Tutte polynomials of fan-like graphs with applications in benzenoid systems
aa r X i v : . [ m a t h . C O ] F e b Tutte polynomials of fan-like graphs with applications inbenzenoid systems
Tianlong Ma, Xian’an Jin, Fuji Zhang
School of Mathematical SciencesXiamen UniversityP. R. China
Email: [email protected], [email protected]@xmu.edu.cn
Abstract
We study the computation of the Tutte polynomials of fan-like graphs andobtain expressions of their Tutte polynomials via generating functions. Asapplications, Tutte polynomials, in particular, the number of spanning trees,of two kinds of benzenoid systems, i.e. pyrene chains and triphenylene chains,are obtained.
Keywords:
Tutte polynomial, fan-like graph, benzenoid system, spanningtree, generating function
1. Introduction
Let G = ( V, E ) be a graph with the set V of vertices and the set E of edges.The Tutte polynomial of the graph G , denoted by T ( G ; x, y ), was introducedby Tutte [19] in 1954 as a generalization of chromatic polynomials, which canbe defined by the closed formula T ( G ; x, y ) = X A ⊆ E ( x − r ( E ) − r ( A ) ( y − | A |− r ( A ) , where r ( A ) = | V | − ω ( V, A ), and ω ( V, A ) denotes the number of connectedcomponents in the graph (
V, A ). It can also be calculated recursively, whichcan be viewed as an alternative definition, by using the following deletion-contraction formula, together with the initial condition that T ( G ; x, y ) = 1 if Preprint submitted to Elsevier February 4, 2021 = ∅ . Let e ∈ E be an edge of the graph G . Then T ( G ; x, y ) = xT ( G/e ; x, y ) , if e is a bridge; yT ( G − e ; x, y ) , if e is a loop; T ( G − e ; x, y ) + T ( G/e ; x, y ) , otherwise,where G − e denotes the graph obtained from G by deleting the edge e and G/e denotes the graph obtained from G by contracting the edge e , that is,deleting e firstly and then identifying its two end-vertices into a new vertex.The Tutte polynomial of a graph contains a large amount of informationabout the graph. For example, T ( G ; 1 ,
1) counts the number of spanning treesof the graph G . It can be specialized to the chromatic and flow polynomials ofa graph, the all terminal reliability probability of a network and the partitionfunction of a q -state Potts model. Moreover, it can also be specialized to theJones polynomial of an alternating knot or link and the weight enumerator ofa linear code over GF( q ).However it is P hard to compute the Tutte polynomial in general [12].Hence, various techniques (transfer matrix method, subgraph-decompositiontrick, etc) have been developed to obtain the Tutte polynomial of many graphfamilies appearing in different fields including the field of mathematical chem-istry. For example, in [9], a recursive formula of the Tutte polynomial ofbenzenoid chains was obtained. In [7], the formula was extended to k -uniformpolygonal chains and when k = 6, it is a benzenoid chain. For more gen-eral polycyclic chains of polygons, a general scheme was discussed in [6] forcomputing many polynomials with the deletion-contraction property. In [10],an explicit expression for the Tutte polynomial of catacondensed benzenoidsystems with exactly one branched hexagon was obtained. In [4], Tutte poly-nomials of alternating polycyclic chains were obtained. Recently, a reductionformula for Tutte polynomial of any catacondensed benzenoid system was ob-tained by three classes of transfer matrices in [18].In [15], the authors studied sextet polynomials of hexagonal systems viagenerating functions, which motivates us to study the Tutte polynomial ofbenzenoid systems with recursive structure via generating functions. This isrealized by computing the Tutte polynomial of fan-like graph families whichare the planar duals of some benzenoid systems with repeated substructures.Fan-like graphs are also used to compute the Kauffman bracket polynomials ofrational links, i.e. 2-bridge links in [13]. As applications, Tutte polynomials,in particular, the number of spanning trees, of two kinds of benzenoid systems,i.e. pyrene chains and triphenylene chains, are obtained.2 . Preliminaries The first two theorems are well-known which can be found in some text-books on graph theory such as [3].
Theorem 2.1.
Let D ( G ) be the planar dual of a plane graph G . Then T ( G ; x, y ) = T ( D ( G ); y, x ) . (1) Theorem 2.2.
Let G ∗ H be a union of two graphs G and H which have onlya common vertex v , as shown in Fig. 1. Then T ( G ∗ H ; x, y ) = T ( G ; x, y ) T ( H ; x, y ) . (2) H G v Fig. 1. G ∗ H . Let S ⊆ V , we use G/S to denote the graph obtained from G by identifyingall the vertices in S into a new vertex (all edges remain preserved). For con-venience, in the following we sometimes abbreviate T ( G ; x, y ) into T ( G ). Thefollowing lemma is contained as a special case in the general splitting-formula[16] of the Tutte polynomial or 2-splitting formula for the Tutte polynomialof signed graphs [14]. H vu H Fig. 2. G . Lemma 2.3.
Let G be the union of two edge-disjoint connected graphs H and H having only two common vertices v and u , as shown in Fig. 2. Let H ′ i = H i / { v, u } for i = 1 , . Then T ( G ) = 1 xy − x − y [( y − T ( H ) T ( H ) + ( x − T ( H ′ ) T ( H ′ ) − T ( H ) T ( H ′ ) − T ( H ′ ) T ( H )] .
3s an application of Theorem 2.2 and Lemma 2.3, we have the followingresult.
Corollary 2.4.
Let G be the union of two edge-disjoint connected graphs H and H such that V ( H ) ∩ V ( H ) = v , V ( H ) ∪ V ( H ) = V ( G ) and E ( H ) ∪ E ( H ) = E ( G − e ) , where e = u u , u ∈ V ( H ) and u ∈ V ( H ) , as shownin Fig. 3. Let H ′ i = H i / { v, u i } for i = 1 , . Then T ( G ) = 1 xy − x − y [( xy − x − T ( H ) T ( H ) + ( x − T ( H ′ ) T ( H ′ ) − T ( H ) T ( H ′ ) − T ( H ′ ) T ( H )] . H H u u v e Fig. 3. G . Proof.
Applying the deletion-contraction formula to the edge e , we have T ( G ) = T ( G − e ) + T ( G/e ). By Theorem 2.2, T ( G − e ) = T ( H ) T ( H ). By Lemma2.3, we have T ( G/e ) = 1 xy − x − y [( y − T ( H ) T ( H ) + ( x − T ( H ′ ) T ( H ′ ) − T ( H ) T ( H ′ ) − T ( H ′ ) T ( H )] . Thus the Corollary holds.The following lemma is not difficult to prove and can be found in [15].
Lemma 2.5. [15]
Let f ( x ) be the generating function of the sequence { a n } ∞ and suppose that f ( x ) = 1 px − qx + 1 . Then a n = λ n +11 − λ n +12 λ − λ , where λ , = q ± p q − p are two roots of the equation λ − qλ + p = 0 . a n also has the following combinatorial expression. Lemma 2.6.
Let f ( x ) be the generating function of the sequence { a n } ∞ andsuppose that f ( x ) = 1 px − qx + 1 . Then a n = ⌊ n ⌋ X j =0 ( − j (cid:18) n − jj (cid:19) q n − j p j x n . Proof. px − qx + 1 = ∞ X i =0 ( q − px ) i x i = ∞ X i =0 i X j =0 ( − j (cid:18) ij (cid:19) q i − j p j x i + j = ∞ X j =0 X i ≥ j ( − j (cid:18) ij (cid:19) q i − j p j x i + ji = n − j ====== ∞ X j =0 X n ≥ j ( − j (cid:18) n − jj (cid:19) q n − j p j x n = ∞ X n =0 ⌊ n ⌋ X j =0 ( − j (cid:18) n − jj (cid:19) q n − j p j x n . Thus a n = ⌊ n ⌋ X j =0 ( − j (cid:18) n − jj (cid:19) q n − j p j .
3. Tutte polynomials of fan-like graphs
We first recall the fan graph and the wheel graph. The join graph of twovertex disjoint graphs G and H , denoted by G ∨ H , is the graph with the vertexset V ( G ) ∪ V ( H ) and the edge set E ( G ) ∪ E ( H ) ∪ { uv | u ∈ V ( G ) , v ∈ V ( H ) } .An n - fan , denoted by F n , is defined as F n = K ∨ P n , where K is the trivialgraph, i.e. has one vertex and no edges, and P n is the path on n vertices. An n - wheel , denoted by W n , is defined as W n = K ∨ C n , where C n is the cycleon n vertices.Tutte polynomials of the fan and wheel graphs were easily obtained byestablishing recursive relations using deletion-contraction formula. See [2] for5he recursion of the wheel graphs. In this section we shall try to generalizethem to the fan-like and wheel-like graphs. In this subsection, we shall define the first kind of fan-like graphs and studytheir Tutte polynomials and Tutte polynomials of related graphs including thewheel-like graphs.Let G be a connected graph with at least two distinct vertices v and u ,and H i be a copy of G for i = 1 , , . . . , n . Then the fan-like graph F n shownas Fig. 4 is defined to be the graph formed by identifying each vertex v of H i ’sinto a new vertex and connecting the vertex u of H i to the vertex u of H i +1 by an edge e i for i = 1 , , . . . , n − F + n is defined to be the graph obtainedfrom F n by adding an edge connecting v and u of H , and F ++ n is defined tobe the graph obtained from F + n by adding another edge connecting v and u of H n , see Fig. 5. In particular, F +1 = G + and F ++1 = G ++ . v u uu uH H n H u v u uu uH H n H u + + e e e n − e e e n − Fig. 4. The fan-like graph F n . Fig. 5. F ++ n . Now we present the first main result of this paper as follows. For con-venience, we write a bivariate function A ( x, y ) as A . In this subsection, wealways assume that A = xxy − x − y (( y − T ( G ) − T ( G/ { v, u } )) ,B = 1 xy − x − y (( x − T ( G/ { v, u } ) − T ( G )) ,C = 1 xy − x − y (( xy − y − T ( G/ { v, u } ) − T ( G )) , and λ , = A + C ± p ( A − C ) + 4 AB . Theorem 3.1.
Let n ≥ be an integer. Then T ( F n ) = T ( G ) λ n − λ n λ − λ + (( B − C ) T ( G ) + BT ( G/ { v, u } )) λ n − − λ n − λ − λ , (3)6 r T ( F n ) =(( B − C ) T ( G ) + BT ( G/ { v, u } )) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + C ) n − j − · ( AB − AC ) j + T ( G ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + C ) n − j − ( AB − AC ) j . (4) Proof.
Let R and R be subgraphs of F n induced by the sets E ( H ) and( S ≤ i ≤ n E ( H i )) ∪ { e } , respectively. Clearly, T ( R ) = T ( G ), T ( R / { v, u } ) = T ( G/ { v, u } ), T ( R ) = xT ( F n − ) and T ( R / { v, u } ) = T ( F + n − ), where u ∈ V ( H ). Let S and S be subgraphs of F + n induced by the sets E ( H ) ∪ { + } and ( S ≤ i ≤ n E ( H i )) ∪{ e } , respectively. Clearly, T ( S ) = T ( G )+ T ( G/ { v, u } ), T ( S / { v, u } ) = yT ( G/ { v, u } ), T ( S ) = xT ( F n − ) and T ( S / { v, u } ) = T ( F + n − ),where u ∈ V ( H ). By Lemma 2.3, we have (cid:26) T ( F n ) = AT ( F n − ) + BT ( F + n − ) ,T ( F + n ) = AT ( F n − ) + CT ( F + n − ) . (5)Let F ( z ) = X n ≥ T ( F n ) z n and G ( z ) = X n ≥ T ( F + n ) z n . By Equation (5), we have F ( z ) = (( B − C ) T ( G ) + BT ( G/ { v, u } )) z + T ( G ) zA ( C − B ) z − ( A + C ) z + 1 . Thus it is clear from Lemma 2.5 to obtain Equation (3). Moreover, by Lemma2.6 we have T ( F n ) =(( B − C ) T ( G ) + BT ( G/ { v, u } )) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + C ) n − j − · ( AB − AC ) j + T ( G ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + C ) n − j − ( AB − AC ) j . Note that F n is reduced to the fan graph F n if G = K . As a corollary, wehave 7 orollary 3.2. Let n ≥ be an integer. Then T ( F n ) = x λ n − λ n λ − λ + y (1 − x ) λ n − − λ n − λ − λ , or T ( F n ) = y (1 − x ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( x + y + 1) n − j − ( − xy ) j + x ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( x + y + 1) n − j − ( − xy ) j , where λ , = x + y + 1 ± p x + ( y − x + 1) . We now consider the Tutte polynomial of F ++ n by using similar method,which will be used in the Section 4. Theorem 3.3.
Let n ≥ be an integer. Then T ( F ++ n ) = T ( G ++ ) λ n − λ n λ − λ − yAT ( G/ { v, u } ) λ n − − λ n − λ − λ , (6) or T ( F ++ n ) = T ( G ++ ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + C ) n − j − ( AB − AC ) j − yAT ( G/ { v, u } ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + C ) n − j − A j ( B − C ) j . (7) Proof.
Let R and R be subgraphs of F + n induced by the sets ( S ≤ i ≤ n − E ( H i )) ∪{ e n − , + } and E ( H n ), respectively. Clearly, T ( R ) = xT ( F + n − ), T ( R / { v, u } ) = T ( F ++ n − ), T ( R ) = T ( G ) and T ( R / { v, u } ) = T ( G/ { v, u } ), where u ∈ V ( H n ).Let S and S be subgraphs of F ++ n induced by the sets ( S ≤ i ≤ n − E ( H i )) ∪{ e n − } ∪ { + } (the left one) and E ( H n ) ∪ { + } (the right one), respectively.Clearly, T ( S ) = xT ( F + n − ), T ( S / { v, u } ) = T ( F ++ n − ), T ( S ) = T ( G ) +8 ( G/ { v, u } ) and T ( S / { v, u } ) = yT ( G/ { v, u } ), where u ∈ V ( H n ). By Lemma2.3, we have (cid:26) T ( F + n ) = AT ( F + n − ) + BT ( F ++ n − ) ,T ( F ++ n ) = AT ( F + n − ) + CT ( F ++ n − ) . (8)Let F ( z ) = X n ≥ T ( F + n ) z n and G ( z ) = X n ≥ T ( F ++ n ) z n . By Equation (8), we have G ( z ) = − yAT ( G/ { v, u } ) z + T ( G ++ ) zA ( C − B ) z − ( A + C ) z + 1 . Thus it is clear from Lemma 2.5 to obtain the Equation (6). Moreover, byLemma 2.6 we have T ( F ++ n ) = − yAT ( G/ { v, u } ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + C ) n − j − A j ( B − C ) j + T ( G ++ ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + C ) n − j − ( AB − AC ) j . We can further define the wheel-like graph as follows. The wheel-like graph W n can be obtained from F n by adding an edge e n connecting u in H and u in H n and n ≥
2, see Fig. 6. If H i = K for i = 1 , , . . . , n then W n is reducedto the wheel graph W n . uu uu H H n H H v e e n e e n − e Fig. 6. The wheel-like graph W n . Theorem 3.4.
Let n ≥ be an integer. Then T ( W n ) = A n − x n − T ( W ) + n − X i =2 A n − i − x n − i − h ( A ( B − C ) T ( G ) + (1 − y ) ABT ( G/ { v, u } ))9 i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( A + C ) i − j − ( AB − AC ) j + (( A + B ) T ( G )+ ( y + 1) BT ( G/ { v, u } )) ⌊ i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( A + C ) i − j − ( AB − AC ) j i . Proof.
Let S and S be subgraphs of W n induced by the sets E ( H n ) and E ( W n ) \ E ( H n ), respectively. Clearly, T ( S ) = T ( G ), T ( S / { v, u } ) = T ( G/ { v, u } ), T ( S ) = xT ( F n − ) + T ( W n − ) and T ( S / { v, u } ) = T ( F ++ n − ), where u ∈ V ( H n ). By Lemma 2.3, we have T ( W n ) = Ax T ( W n − ) + AT ( F n − ) + BT ( F ++ n − )= A n − x n − T ( W ) + n − X i =2 A n − i − x n − i − ( AT ( F i ) + BT ( F ++ i )) . By Equations (4) and (7) we have T ( W n ) = A n − x n − T ( W ) + n − X i =2 A n − i − x n − i − ( AT ( F i ) + BT ( F ++ i ))= A n − x n − T ( W ) + n − X i =2 A n − i − x n − i − h ( A ( B − C ) T ( G ) + ABT ( G/ { v, u } )) ⌊ i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( A + C ) i − j − ( AB − AC ) j + AT ( G ) ⌊ i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( A + C ) i − j − ( AB − AC ) j − yABT ( G/ { v, u } ) ⌊ i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( A + C ) i − j − ( AB − AC ) j + BT ( G ++ ) ⌊ i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( A + C ) i − j − ( AB − AC ) j i = A n − x n − T ( W ) + n − X i =2 A n − i − x n − i − h ( A ( B − C ) T ( G ) + (1 − y ) ABT ( G/ { v, u } ))10 i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( A + C ) i − j − ( AB − AC ) j + ( AT ( G ) + BT ( G ++ )) ⌊ i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( A + C ) i − j − ( AB − AC ) j i . Note that T ( G ++ ) = T ( G ) + ( y + 1) T ( G/ { v, u } ). Thus the result holds.As an immediate consequence of Theorem 3.4, we have Corollary 3.5.
Let n ≥ be an integer. Then T ( W n ) = n − X i =2 h ( xy (1 − x − y )) ⌊ i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( x + y + 1) i − j − ( − xy ) j + ( x + x + y + y ) ⌊ i − ⌋ X j =0 (cid:18) i − j − j (cid:19) ( x + y + 1) i − j − ( − xy ) j i + x + x + xy + y + y . In the following, we consider another kind of more complicated fan-likegraphs.Let G be a connected graph with at least three distinct vertices v , u and w , and H i be a copy of G for i = 1 , , . . . , n . Then the fan-like graph G n isdefined to be the graph formed by identifying each vertex v of H i ’s into a newvertex and connecting the vertex w of H i to the vertex u of H i +1 by an edge e i for i = 1 , , . . . , n −
1, see Fig. 7. Moreover, we define + G n and + G + n similarly.The graph + G n is obtained from G n by adding an edge connecting v and u of H , and the graph + G + n is obtained from + G n by adding an edge connecting v and w of H n , see Fig. 8. In particular, + G = + G and + G +1 = + G + . v u uu ww wH H n H wu v u uu ww wH H n H wu + + e e e n − e e e n − Fig. 7. The fan-like graph G n . Fig. 8. + G + n . heorem 3.6. Let n ≥ be an integer. Then T ( G n ) = T ( G ) λ n − λ n λ − λ + ( BT ( G/ { v, u } ) − DT ( G )) λ n − − λ n − λ − λ , (9) or T ( G n ) =( BT ( G/ { v, u } ) − DT ( G )) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j + T ( G ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j , where A = 1 xy − x − y (( xy − x − T ( G ) − T ( G/ { v, w } )) ,B = 1 xy − x − y (( x − T ( G/ { v, w } ) − T ( G )) ,C = 1 xy − x − y (( xy − x − T ( G/ { v, u } ) − T ( G/ { v, u, w } )) ,D = 1 xy − x − y (( x − T ( G/ { v, u, w } ) − T ( G/ { v, u } )) , and λ , = A + D ± p ( A − D ) + 4 BC . Proof.
Let G ′ n = G n / { v, u } , where u ∈ V ( H ). From Corollary 2.4, conductingdeletion-contraction operation for the edge e = wu connecting H and H in G n and G ′ n , respectively, we have (cid:26) T ( G n ) = AT ( G n − ) + BT ( G ′ n − ) ,T ( G ′ n ) = CT ( G n − ) + DT ( G ′ n − ) . (10)Let F ( z ) = X n ≥ T ( G n ) z n and G ( z ) = X n ≥ T ( G ′ n ) z n . By Equation (10), we have F ( z ) = ( BT ( G/ { v, u } ) − DT ( G )) z + T ( G ) z ( AD − BC ) z − ( A + D ) z + 1 . T ( G n ) =( BT ( G/ { v, u } ) − DT ( G )) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j + T ( G ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j . The next goal is to get the Tutte polynomials of + G + n . Theorem 3.7.
Let n ≥ be an integer. Then T ( + G + n ) = T ( + G + ) λ n − λ n λ − λ + ( CT ( + G ) − AT ( + G + )) λ n − − λ n − λ − λ , (11) or T ( + G + n ) =( CT ( + G ) − AT ( + G + )) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j + T ( + G + ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j , where A = xxy − x − y (( y − T ( G ) − T ( G/ { v, u } )) ,B = 1 xy − x − y (( x − T ( G/ { v, u } ) − T ( G )) ,C = xxy − x − y (( y − T ( G + ) − T ( G + / { v, u } )) ,D = 1 xy − x − y (( x − T ( G + / { v, u } ) − T ( G + )) , and λ , = A + D ± p ( A − D ) + 4 BC . Proof.
Let R and R be subgraphs of + G n induced by the sets ( S ≤ i ≤ n − E ( H i )) ∪{ e n − , + } and E ( H n ), respectively. Clearly, T ( R ) = xT ( + G n − ), T ( R / { v, u } ) = T ( + G + n − ), T ( R ) = T ( G ) and T ( R / { v, u } ) = T ( G/ { v, u } ), where u ∈ V ( H n ).13et S and S be subgraphs of + G + n induced by the sets ( S ≤ i ≤ n − E ( H i )) ∪{ e n − } ∪ { + } (the left one) and E ( H n ) ∪ { + } (the right one), respectively.Clearly, T ( S ) = xT ( + G n − ), T ( S / { v, u } ) = T ( + G + n − ), T ( S ) = T ( G + ) and T ( S / { v, u } ) = T ( G + / { v, u } ), where u ∈ V ( H n ). From Lemma 2.3, we have (cid:26) T ( G + n ) = AT ( G + n − ) + BT ( G ++ n − ) ,T ( G ++ n ) = CT ( G + n − ) + DT ( G ++ n − ) . (12)Let F ( z ) = X n ≥ T ( + G n ) z n and G ( z ) = X n ≥ T ( + G + n ) z n . By Equation (12), we have G ( z ) = ( CT ( G + ) − AT ( G ++ )) z + T ( G ++ ) z ( AD − BC ) z − ( A + D ) z + 1 . Thus it is clear from Lemma 2.5 to obtain Equation (11). Moreover, by Lemma2.6 we have T ( + G + n ) =( CT ( + G ) − AT ( + G + )) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j + T ( + G + ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j . This theorem will be used to obtain the Tutte polynomials of two familiesof benzenoid systems in the next section.
4. Applications
A benzenoid system is a finite connected plane graph without cut verticesin which every interior face is bounded by a regular hexagon of side length 1. Abenzenoid system is called catacondensed if there are no three hexagons sharingone common vertex. As mentioned in the Introduction, the Tutte polynomialsof several classes of benzenoid systems are derived. One idea computing theTutte polynomials of benzenoid systems is to compute the Tutte polynomialsof their duals. 14 .1. Benzenoid chains A benzenoid chain , denoted by L n , is a catacondensed benzenoid systemin which no hexagon is adjacent to three hexagons and there exist exact twohexagons that are adjacent to one hexagon, see Fig. 9. Fig. 9. Two benzenoid chains with 6 hexagons.
In[9] and [7], the Tutte polynomial of benzenoid chains was obtained.
Theorem 4.1. [9, 7]
Let n ≥ be an integer. Then T ( L n ) = A J + √ ∆2 ! n + B J − √ ∆2 ! n , where J = X i =0 x i + y, ∆ = X i =0 x i ! + 2 y X i =0 x i + y − x y,A = 2 T ( L ) − T ( L )( J − √ ∆)∆ + J √ ∆ , B = 2 T ( L ) − T ( L )( J + √ ∆)∆ − J √ ∆ . As an application of Theorem 3.3, we can also obtain T ( L n ) in anothertwo forms. Corollary 4.2.
Let n ≥ be an integer. Then T ( L n ) = X i =1 x i + y ! λ n − λ n λ − λ − x y λ n − − λ n − λ − λ , where λ , = P i =0 x i + y ± r y P i =0 x i + (cid:16)P i =0 x i − y (cid:17) or T ( L n ) = X i =1 x i + y ! ⌊ n − ⌋ X j =0 ( − x y ) j (cid:18) n − j − j (cid:19) X i =0 x i + y ! n − j − − x y ⌊ n − ⌋ X j =0 ( − x y ) j (cid:18) n − j − j (cid:19) X i =0 x i + y ! n − j − . .2. Pyrene chains Let R n denote the pyrene chain as shown in Fig. 10. In [17], the recur-sive relation of the sextet polynomial for several classes of benzenoid systemsincluding pyrene chains was obtained. Recently, zeros of sextet polynomi-als for pyrene chains were analyzed in [15] and the forcing and anti-forcingpolynomials of perfect matchings of pyrene chains were studied in [5]. n Fig. 10. The pyrene chain R n . Note D ( R n ) = + G + n when G is the graph as shown in Fig. 11 (the labelnear the edge denotes the number of parallel edges). n v wu G Fig. 11. The dual of R n and the corresponding G . As an application of Theorem 3.7, we have
Corollary 4.3.
Let n ≥ be an integer. Then T ( R n ) = ( I + 2 J + K ) λ n − λ n λ − λ + ( C ( I + J ) − A ( I + 2 J + K )) λ n − − λ n − λ − λ , or T ( R n ) = ( I + 2 J + K ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j + ( C ( I + J ) − A ( I + 2 J + K )) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j , here A = y (( x − I − J ) xy − x − y , C = y (( x − I + J ) − ( J + K )) xy − x − y ,B = ( y − J − Ixy − x − y , D = ( y − J + K ) − ( I + J ) xy − x − y ,I = x + 4 x + 10 x + 20 x + 2 x y + 33 x + 8 x y + 46 x + 18 x y + 56 x + x y + 31 x y + 60 x + 6 x y + 42 x y + 56 x + 11 x y + 49 x y + 44 x + 2 x y +17 x y + 44 x y + 29 x + 2 x y + 17 x y + 34 x y + 15 x + 4 xy + 17 xy +19 xy + 4 x + y + 5 y + 8 y + 4 y , J = x ( x + 3 x + 6 x + 10 x + x y + 14 x + 4 x y + 16 x + 7 x y + 16 x +10 x y + 14 x + 2 x y + 11 x y + 10 x + 2 x y + 10 x y + 6 x + 3 x y + 7 x y +3 x + 3 xy + 4 xy + x + y + 2 y + y ) , K = x ( x + x + x + x + x + y ) ,and λ , = A + D ± p ( A − D ) + 4 BC . Let T n denote the triphenylene chain as shown in Fig. 12. Note that D ( T n ) = + G + n when G is the graph as shown in Fig. 13. n Fig. 12. The triphenylene chain T n . n Gv wu
Fig. 13. The dual of T n and the corresponding G As another application of Theorem 3.7, we have
Corollary 4.4.
Let n ≥ be an integer. Then T ( T n ) = ( I + 2 J + K ) λ n − λ n λ − λ + ( C ( I + J ) − A ( I + 2 J + K )) λ n − − λ n − λ − λ , or T ( T n ) = ( I + 2 J + K ) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j + ( C ( I + J ) − A ( I + 2 J + K )) ⌊ n − ⌋ X j =0 (cid:18) n − j − j (cid:19) ( A + D ) n − j − ( BC − AD ) j , where A = y (( x − I − J ) xy − x − y , C = y (( x − I + J ) − ( J + K )) xy − x − y ,B = ( y − J − Ixy − x − y , D = ( y − J + K ) − ( I + J ) xy − x − y ,I = y + y x + 3 y x + 4 y x + 4 y x + 3 y + 3 y x + 9 y x + 15 y x + 20 y x +21 y x + 15 y x + 9 y x + 3 y + 2 yx + 8 yx + 18 yx + 32 yx + 46 yx + 53 yx +52 yx + 43 yx + 28 yx + 15 yx + 6 yx + y + x + 4 x + 10 x + 20 x + 33 x +46 x + 56 x + 60 x + 56 x + 46 x + 33 x + 20 x + 10 x + 4 x + x , J = x ( y + y x + 3 y x + 3 y x + 3 y x + 2 y + yx + 4 yx + 7 yx + 10 yx +12 yx + 10 yx + 7 yx + 4 yx + y + x + 3 x + 6 x + 10 x + 14 x + 16 x +16 x + 14 x + 10 x + 6 x + 3 x + x ) , K = x ( y + x + ( x + x + x + x + y ) + x + x + x + x + x + x + x + x ) ,and λ , = A + D ± p ( A − D ) + 4 BC . .4. Number of spanning trees We denoted by τ ( G ) = T ( G ; 1 ,
1) the number of spanning trees of a graph G . By Corollary 4.2, we have Corollary 4.5. [8, 11]
Let n ≥ be an integer. Then τ ( L n ) = 4 + 3 √
28 (3 + 2 √ n + 4 − √
28 (3 − √ n . By Corollaries 4.3 and 4.4, we have
Corollary 4.6.
Let n ≥ be an integer. Then τ ( R n ) = 240 + 47 √ √ n + 240 − √ − √ n ,τ ( T n ) = 1329265 + 1223 √ √ ! n +1329265 − √ − √ ! n .
5. Discussions
In this paper we mainly obtain expressions of Tutte polynomial of twofamilies of fan-like graphs. As applications, the Tutte polynomials, in partic-ular the number of spanning trees, of several families of recursive benzenoidsystems such as pyrene chains and triphenylene chains are obtained, which asfar as we know are both unknown. Some numerical results are listed in theAppendixes 1-3. It is interesting to find that τ ( T n ) is greater than τ ( R n ), andit seems that the difference becomes bigger and bigger when n increases. Wealso note that dual graphs of some corona-condensed hexagonal systems arebipyramid-like graphs. For example, the Kekulene as shown in Fig. 14 and theprimitive coronoid in [20]. The Tutte polynomials of cones over a graph G , i.e. K ∨ G and bipyramids K ∨ C n was studied long long ago, see [1]. In order tocompute the Tutte polynomials of corona-condensed hexagonal systems, thecomputation of the Tutte polynomials of bipyramid-like graphs is deserved forfurther study. 19 ig. 14. Kekulene. ReferencesReferences [1] N. Biggs,
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Tutte polynomials of pyrene chains. T ( R n , x, y ) n = 1 x + 4 x + 10 x + 20 x + 35 x + 4 x y + 52 x + 12 x y + 68 x + 24 x y +80 x + 40 x y + 85 x + 5 x y + 55 x y + 80 x + 11 x y + 62 x y + 68 x + 17 x y +63 x y + 50 x + 2 x y + 23 x y + 52 x y + 31 x + 4 x y + 21 x y + 36 x y +15 x + 4 xy + 17 xy + 19 xy + 4 x + y + 5 y + 8 y + 4 y n = 2 x + 8 x + 36 x + 120 x + 330 x + 8 x y + 784 x + 56 x y + 1652 x +224 x y + 3144 x + 672 x y + 5475 x + 11 x y + 1669 x y + 8800 x +83 x y + 3574 x y + 13140 x + 333 x y + 6773 x y + 18316 x + 4 x y +973 x y +11560 x y +23916 x +30 x y +2316 x y +17976 x y +29318 x +144 x y +4698 x y +25660 x y +33788 x +2 x y +470 x y +8336 x y +33812 x y + 36624 x + 12 x y + 1180 x y + 13188 x y + 41266 x y +37323 x + 59 x y + 2455 x y + 18807 x y + 46729 x y + 35714 x +202 x y +4360 x y +24348 x y +49112 x y +32016 x +2 x y +512 x y +6768 x y +28732 x y +47842 x y +26796 x +24 x y +1049 x y +9282 x y +30924 x y +43076 x y +20838 x +80 x y +1769 x y +11319 x y +30333 x y +35685 x y +14950 x +188 x y +2556 x y +12332 x y +27002 x y +26990 x y +9796 x + 10 x y + 362 x y + 3190 x y + 11938 x y + 21626 x y + 18434 x y +5780 x + 28 x y + 532 x y + 3409 x y + 10194 x y + 15404 x y + 11188 x y +3005 x +53 x y +655 x y +3151 x y +7583 x y +9556 x y +5876 x y +1330 x +4 x y +89 x y +670 x y +2438 x y +4758 x y +4990 x y +2562 x y +473 x +6 x y +91 x y +525 x y +1526 x y +2420 x y +2071 x y +857 x y +120 x +8 xy + 81 xy + 336 xy + 733 xy + 894 xy + 592 xy + 184 xy + 16 x + y + 11 y +51 y + 129 y + 192 y + 168 y + 80 y + 16 y Appendix 2.
Tutte polynomials of triphenylene chains. T ( T n , x, y ) n = 1 x + 4 x + 10 x + 20 x + 35 x + 4 x y + 52 x + 12 x y + 68 x + 24 x y +80 x + 40 x y + 85 x + 3 x y + 57 x y + 80 x + 9 x y + 66 x y + 68 x + 15 x y +67 x y +52 x +21 x y +60 x y +35 x +3 x y +24 x y +45 x y +20 x +3 x y +21 x y + 28 x y + 10 x + 4 x y + 15 x y + 15 x y + 4 x + 4 xy + 9 xy + 6 xy + x + y + 3 y + 3 y + y n = 2 x + 8 x + 36 x + 120 x + 330 x + 8 x y + 784 x + 56 x y + 1652 x +224 x y + 3144 x + 672 x y + 5475 x + 7 x y + 1673 x y + 8800 x +63 x y + 3598 x y + 13140 x + 273 x y + 6853 x y + 18320 x + 833 x y +11768 x y +23940 x +8 x y +2050 x y +18438 x y +29400 x +66 x y +4294 x y + 26558 x y + 34000 x + 272 x y + 7858 x y + 35378 x y +37080 x +780 x y +12818 x y +43778 x y +38165 x +7 x y +1793 x y +18893 x y + 50463 x y + 37080 x + 60 x y + 3478 x y + 25358 x y +54278 x y + 34000 x + 218 x y + 5842 x y + 31166 x y + 54538 x y +29400 x +551 x y +8668 x y +35228 x y +51198 x y +23940 x +6 x y +1123 x y + 11515 x y + 36693 x y + 44863 x y + 18320 x + 46 x y +1918 x y + 13770 x y + 35228 x y + 36638 x y + 13140 x + 132 x y +2800 x y + 14888 x y + 31166 x y + 27818 x y + 8800 x + 272 x y +3577 x y + 14596 x y + 25358 x y + 19558 x y + 5475 x + 6 x y + 462 x y +4039 x y + 12957 x y + 18893 x y + 12663 x y + 3144 x + 25 x y + 646 x y +4019 x y + 10370 x y + 12818 x y + 7498 x y + 1652 x + 50 x y + 754 x y +3532 x y + 7452 x y + 7858 x y + 4018 x y + 784 x + 77 x y + 760 x y +2737 x y +4760 x y +4294 x y +1918 x y +330 x +4 x y +101 x y +655 x y +1837 x y +2651 x y +2050 x y +798 x y +120 x +6 x y +97 x y +464 x y +1042 x y + 1258 x y + 833 x y + 280 x y + 36 x + 8 x y + 77 x y + 273 x y +490 x y + 490 x y + 273 x y + 77 x y + 8 x + 8 xy + 49 xy + 126 xy + 175 xy +140 xy + 63 xy + 14 xy + x + y + 7 y + 21 y + 35 y + 35 y + 21 y + 7 y + y ppendix 3. Numbers of spanning trees of pyrene chains and tripheny-lene chains. n = 1 n = 2 n = 3 n = 4 R n T nn