IMFM
Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia
Preprint series Vol. 50 (2012), 1184
ISSN 2232-2094
COMPUTING HOSOYA POLYNOMIALS OF GRAPHS FROM PRIMARY SUBGRAPHS
Emeric Deutsch Sandi Klavžar
Ljubljana, December 14, 2012
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Computing Hosoya polynomials of graphs from primary
subgraphs
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Emeric Deutsch Polytechnic Institute of New York University United States emericdeutsch@msn.com
Sandi Klavzar
Faculty of Mathematics and Physics, University of Ljubljana Jadranska 19, 1000 Ljubljana, Slovenia and
Faculty of Natural Sciences and Mathematics, University of Maribor Koroska 160, 2000 Maribor, Slovenia
sandi.klavzar@fmf.uni-li.si
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December 13, 2012
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Abstract
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The Hosoya polynomial of a graph encompasses many of its metric prop-CSI	erties, for instance the Wiener index (alias average distance) and the hyper-
CSI	Wiener index. An expression is obtained that reduces the computation of the
Hosoya polynomials of a graph with cut vertices to the Hosoya polynomial of the so-called primary subgraphs. The main theorem is applied to specific constructions including bouquets of graphs, circuits of graphs and link of graphs.
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This is in turn applied to obtain the Hosoya polynomial of several chemically relevant families of graphs. In this way numerous known results are generalized and an approach to obtain them is simplified. Along the way several misprints from the literature are corrected.
Key words: Hosoya polynomial; graph decomposition; Wiener index; spiro-chain; nanostar dendrimer; triangulene
AMS subject classification (2010): 05C31, 92E10, 05C12
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1 Introduction
The Hosoya polynomial of a graph was introduced in the Hosoya's seminal paper [22] back in 1988 and received a lot of attention afterwards. The polynomial was later
CD independently introduced and considered by Sagan, Yeh, and Zhang [31] under the name Wiener polynomial of a graph. Both names are still used for the polynomial but the term Hosoya polynomial is nowadays used by the majority of researchers.
The main advantage of the Hosoya polynomial is that it contains a wealth of information about distance based graph invariants. For instance, knowing the Hosoya polynomial of a graph, it is straightforward to determine the famous Wiener index of a graph as the first derivative of the polynomial at point 1. Cash [5] noticed that the hyper-Wiener can be obtained from the Hosoya polynomial in a similar simple manner.
Among others, the Hosoya polynomial has been by now investigated on (in the historical order) trees [4, 33, 19], composite graphs [32, 12, 13], benzenoid graphs [20, 36], tori [11], zig-zag open-ended nanotubes [38], certain graph decorations [39], armchair open-ended nanotubes [34], zigzag polyhex nanotorus [14], TUC4C8(S) nanotubes [37], pentachains [1], polyphenyl chains [26], as well as on Fibonacci and Lucas cubes [25] and Hanoi graphs [30]. For relations to other graph polynomials see [21, 3].
In this paper we consider the Hosoya polynomial on graphs that contain cut-vertices. Such graphs can be decomposed into subgraphs that we call primary subgraphs. Blocks of graphs are particular examples of primary subgraphs, but a primary subgraph may consist of several blocks. In our main result, the Hosoya polynomial of a graph is expressed in terms of the Hosoya polynomials of the corre-СЧ sponding primary subgraphs. A related result for the Wiener index of a graph (in СЧ terms of the block-cut-vertex tree of the graph) was obtained in [2]. Our main result can be thus considered as an extension (and a simplification) of [2, Theorem 1]. In the case when a graph is decomposed into two primary subgraphs, our result is a special case of [35, Theorem 2.1] where a formula is given for the Hosoya polynomial of the gated amalgamation of two graphs, which is in turn a generalization of the corresponding result on the Wiener index [24]. On the other hand, [35, Corollary 2.1] is a special case of our main result.
We point out that our formulae require the knowledge of the Hosoya polynomials of the primary subgraphs, the so-called partial Hosoya polynomials, and specific distances. In many cases these are known or easy to find; especially in the case of bouquets, chains, and links when—to make things easier—the blocks are very often identical graphs. Very often authors go through several pages of computations to find only the Wiener index of a family of graphs; one of the point of the present paper is to show that with much less effort one can find the Hosoya polynomial.
We proceed as follows. In the rest of this section the Hosoya polynomial and other concepts needed are formally introduced, while in the next section the main
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result is stated and proved. In Section 3 the result is applied to bouquets of graphs, circuits of graphs, chains of graphs, and links of graphs. These results are then applied in the final section to several families of graphs that appear in chemistry. The Wiener index and the hyper-Wiener index of them is obtained as a side product.
Let G be a connected graph and let d(G,k), k > 0, be the number of vertex pairs at distance k. Then the Hosoya polynomial [22] of G is defined as
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H (G, t) = £ d(G, k) tk
k> i
Before we continue we point out that some authors define the Hosoya polynomial by adding in the above expression also the constant term d(G, 0) = |V(G)|. For our purposes the present definition is more convenient. Clearly, no matter which definition is selected, the considerations are equivalent.
We will write dG(u, v) for the usual shortest-path distance between u and v in G. If there will be only one graph in question, we will shorten the notation to d(u, v). Let Hi and H2 be subgraphs of a connected graph G. Then the distance da(H\, H2) between Hi and H2 is min{d(u, v) | u G V(H1),v G V(H2)}. The diameter of G is defined as diam(G) = maxu veV(G) d(u, v). For a finite set A and a nonegative
integer k let (A) denote the set of all k-subsets of A. Note that (A) = |). With these notations in hand H (G, t) can be more specifically written as
diam(G)
H (G, t) = £ d(G, k) tk = £ td(u,v) . k=1	Kv}e(V(2G))
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Recall that the Wiener index W(G) of G is defined by
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W(G) = £ d(u, v),
Me(V 2G))
and that the hyper-Wiener index WW(G) is
1
WW(G) = - £ (d(u,v) + d(u,v)2) .
J	Idi u. v I + dl u. v l2^
2 Kv}€(V2G))
The relations between the Hosoya polynomial and these two indices are then
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W(G) = dH(G,t)| and WW(G) = dH(G,t)| i d2H(G,t)| W(G)= dt |t=i and WW(G)= dt |t=i + 2 dt2 |t=i.
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Finally, for a positive integer n we will use the notation [n] = {1,2,.... n}.
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2 Main result
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Let G be a connected graph and let u G V (G). Then the partial Hosoya polynomial with respect to u is
Hu(G,t) = ^ td(u'v) .
vev (g)
This concept was used by DoSliC in [12] under the name partial Wiener polynomial.
Let G be a connected graph constructed from pairwise disjoint connected graphs G1,... ,Gk as follows. Select a vertex of Gì, a vertex of G2, and identify these two vertices. Then continue in this manner inductively. More precisely, suppose that we have already used G1,..., Gi in the construction, where 2 < i < k — 1. Then select a vertex in the already constructed graph (which may in particular be one of the already selected vertices) and a vertex of Gi+1; we identify these two vertices. Note that the graph G constructed in this way has a tree-like structure, the Gi's being its building stones (see Fig. 1). We will briefly say that G is obtained by point-attaching from G1 ,...,Gk and that Gi's are the primary subgraphs of G. A particular case of this construction is the decomposition of a connected graphs into blocks.
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Figure 1: Graph G obtained by point-attaching from G1,..., Gk
Let G be a graph obtained by point-attaching from G1,..., Gk. Then let = dG(Gi, G j ). This distance is realized by precisely one vertex from Gi and one vertex from G j, denote them with and respectively; see Fig. 1 where the distance
between Gi and Gj is indicated with a dashed line.
Note that if Gi and G j share a
vertex x, then x = x.
i^j
= x
and Sij = 0.
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Now everything is ready for our main result.
Theorem 1 Let G be a connected graph obtained by point-attaching from G1,..., Gk, and let xi^j and 6ij be as above. Then
H (G, t) = E H (Gi, t) + E (я*^. (Gi, t) ■ j (Gj, t) ■ tj
{i,jK([2])
(1)
i=1
Proof. Let u = v be arbitrary vertices of G. We need to show that their contribution to the claimed expression is
Suppose first that u and v belong to the same primary subgraph, say u, v G Gi. Then dG(u, v) = dGi (u, v) and hence td(u'v) is included in the corresponding term of the first sum of the theorem.
Assume next that u and v do not belong to the same primary subgraph. If u or v is an attaching vertex, then it belongs to more than one primary subgraph. Hence select primary subgraphs Gi and Gj with u G Gi and v G Gj such that öj = dG(Gi, Gj). By our assumption i = j and hence
dG(u, v) = dGi(u,xi^j) + öij + dGj(xj^i, v),
cf. Fig. 1 again. It is possible that öij = 0, that is, xi^j = xj^i, but in any case td(u'v) is a term in the product (Gi,t) ■ t^ ■ H^(Gj,t).
We have thus proved that for any distinct vertices u and v, the term
td(U'V) is
included in the claimed expression. To complete the argument we need to show that no such term is included more than once. To verify this it suffices to prove that the total number of pairs of vertices considered in (1) is equal to the total number of pairs of vertices. Set ni = |V(Gi)| — 1, 1 < i < k, and note that then |V(G)| = 1 + k=1 ni. Then the first term of (1) involves
k
a = E
i=1
ni + 1
2
pairs of vertices, while the second sum involves
B = e n
{«KM)
pairs of vertices of G. Then
2(A + В) = > ni
En« + E ni + E
{i,jK(W)
2ninj
i=1
i=1
E
ni
1 + E ni
(|V=(G)| — 1) ■ |V(G)|.
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We conclude that A + B = (|У(2С)|), that is, the number of pairs of vertices involved
in (1) is equal to the number of all pairs.
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As an example consider the graph Q(m, n) constructed in the following manner: denoting by Kq the complete graph with q vertices, consider the graph Km and m copies of Kn. By definition, the graph Q(m, n) is obtained by identifying each vertex of Km with a vertex of a unique Kn. The graph Q(6,4) is shown in Fig. 2.
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Figure 2: Q(6, 4)
Clearly, the Hosoya polynomial of Kq is 2q(q — 1)t and the partial Hosoya polynomial with respect to any of its vertices is (q — 1)t. The distance between the central Km and a Kn is 0, while the distance between any two distinct Kn's is 1. Now, Theorem 1 gives, after elementary calculations,
H (Q(m,n),t) = 1 m(m + n2 — n — 1)t
+m(m — 1)(n — 1)t2 + 1 m(m — 1)(n — 1)2t3 . For the Wiener index and the hyper-Wiener index we obtain W(Q(m,n)) = 1 mn (3mn — 2m — 2n + 1) ,
HW(Q(m, n)) = 1 m (6mn2 — 6mn — 5n2 + m + 5n — 1) . Notice that the Wiener index W(Q(m, n)) is symmetric in m and n.
3 Specific constructions
In this section we present several constructions of graphs to which our main result can be applied. These constructions will in turn be used in the next section where chemical applications will be given.
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3.1 Bouquet of graphs
Let С1,С2,... ,Gk be a finite sequence of pairwise disjoint connected graphs and let xi G V (G i). By definition, the bouquet G of the graphs {Gi}k=1 with respect to the vertices {xi}k=1 is obtained by identifying the vertices x1,x2,... , xk (see Fig. 3 for k = 3).
x1 = x2 = x3 = x
Figure 3: A bouquet of three graphs
Clearly, we have a graph obtained by point-attaching from G1, G2,..., Gk and formula (1) holds with = 0 and = xj^i = x, where x is the vertex obtained from the identification of the xi's. Formula (1) becomes
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H (G,t) = Y H (Gi,t)+ Y Hx(Gi,t) • Hx(Gj ,t).	(2)
i=1	{i,j}e([2])
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Consider the following special case of identical Gi's. Let X be a connected graph and let x G V(X). Take Gi = X and xi = x for i G [k]. Formula (2) becomes
H (G, t) = kH (X, t) + 1 k(k - 1)H2(X, t).	(3)
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3.2 Circuit of graphs
Let G1, G2, ..., Gfc be a finite sequence of pairwise disjoint connected graphs and let xi G V(Gi). By definition, the circuit G of the graphs {Gi}k=1 with respect to the vertices {xi}k=1 is obtained by identifying the vertex xi of the graph Gi with the i-th vertex of the cycle graph Ck (see Fig. 4 for k = 4). The Hosoya polynomial of G is given by
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k
H (G, t) = Y H (Gi, t) + Y tmin(j-i'n-j+i) (1 + Hxi (Gi, t)) (1 + Hxj (G j ,t)) .
i=1	{i,j}e([2])
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Figure 4: A circuit of four graphs
This can be derived from Theorem 1 by viewing G as a graph obtained by pointattaching from the k + 1 graphs G1,G2,...,Gk, and Ck. However, we prefer to give a direct proof.
Let u = v be arbitrary vertices in G. Suppose first that u and v belong to the same graph G%, In this case,	is included in the corresponding term of
the first sum in (4). Assume now that u G Gi and v G G j, i < j. Then d(u,v) = d(u,x) + d(xi,xj) + d(xj, v), where d(xi,xj) = min(j — i, k — j + i). The first and the last term of this sum may be equal to 0. It follows that td(u'v) is a term in the product under the 2nd sum in (4). To complete the argument we need to show that no such term is included more than once. To verify this it suffices to prove that the total number of pairs of vertices considered in (4) is equal to the total number of pairs of vertices. Setting ni = |V(Gj)|, the number of pairs of vertices involved in the right-
hand side of (4) is 2 YÌ= i ni(ni—1)+E{i;j}e([k]) ninj = 1 (Ei=i m) (Ei=I m — ^,
i.e. the number of all unordered pairs of distinct vertices in G.
Consider the following special case of identical Gi's. Let X be a connected graph and let x G V(X). Take Gi = X, xi = x for i G [k]. Then formula (4) becomes
H (G, t) = kH (X, t) + (1 + Hx(X, t))2H (Ck, t).	(5)
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3.3 Chain of graphs
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Let G1,G2,... ,Gk be a finite sequence of pairwise disjoint connected graphs and let xi,yi G V(Gi). By definition (see [28, 29]) the chain G of the graphs {Gi}k=1 with respect to the vertices {xi,yi}ik=1 is obtained by identifying the vertex yi with the vertex xi+1 for i G [k — 1] (see Fig. 5 for k = 4). Denoting di = d(xi,yi), we define
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xi	yi = X2 У2 = Хз У3 = X4	У4
Figure 5: A chain of graphs
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_ I di+i + di+2 +-----+ dj-i if j — i > 2 ,
0	otherwise .
(6)
Clearly, we have a graph obtained by point-attaching from Gi,..., Gk and formula (1) holds with = yi,xj^i = xj, and 6ij = sij. Consequently, we have
k
H (G, t) = £ H (Gi,t)+ £ Hyi (Gi,t)Hxj (Gj ,t)tSij.	(7)
i=i	(i,j}e([2])
Consider the following special case of identical Gi's. Let X be a connected graph and let x,у g V(X). Take Gi = X, xi = x, yi = у for i G [k]. Then, denoting d = d(x, y), we have si,j = (j — i — 1)d and formula (7) becomes
tkd_ktd + k — 1
H (G, t) = kH (X, t) + Hx(X,t)Hy (X, t)-1)k-.	(8)
We add that in [29] long expressions with long proofs are given for the Wiener index (pp. 86-89) and for the hyper-Wiener index (pp. 93-94) for the chain of graphs.
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3.4 Link of graphs
Let Gi,G2,... ,Gk be a finite sequence of pairwise disjoint connected graphs and let xi,yi G V(Gi). By definition (see [16]), the link G of the graphs {Gi}k=i with respect to the vertices {xi,yi}k=i is obtained by joining by an edge the vertex yi of Gi with the vertex xi+i of Gi+i for all i = 1,2,..., k — 1 (see Fig. 6 for k = 4).
Figure 6: A link of graphs
The Hosoya polynomial of G is given by
k
H (G, t) = £ H (Gi, t) + Y (1 + Hyi (Gi, t)) (1 + Hxj (Gj, t)) tj-i+Sij , (9)
i= i
{i,jK([2])
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where Sj is defined in (6), with dl = d(xl,yl).
Formula (9) can be derived from Theorem 1 by viewing G as a chain of 2k — 1 graphs: the k Gì s alternating with the k — 1 K2's (edges). This derivation is rather cumbersome and, consequently, we prefer to give a direct proof.
Let u = v be arbitrary vertices in G. Suppose first that u and v belong to the same graph Gì, In this case,	is included in the corresponding term of the first
sum in (9). Assume now that u G Gì and v G G j, i < j. We break up d(u, v) into three parts: d(u, yì),d(yì,Xj) = j — i + sì)j-, and d(xj, v). The first and the last part may be equal to 0. It follows that td(u, v) is a term in the product under the 2nd sum in (9). Using the same reasoning as for the circuit of graphs we then infer that the number of pairs of vertices involved in the right-hand side in (9) is equal to the number of all unordered pairs of distinct vertices in G.
Consider the following special case of identical Gi's. Let X be a connected graph and let x,y G V(X). Take Gì = X, xì = x, yì = y for all i G [k]. Then, denoting d = d(x, y), we have di = d2 = ■ ■ ■ = dk = d and formula (9) becomes
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tfcd+fc+1 _ktd+2 + kt — t
H (G, t) = kH (X, t) + (1 + Hx(X, t)) (1 + Hy (X, t))- 1 — +-. (10)
4 Chemical applications
In this section we apply our previous results in order to obtain the Hosoya polynomial of families of graphs that are of importance in chemistry. As already pointed out, numerous distance-based invariants such as the Wiener and the hyper-Wiener index
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can then be routinely derived. 4.1 Spiro-chains
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Spiro-chains are defined in [10, p.114]. Making use of the concept of chain of graphs, a spiro-chain can be defined as a chain of cycles. We denote by Sq,h,k the chain of k cycles Cq in which the distance between two consecutive contact vertices is h (see Se,2,5 in Fig. 7).
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Figure 7: Spiro-chain Se,2,5
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The Hosoya polynomial of Sq>h)k can be easily obtained from (8). We distinguish two cases: q odd and q even.
Assume q is odd: q = 2r + 1 (r > 1). In (8) we take g = Cq and d = h. We have H (Cq, t) = (2r + 1) Yj=! tj [31] and Hx(Cq, t) = 2 £r=s tj for any vertex x of Cq. Now, (8) yields
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ufo	, k(2r + 1)t(tr - 1) 4t2(tr - 1)2(tkh - kth + k - 1)
H (S2r+1,h,k, t) = ---:--+
t - 1 ^ (t - 1)2(th - 1)2
For the Wiener index and the hyper-Wiener index we obtain
W(S2r+i,h,k) = 1 kr[3(r + 1)(1 - 2r + 4kr)+4rh(k - 1)(k - 2)] , (11) 6
00	WW(S„ . -, , , ) = 1
+2rh(k - 1)(k - 2)(2r + 3) + rh2(k - 1)2(k - 2)]. (12)
WW(S2r+ihk) = 1 kr[(r + 1)(2 - 6r + 11kr + 7kr2 - 5r2) 6
Assume q is even: q = 2r (r > 1). Again in (8) we take g = Cq and d = h. We have H (Cq, t) = 2r tj + rtr [31] and Hx(Cq, t) = 2 tj + tr for any vertex x of Cq. Now, (8) yields
^ kr(tr+s + tr - 2t) (tr+s + tr - 2t)2(tkh - kth + k - 1)
H (S2r,h,k, t) = ---:--+
t - ^ (t - 1)2(th - 1)2 For the Wiener index and the hyper-Wiener index we obtain
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W(S2r+i)h,k) = 1 k[h(2r - 1)2(k - 1)(k - 2) + 6r2(1 - r + 2rk - k)], (13) 6
WW(S2r+i)h,k) = 1 kr[(r + 1)(2 - 6r + 11kr + 7kr2 - 5r2) 6
+2rh(k - 1)(k - 1)(2r + 3) + rh2(k - 1)2(k - 2)]. (14)
Sq
From Eqs. (11), (12), (13), (14), setting q = 3, 4, 5, 6 and h G {1, 2,..., |_2J}, we recover all the expressions in Table 4.2 of [10, p.115] (they occur also in [9]). Incidentally, there is a typo in the last expression of Table 4.2 of [10]: 847 should be changed to 874. The corresponding expression in [9, Eq. (64)] is correct.
4.2 Polyphenylenes
Similarly to the above definition of the spiro-chain Sq,h,k, we can define the graph Lq,h,k as the link of k cycles Cq in which the distance between the two contact vertices in the same cycle is h. See Fig. 8 for Ьб,2,5.
We consider here only the case of hexagons (q = 6), the so-called ortho-, meta-, or para-polyphenyl chains, corresponding to h = 1,2 or 3, respectively (see [7, 8]).
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Figure 8: £5,2,5
If in (10) we take H(X, t) = 6t + 612 + 3t3 (the Hosoya polynomial of C6) and Hx(X, t) = Hy(X, t) = 2t + 2t2 + t3 (the relative Hosoya polynomial of C6 with respect to any of its vertices), then (10) becomes
H(L6,h,k, t) = 3kt(2 + 2t + t2) +
(t + 1)2(t2 +1 + 1)2(tkh+k+1 - kth+2 + kt - t) th+i - 1)2
The expression obtained from here for all the possible values h = 1,2,3 have been obtained by a different method in [26] (Theorems 2.1, 2.2, and 2.3). Now, for the Wiener index and the hyper-Wiener index we obtain
W(L6,h,k) = 3k[4h - 11 + 6k(3 - h)+2k2(1 + h)], 3
WW(L6,h,k) = 3k[-2h2 + 32h - 69 + k(5h2 - 44h + 82)
(15)
-2k2(h + 1)(2h - 7) + k3(h + 1)2].
Setting h = 1,2,3 in (15), we recover the expressions given in [8, Corollary 3.3]. The Wiener index of L6,3,k is found also in [7, p.1233]. However, the formulation of the final result has a typo: the binomial ("+1) should be preceded by 144.
The reader may be interested to find in the same way the Hosoya polynomial, the Wiener index, and the hyper-Wiener index of Lq,h,k.
4.3 Nanostar dendrimers
We intend to derive the Hosoya polynomial of the nanostar dendrimer Dk defined pictorially in [17]. A better pictorial definition can be found in [18]. In order to define Dk, first we define recursively an auxiliary family of rooted dendrimers G k (k > 1). We need a fixed graph F defined in Fig. 9; we consider one of its endpoint to be the root of F. The graph G1 is defined in Fig. 10, the leaf being its root. Now we define Gk (k > 2) as the bouquet of the following 3 graphs: Gk-1, Gk-1, and F with respect to their roots; the root of Gk is taken to be its unique leaf (see G2 and G3 in Fig. 11). Finally, we define Dk (k > 1) as the bouquet of 3 copies of Gk with respect to their roots (D2 is shown in Fig. 12, where the circles represent hexagons).
Let s denote the partial Hosoya polynomial of the graph F with respect to its root and let p denote the Hosoya polynomial of F. Direct computation yields
s = t9 + t(1+1)(1+1 +12)(1 + t4),
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Figure 9: Graph F
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Figure 10: Graph G1
Figure 11: Graphs G2 and G3
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p = 15t + 20t2 + 18t3 + 12t4 + 10t5 + 8t6 + 5t7 + 2t8 + t9 .
(16)
Let rk denote the partial Hosoya polynomial of Gk with respect to its root. It is straightforward to find ri = t(1 + t)(1 + t + t2) and the recurrence relation rk = s + 2t9rk-1; they lead to
rk
(2t9)1 - 1 2t9 - 1
+ (2t9)k-1t(1 + t)(1 + t + t2).
(17)
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Figure 12: Nanostar D2
Now from (2) we obtain a recurrence relation for H (Gk , t):
H (G k, t) = 2H (Gk-i, t) + p + 2srfc_i + r2_i,
the initial condition being
H (Gi,t) = 7t + 8t2 + 5t3 +14
The solution is
ki
H(Gk,t) = 2k-1(p + H(Gi,t)) - p ^E2k-1_jr,(2s + r,-)
j=i
(18)
(19)
where p and H(Gi,t) are given in (16) and (18), respectively.
Although not required in the sequel, we give the Wiener index and the hyperWiener index of Gk :
W(Gk) = 1323 + 2k-i3735 - 22k-212711 + 2k2223k +22k_23249k, WW (Gk) = -45867 - 2k-i173401 + 22k_31060083 - 2k-i132777k -22k_3454347k + 20007k22k_i + 29241k222k_2 .
Since Dk is a bouquet of three copies of Gk with respect to their roots, from (3) we have
H (Dk ,t) =3H (Gk ,t) + 3r2 ,
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where the terms in the right-hand side are given in (19) and (17). For the Wiener index and the hyper-Wiener index of Dk we obtain
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W(Dk) = -9369 - 22k-275411 + 22k-229241k + 2k-156205 , WW(Dk) = 116340 - 2k-11429983 + 22k-34790367 - 22k-32685555k
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+22k-2263169k2.
Incidentally, the formula given in [17, p.62] for the Wiener index of Dn contains some misprints; for example, for D1 it does not give the correct value 666 found on p. 60.
4.4 Triangulenes
We intend to derive the Hosoya polynomial of the triangulane Tk defined pictorially in [23]. We define Tk recursively in a manner that will be useful in our approach. First we define recursively an auxiliary family of triangulanes Gk (k > 1). Let G1 be a triangle and denote one of its vertices by y1. We define Gk (k > 2) as the circuit of the graphs Gk-1 ,Gk-1, and K1 and denote by yk the vertex where K1 has been placed. The graphs G1,G2,G3 are shown in Fig. 13.
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Figure 13: Graphs G1, G2, G3
Now, Tk is defined as the circuit of 3 copies of Gk with respect to their vertices yk (T2 is shown in Fig. 14).
Let rk denote the partial Hosoya polynomial of Gk with respect to the vertex yk. It is straightforward to derive that
2k+1tk+1 _ i
1 + rk = 2 * , 1 .	(20)
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Since Gk is the circuit of the graphs Gk-1,Gk-1, and K1, from (4) we obtain the recurrence equation
H (Gk ,t) = 2H (Gk-1 ,t) +1(1 + rk-1)2 + 2t(1 + rk-1).
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Figure 14: Graph T2
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The initial condition is H(Gi,t) = 3t and the solution is found to be
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H(G t. = 2k+2tk+2 + 4t2 - 3t 2k (4t2 + 3t) 22fc+1t2fc+3	,91.
H(Gk,t)= (2t - 1)2 2t2 - 1 +(2t - 1)2(2t2 - 1) • (21)
Although not required in the sequel, we give the Wiener index and the hyper-Wiener index of Gk :
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W(Gk) = 22k+1(2k - 5) + 2k (4k + 9) + 1, WW (Gk ) = 22k+1(2k2 - 9k + 16) + 2k (2k2 - 6k - 29) - 3 .
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Since Tk is a circuit of 3 copies of Gk with respect to the vertices yk, from (5) we obtain
H (Tk, t) = 3H (Gk, t) + 3t(1 + rk )2 ,
where the expressions occurring in the right-hand side are given in (21) and (20). We obtain easily
6t 2n3t(4t + 3) 22n+13t2n+3(2t + 1) H(Tk,t) = 2t2 - 1 + (2t - 1)(2t2 - 1) .
For the Wiener index of Tk we obtain
W(Tk) = 22n+13(6n - 7) + 2n51 - 6.
This expression can be found in [23, Theorem 1] and in [6, p.37, Theorem 5]. For the hyper-Wiener index of Tk we have
WW (Tk ) = 22n+13(6n2 - 11n + 20) - 2n123 + 6 .
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Acknowledgments
This work has been financed by ARRS Slovenia under the grant P1-0297. The second author is also with the Institute of Mathematics, Physics and Mechanics,
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