3.3)weﬁnd

k − n k − n t j(t+1) t jt (−n)j(t+1) = Πk= (t +1) ,(−n)jt = Πk= (t) . (3.5) 0 (t +1)j 0 (t)j

Then, substituting (3.5)in(3.4), we obtain n/ t t [ ( +1)] k − n / t t+1 j n k=0 ( ) ( +1) j (t +1) 1 Mt(Pn) = x j t−1 k − n /t tt xt+1 j=0 ! k=0 ( ) j (3.6) t+1 n (t +1) = x t Ft a;b; , +1 ttxt+1 where

n 1 − n t − n n 1 − n t − 1 − n a = − , ,..., , b = − , ,..., . (3.7) t +1 t +1 t +1 t t t Oswaldo Araujo et al. 1573

In the particular case of t = 1, (3.6) reduces to the known result

n n (1 − n) 4 x M Pn = x F − , ;−n; = Un , (3.8) 2 1 2 2 x2 2 where Un(z) are the Chebyshev polynomials of the second kind [2].

3.2. Cycles Cn. AsinthecaseofPn,(2.12)canbewrittenas n/ t t [ ( +1)] k − n / t t+1 j t+1 n k=0 ( ) ( +1) j (t +1) 1 n (t +1) Mt Cn = x = x t Ft a;b; , j t−1 k − n /t tt xt+1 +1 ttxt+1 j=0 ! k=0 ( +1 ) j (3.9) where

n 1 − n t − n 1 − n 2 − n t − n a = − , ,..., , b = , ,..., . (3.10) t +1 t +1 t +1 t t t

In the particular case of t = 1, (3.9)reducesto

n n 1 − n 4 x M Cn = M Cn = x F − , ;1− n; = 2Tn , (3.11) 1 2 1 2 2 x2 2 where Tn(z) are the Chebyshev polynomials of the ﬁrst kind [2].

3.3. Complete graphs Kn. Equation (2.21)canbewrittenas n/ t t [ ( +1)] k − n / t t+1 j n k=0 ( ) ( +1) j (t +1) 1 Mt Kn = x j − t xt+1 j=0 ! 2( 1) (3.12) t+1 n (t +1) = x t F a;·; , +1 0 2(−1)txt+1 where

−n 1 − n t − n a = , ,..., . (3.13) t +1 t +1 t +1

In the particular case of t = 1, (3.12) reduces to the known result

n n 1 − n 2 M Kn = M Kn = x F − , ;·;− 1 2 0 2 2 x2

2 (3.14) n− / 1 − n 3 x −n/ x = 2( 1) 2x F ; ; = 2 2Hn √ = Hen(x), 1 1 2 2 2 2 where Hen(z) are the Hermite polynomials [2]. 1574 The higher-order matching polynomial of a graph

3.3.1. Complete bipartite graphs Kn,m, n ≥ m,witht odd. Bythesameprocedureused before, (2.28)canbewrittenas [2m/(t+1)] (t− 1)/2 t+1 j n m 1 2(i − n) 2(i − m) (t +1)/2 Mt Kn m = x + − , j t t xt+1 j= ! i= +1 j +1 j 0 0 (3.15) t+1 n m (t +1)/2 = x + t F a;·;− , +1 0 xt+1 where 2n 2(1 − n) 2 (t − 1)/2 − n a = − , ,..., , t +1 t +1 t +1 (3.16) 2m 2(1 − m) 2 (t − 1)/2 − m − , ,..., . t +1 t +1 t +1 In the particular case of t = 1, (3.15)reducesto

n m 1 M Kn m = M Kn m = x + F − n,−m;·;− 1 , , 2 0 x2 (3.17) n−m 2 m n−m (n−m) 2 = x U − m;1− m + n;x = (−1) x m!Lm x ,

(β) where U(a,b,z) is the conﬂuent hypergeometric function of the second kind and Lα (z) are the generalized Laguerre polynomials [2]. When n = m,(3.17)yields

n 1 n M Kn n = M Kn n = x2 F − n,−n;·;− = U − n;1;x2 = (−1) n!Ln x2 , 1 , , 2 0 x2 (3.18)

(0) where Ln(x) = Ln (x)isaLaguerrepolynomial[2].

3.4. Complete bipartite graphs Kn,n with t even. Equation (2.27)canbewrittenas [2n/(t+1)] (t/2)−1 (t/2)−1 j= 2(j − n)/t j − n 2n 1 0 k 2( +1) Mt Kn,n = x t− k 1 j − n /t t k k=0 ! j=0 ( +1 2 ) k j=0 (3.19) t j − n t 1+t k 2 1+ 2n × − = x 2t+1Ft(a;b;z), t k x j=0 1+ 2 where

n n − n n n t − n = − 2 ... − 2(1 + ) 2(1 ) ... − 2 − 2 ... 2 a t , ,1 t , t , ,1 t , t , , t , 1+ 1+ − n n = 1 2 ... − 2 b t , ,1 t , (3.20)

t 1+t z =− 1+ . 2x Oswaldo Araujo et al. 1575

Acknowledgments We are grateful to Professor M. Randic´ for his original suggestion of the possible use of the higher-order Hosoya index to construct a new higher-order matching polynomial. We thank Professor I. Gutman for useful comments and suggestions and also the anonymous referee for his careful reading of the manuscript. M. Estrada thanks the CIEN, Universi- dad de Antioquia, for partial support of his research. This work was partially supported by Consejo de Desarrollo Cient´ıﬁco, Human´ıstico y Tecnologico´ (CDCHT), Universidad de Los Andes.

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[18] M. Randic,´ D. A. Morales, and O. Araujo, Higher-order Fibonacci numbers, J. Math. Chem. 20 (1996), no. 1-2, 79–94. [19] , Higher-order Lucas numbers, unpublished. [20] A. P. Stakhov, Golden Ratio Codes, Cybernetics, Radio i Svyaz’, Moscow, 1984.

Oswaldo Araujo: Departamento de Matematicas,´ Facultad de Ciencias, Universidad de Los Andes, Merida´ 5101, Venezuela E-mail address: [email protected]

Mario Estrada: ICIMAF, La Habana, Cuba E-mail address: [email protected]

Daniel A. Morales: Facultad de Ciencias, Universidad de Los Andes, Apartado Postal A61, La Hechicera, Merida´ 5101, Venezuela E-mail address: [email protected]

Juan Rada: Departamento de Matematicas,´ Facultad de Ciencias, Universidad de Los Andes, Merida´ 5101, Venezuela E-mail address: [email protected] Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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