Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 6 (2013) 211–219
Wiener index of iterated line graphs of trees
homeomorphic to the claw K1,3
Martin Knor
Department of Mathematics, Faculty of Civil Engineering, Slovak University of
Technology, Radlinského 11, 813 68, Bratislava, Slovakia
Primož Potočnik , Riste Škrekovski
Faculty of Mathematics and Physics, University of Ljubljana, and Institute of
Mathematics, Physics and Mechanics, Jadranska 21, 1111 Ljubljana, Slovenia
Received 24 October 2011, accepted 20 March 2012, published online 26 July 2012
Abstract
Let G be a graph. Denote by Li(G) its i-iterated line graph and denote by W (G) its
Wiener index. Dobrynin, Entringer and Gutman stated the following problem: Does there
exist a non-trivial tree T and i ≥ 3 such that W (Li(T )) = W (T )? In a series of five
papers we solve this problem. In a previous paper we proved that W (Li(T )) > W (T ) for
every tree T that is not homeomorphic to a path, claw K1,3 and to the graph of “letter H”,
where i ≥ 3. Here we prove that W (Li(T )) > W (T ) for every tree T homeomorphic to
the claw, T 6= K1,3 and i ≥ 4.
Keywords: Wiener index, iterated line graph, tree, claw.
Math. Subj. Class.: 05C12
1 Introduction
LetG be a graph. For any two of its vertices, say u and v, denote by dG(u, v) (or by d(u, v)
if no confusion is likely) the distance from u to v in G. The Wiener index of G, W (G), is
defined as
W (G) =
∑
u6=v
d(u, v),
where the sum is taken through all unordered pairs of vertices of G. Wiener index was
introduced by Wiener in [12]. It is related to boiling point, heat of evaporation, heat of
E-mail addresses: knor@math.sk (Martin Knor)primoz.potocnik@fmf.uni-lj.si (Primož Potočnik),
skrekovski@gmail.com (Riste Škrekovski),
Copyright c© 2013 DMFA Slovenije
212 Ars Math. Contemp. 6 (2013) 211–219
formation, chromatographic retention times, surface tension, vapour pressure, partition co-
efficients, total electron energy of polymers, ultrasonic sound velocity, internal energy, etc.,
see [8]. For this reason Wiener index is widely studied by chemists. The interest of mathe-
maticians was attracted in 1970’s. It was reintroduced as the distance and transmission, see
[5] and [11], respectively. Recently, there are whole special issues of journals devoted to
(mathematical properties) of Wiener index, see [6] and [7], as well as several surveys, see
e.g. [3] and [4].
By definition, if G has a unique vertex, then W (G) = 0. In this case, we say that the
graph G is trivial. We set W (G) = 0 also when the set of vertices (and hence also the set
of edges) of G is empty.
The line graph of G, L(G), has vertex set identical with the set of edges of G. Two
vertices of L(G) are adjacent if and only if the corresponding edges are adjacent in G.
Iterated line graphs are defined inductively as follows:
Li(G) =
{
G if i = 0,
L(Li−1(G)) if i > 0.
In [1] we have the following statement.
Theorem 1.1. Let T be a tree on n vertices. Then W (L(T )) = W (T )−
(
n
2
)
.
Since
(
n
2
)
> 0 if n ≥ 2, there is no nontrivial tree for whichW (L(T )) = W (T ). How-
ever, there are trees T satisfying W (L2(T )) = W (T ), see e.g. [2]. In [3], the following
problem was posed:
Problem 1.2. Is there any tree T satisfying the equality W (Li(T )) = W (T ) for some
i ≥ 3?
As observed above, if T is a trivial tree then W (Li(T )) = W (T ) for every i ≥ 1,
although here the graph Li(T ) is empty.
Denote by H the tree on six vertices out of which two have degree 3 and four have
degree 1. Since H can be drawn to resemble the letter H , it is often called the H-graph.
Graphs G1 and G2 are homeomorphic if and only if the graphs obtained from G1 and G2,
respectively, by substituting the vertices of degree two together with the two incident edges
with a single edge, are isomorphic. In [10] we proved the following:
Theorem 1.3. Let T be a tree, not homeomorphic to a path, claw K1,3 and the graph H .
Then W (Li(T )) > W (T ) for all i ≥ 3.
Since the case when T is a path is trivial, it remains to consider graphs homeomorphic
to the claw K1,3 and those homeomorphic to H . In this paper we concentrate on graphs
homeomorphic to the clawK1,3. The remaining two cases, namely the trees homeomorphic
to H for i ≥ 3 and trees homeomorphic to K1,3 for i = 3, are dealt with in a forthcoming
paper.
First, consider the case of the claw K1,3 itself. Then Li(K1,3) is a cycle of length 3 for
every i ≥ 1. Since W (K1,3) = 9 and the Wiener index of the cycle of length 3 is 3, we
have W (Li(K1,3)) < W (K1,3) for every i ≥ 1. For other trees homeomorphic to K1,3,
we prove the opposite inequality, provided that i ≥ 4:
Theorem 1.4. Let T be a tree homeomorphic to K1,3, such that T 6= K1,3. Then it holds
that W (Li(T )) > W (T ) for every i ≥ 4.
M. Knor et al.: Wiener index of iterated line graphs of trees homeomorphic to the claw K1,3 213
In [9] we proved the following statement:
Theorem 1.5. Let G be a connected graph. Then fG(i) = W (Li(G)) is a convex function
in variable i.
Hence, to prove Theorem 1.4 it suffices to prove:
Theorem 1.6. Let T be a tree homeomorphic to K1,3, such that T 6= K1,3. Then it holds
W (L4(T )) > W (T ).
2 Proofs
Let a, b, c ≥ 1. Denote by Ca,b,c a tree that has three paths of lengths a, b and c, starting
at a common vertex of degree 3. Obviously, Ca,b,c is homeomorphic to K1,3 and C1,1,1 =
K1,3. By symmetry, we may assume a ≥ b ≥ c, see Figure 1 for C5,4,3.
Figure 1: The graph C5,4,3.
Denote
∆Ca,b,c = W (L
4(Ca,b,c))−W (Ca,b,c).
Our aim is to prove ∆Ca,b,c > 0 if a ≥ 2. We start with the case a ≤ 3. This case will
serve as the base of induction in the proof of Theorem 1.6.
Lemma 2.1. Let 3 ≥ a ≥ b ≥ c ≥ 1 and a 6= 1. Then ∆Ca,b,c > 0.
Proof. Since 3 ≥ a ≥ b ≥ c ≥ 1 and a 6= 1, there are 9 cases to consider. In Table 1
we present ∆Ca,b,c for each of these cases. The results were found by a computer, though
it is rather easy to find W (Ca,b,c) by hand, and W (L4(Ca,b,c)) can be found by applying
Proposition 2.3 to L2(Ca,b,c).
214 Ars Math. Contemp. 6 (2013) 211–219
(a, b, c) W (Ca,b,c) W (L
4(Ca,b,c)) ∆Ca,b,c
(3, 3, 3) 138 642 504
(3, 3, 2) 102 533 431
(3, 3, 1) 75 257 182
(3, 2, 2) 72 435 363
(3, 2, 1) 50 192 142
(3, 1, 1) 32 65 33
(2, 2, 2) 48 348 300
(2, 2, 1) 31 138 107
(2, 1, 1) 18 38 20
Table 1: ∆Ca,b,c for a ≤ 3.
In what follows we assume that a ≥ 4. Denote
δ0(a, b, c) = W (Ca,b,c)−W (Ca−1,b,c)
δ4(a, b, c) = W (L
4(Ca,b,c))−W (L4(Ca−1,b,c)).
Then
∆Ca,b,c −∆Ca−1,b,c = δ4(a, b, c)− δ0(a, b, c), (2.1)
so if we prove δ4(a, b, c)− δ0(a, b, c) ≥ 0, we obtain ∆Ca,b,c ≥ ∆Ca−1,b,c.
We distinguish 4 vertices in Ca,b,c. Denote by y the vertex of degree 3, and denote by
x1, x2 and x3 the pendant vertices so that d(x1, y) = a, d(x2, y) = b and d(x3, y) = c, see
Figure 1. As is the custom, by V (G) we denote the vertex set of G.
Lemma 2.2. Let a, b, c ≥ 1. Then
δ0(a, b, c) =
(
a+ b+ 1
2
)
+
(
a+ c+ 1
2
)
−
(
a+ 1
2
)
.
Proof. Since Ca−1,b,c is a subgraph of Ca,b,c with V (Ca,b,c) − V (Ca−1,b,c) = {x1}, we
have
δ0(a, b, c) = W (Ca,b,c)−W (Ca−1,b,c) =
∑
u
d(u, x1),
where the sum goes through all u ∈ V (Ca,b,c) \ {x1}. For vertices u of the x1 − x2 path,
the sum of all d(u, x1) is 1 + 2 + · · ·+ (a+b) =
(
a+b+1
2
)
. For vertices of the x1 − x3 path
which do not lay on x1 − x2 path, the sum of d(u, x1) is (a+1) + (a+2) + · · ·+ (a+c) =(
a+c+1
2
)
−
(
a+1
2
)
, see Figure 1. Since the paths x1− x2 and x1− x3 contain all vertices of
Ca,b,c, we have δ0(a, b, c) =
(
a+b+1
2
)
+
(
a+c+1
2
)
−
(
a+1
2
)
.
For two subgraphs S1 and S2 of G, by d(S1, S2) we denote the shortest distance in
G between a vertex of S1 and a vertex of S2. If S1 and S2 share an edge then we set
d(S1, S2) = −1.
Analogously as a vertex of L(G) corresponds to an edge of G, a vertex of L2(G)
corresponds to a path of length two in G. For x ∈ V (L2(G)) we denote by B2(x) the
corresponding path in G. Let x and y be two distinct vertices of L2(G). It was proved in
[9] that
dL2(G)(x, y) = dG(B2(x), B2(y)) + 2.
M. Knor et al.: Wiener index of iterated line graphs of trees homeomorphic to the claw K1,3 215
Let u, v ∈ V (G), u 6= v. Denote by βi(u, v) the number of pairs x, y ∈ V (L2(G)),
with u being the center of B2(x) and v being the center of B2(y), such that d(B2(x),
B2(y)) = d(u, v) − 2 + i. Since d(u, v) − 2 ≤ d(B2(x), B2(y)) ≤ d(u, v), we have
βi(u, v) = 0 for all i /∈ {0, 1, 2}. Denote by deg(w) the degree of w in G. In [9] we have
the following statement:
Proposition 2.3. Let G be a connected graph. Then
W (L2(G)) =
∑
u 6=v
[(
deg(u)
2
)(
deg(v)
2
)
d(u, v) + β1(u, v) + 2β2(u, v)
]
+
∑
u
[
3
(
deg(u)
3
)
+ 6
(
deg(u)
4
)]
, (2.2)
where the first sum goes through unordered pairs u, v ∈ V (G) and the second one goes
through u ∈ V (G).
We apply Proposition 2.3 to L2(Ca,b,c) and L2(Ca−1,b,c). This enables us to calculate
δ4(a, b, c) using degrees and distances of the second iterated line graph.
Figure 2: The graph L2(C5,4,3).
Denote by w1 the pendant vertex of L2(Ca,b,c) corresponding to the path of length 2
terminating at x1. Since a ≥ 4, the unique neighbour of w1 has degree 2. Denote by w this
neighbour, see Figure 2. For every vertex u ∈ V (L2(Ca,b,c)) \ {w,w1}, denote by n(u)
the number of neighbours of u, whose distance to w is at least d(u,w). We have:
Lemma 2.4. Let a ≥ 4 and b, c ≥ 1. Then
δ4(a, b, c) =
∑
u
[(
deg(u)
2
)
d(u,w) +
(
n(u)
2
)]
,
where the sum goes through all vertices of V (L2(Ca,b,c)) \ {w,w1}.
PROOF. Observe that L2(Ca−1,b,c) is a subgraph of L2(Ca,b,c) and V (L2(Ca,b,c)) \
V (L2(Ca−1,b,c)) = {w1}. Since deg(w1) = 1, the vertex w1 cannot be the center of
a path of length 2, implying that βi(u,w1) = 0 for every u and i. Since
(
deg(w1)
2
)
=
216 Ars Math. Contemp. 6 (2013) 211–219
0, all summands of (2.2) containing w1 contribute 0 to W (L4(Ca,b,c)). The vertices of
L2(Ca−1,b,c), exceptw, have the same degree inL2(Ca,b,c) as inL2(Ca−1,b,c). The degree
of w is 1 in L2(Ca−1,b,c), and it is 2 in L2(Ca,b,c). Therefore
∑
u[3
(
deg(u)
3
)
+ 6
(
deg(u)
4
)
]
has the same value in L2(Ca,b,c) as in L2(Ca−1,b,c), so these sums will cancel out. Thus,
we have
δ4(a, b, c) = W (L
2(L2(Ca,b,c)))−W (L2(L2(Ca−1,b,c)))
=
∑
u
[(
deg(u)
2
)(
2
2
)
d(u,w) + β1(u,w) + 2β2(u,w)
]
,
where the sum goes through u ∈ V (L2(Ca−1,b,c)) \ {w}.
Let u ∈ V (L2(Ca−1,b,c)) \ {w}. Since deg(w1) = 1 and deg(w) = 2 in L2(Ca,b,c),
the unique path of length 2 centered at w contains an endvertex closer to u than w. Hence,
β2(u,w) = 0. Consequently, β1(u,w) equals the number of paths of length 2 centered at
u, both endvertices of which have distance tow at least d(u,w). Hence, β1(u,w) =
(
n(u)
2
)
,
which completes the proof.
Using Lemma 2.4 we prove the induction step.
Lemma 2.5. Let a ≥ b ≥ c ≥ 1 and a ≥ 4. Then δ4(a, b, c) ≥ δ0(a, b, c).
Proof. We distinguish 8 more vertices in L2(Ca,b,c). Denote by w2 and w3 pendant ver-
tices corresponding to the paths of length 2 containing x2 and x3, respectively, see Figure 1
and 2. Denote by z1, z2 and z3 the vertices corresponding to the paths of length 2, whose
endvertex is y; and denote by z4, z5 and z6 the vertices corresponding to the paths of length
2 centered at y. Of course, if b ≤ 2 or c ≤ 2, then some of these vertices are not defined.
For u ∈ V (L2(Ca−1,b,c)) \ {w}, denote
h(u) =
(
deg(u)
2
)
d(u,w) +
(
n(u)
2
)
.
By Lemma 2.4, we have δ4(a, b, c) =
∑
u h(u), where the sum goes through all vertices of
V (L2(Ca,b,c)) \ {w,w1}. If u ∈ {w2, w3} then deg(u) = 1 and n(u) = 0, so h(u) = 0.
Thus, vertices of degree 1 contribute 0 to
∑
u h(u). Denote
Si =
∑
u
h(u),
where the sum is taken over all interior vertices u of thewi−zi path, u 6= w and 1 ≤ i ≤ 3.
Then δ4(a, b, c) =
∑3
i=1 Si +
∑6
i=1 h(zi).
Regarding the values of a, b and c, we distinguish 4 cases:
Case 1. a ≥ 4 and b, c ≥ 3.
If u is a vertex of degree 2, then n(u) = 1 and
(
deg(u)
2
)
= 1. Hence h(u) = d(u,w).
Thus,
S1 = 1 + 2 + · · ·+ (a−4) =
(a− 3
2
)
S2 = a+ (a+1) + · · ·+ (a+b−4) =
(a+ b− 3
2
)
−
(a
2
)
S3 = a+ (a+1) + · · ·+ (a+c−4) =
(a+ c− 3
2
)
−
(a
2
)
.
M. Knor et al.: Wiener index of iterated line graphs of trees homeomorphic to the claw K1,3 217
If u ∈ {z1, z2, z3}, then deg(u) = 3 and n(u) = 2. Thus h(u) = 3d(u,w) + 1. If
u ∈ {z4, z5}, then deg(u) = 4 and n(u) = 3, so h(u) = 6d(u,w) + 3. Finally, if u = z6,
then deg(u) = 4 and n(u) = 2, so h(u) = 6d(u,w) + 1. This gives
h(z1) = 3(a−3) + 1 h(z4) = h(z5) = 6(a−2) + 3
h(z2) = h(z3) = 3(a−1) + 1 h(z6) = 6(a−1) + 1.
Since δ4(a, b, c) =
∑3
i=1 Si +
∑6
i=1 h(zi), we have
δ4(a, b, c) =
(a− 3
2
)
+
(a+ b− 3
2
)
+
(a+ c− 3
2
)
− 2
(a
2
)
+(3a−8) + 2(3a−2) + 2(6a−9) + (6a−5).
Denote P = δ4(a, b, c) − δ0(a, b, c). By Lemma 2.2 we have δ0(a, b, c) =
(
a+b+1
2
)
+(
a+c+1
2
)
−
(
a+1
2
)
. Expanding the terms we get
P = 17a− 4b− 4c− 17.
Since a ≥ b and a ≥ c, we have P ≥ 9a − 17. Finally, since a ≥ 4, we have P =
δ4(a, b, c)− δ0(a, b, c) ≥ 0.
Case 2. a ≥ 4, b ≥ 3 and c ≤ 2.
We calculate first δ4(a, b, 1). In L2(Ca,b,1) we have S3 = 0; note that z3 is not defined
here and that deg(z5) = deg(z6) = 3 (see Figure 2). Analogously as in Case 1 we get:
S1 =
(
a−3
2
)
h(z2) = 3(a−1) + 1
S2 =
(
a+b−3
2
)
−
(
a
2
)
h(z4) = 6(a−2) + 3
S3 = 0 h(z5) = 3(a−2) + 1
h(z1) = 3(a−3) + 1 h(z6) = 3(a−1)
since n(z5) = 2 and n(z6) = 1. Thus,
δ4(a, b, 1) =
(a− 3
2
)
+
(a+ b− 3
2
)
−
(a
2
)
+ (3a−8)
+(3a−2) + (6a−9) + (3a−5) + (3a−3).
Denote P = δ4(a, b, 1) − δ0(a, b, 2). By Lemma 2.2 we have δ0(a, b, 2) =
(
a+b+1
2
)
+(
a+3
2
)
−
(
a+1
2
)
. Expanding the terms we get
P = 9a− 4b− 18.
Since a ≥ b, we have P ≥ 5a − 18, and as a ≥ 4, we have P ≥ 0. Since δ4(a, b, 2) ≥
δ4(a, b, 1) and δ0(a, b, 2) ≥ δ0(a, b, 1), we conclude δ4(a, b, i) − δ0(a, b, i) ≥ P ≥ 0 for
i ∈ {1, 2}.
Case 3. a ≥ 4, b = 2 and c ≤ 2.
We find δ4(a, 2, 1). In L2(Ca,2,1) we have S2 = S3 = 0. Again, the vertex z3 is not
defined here, deg(z2) = 2 and deg(z5) = deg(z6) = 3 (see Figure 2). Analogously as in
the previous cases we get:
S1 =
(
a−3
2
)
h(z4) = 6(a−2) + 3
h(z1) = 3(a−3) + 1 h(z5) = 3(a−2) + 1
h(z2) = (a−1) h(z6) = 3(a−1)
218 Ars Math. Contemp. 6 (2013) 211–219
since n(z2) = 1, n(z5) = 2 and n(z6) = 1. Thus,
δ4(a, 2, 1) =
(a− 3
2
)
+ (3a−8) + (a−1) + (6a−9) + (3a−5) + (3a−3).
Denote P = δ4(a, 2, 1)−δ0(a, 2, 2). By Lemma 2.2 we have δ0(a, 2, 2) = 2
(
a+3
2
)
−
(
a+1
2
)
.
Expanding the terms we get
P = 8a− 26.
Since a ≥ 4, we have P ≥ 0. Since δ4(a, 2, 2) ≥ δ4(a, 2, 1) and δ0(a, 2, 2) ≥ δ0(a, 2, 1),
we conclude δ4(a, 2, i)− δ0(a, 2, i) ≥ P ≥ 0 for i ∈ {1, 2}.
Case 4. a ≥ 4 and b = c = 1.
In L2(Ca,1,1) we have S2 = S3 = 0. Note that the vertices z2 and z3 are not defined,
while deg(z4) = deg(z5) = 3 and deg(z6) = 2 (see Figure 2). Analogously as in the
previous cases we get:
S1 =
(
a−3
2
)
h(z4) = h(z5) = 3(a−2) + 1
h(z1) = 3(a−3) + 1 h(z6) = (a−1)
since n(z4) = n(z5) = 2 and n(z6) = 0. Thus,
δ4(a, 1, 1) =
(a− 3
2
)
+ (3a−8) + 2(3a−5) + (a−1).
Denote P = δ4(a, 1, 1)−δ0(a, 1, 1). By Lemma 2.2 we have δ0(a, 1, 1) = 2
(
a+2
2
)
−
(
a+1
2
)
.
Expanding the terms we get
P = 4a− 15.
Since a ≥ 4, we have P ≥ 0, and hence δ4(a, 1, 1)− δ0(a, 1, 1) ≥ P ≥ 0.
Proof of Theorem 1.6. Let T be the tree Ca,b,c with a ≥ b ≥ c ≥ 1, such that a 6= 1. If
a ≤ 3, then ∆Ca,b,c = W (L4(Ca,b,c))−W (Ca,b,c) > 0, by Lemma 2.1.
Now suppose that a ≥ 4. Consider lexicographical ordering of triples (a′, b′, c′) with
a′ ≥ b′ ≥ c′ ≥ 1 and a′ ≥ 2. Further, assume that ∆Ca′,b′,c′ > 0 for every triple
(a′, b′, c′), such that a′ ≥ b′ ≥ c′ ≥ 1 and a′ ≥ 2, which is in the lexicographical ordering
smaller than (a, b, c).
Let (a∗, b∗, c∗) be ordering of triple (a−1, b, c), such that the multisets {a∗, b∗, c∗} and
{a−1, b, c} are the same and a∗ ≥ b∗ ≥ c∗. Then Ca−1,b,c and Ca∗,b∗,c∗ are isomorphic
graphs. Moreover, since a ≥ 4, we have a∗ ≥ 3. By (2.1) and Lemma 2.5 we see that
∆Ca,b,c −∆Ca∗,b∗,c∗ = ∆Ca,b,c −∆Ca−1,b,c
= δ4(a, b, c)− δ0(a, b, c)
≥ 0.
Since (a∗, b∗, c∗) is in the lexicographical ordering smaller than (a, b, c) and a∗ ≥ 2, by
the induction hypothesis we have ∆Ca∗,b∗,c∗ > 0. Hence, ∆Ca,b,c = W (L4(Ca,b,c)) −
W (Ca,b,c) > 0.
M. Knor et al.: Wiener index of iterated line graphs of trees homeomorphic to the claw K1,3 219
Acknowledgements
The first author acknowledges partial support by Slovak research grants VEGA 1/0871/11
and APVV-0223-10. All authors acknowledge the support of bilateral Slovak-Slovenian
grant.
References
[1] F. Buckley, Mean distance in line graphs, Congr. Numer. 32 (1981), 153–162.
[2] A. A. Dobrynin, Distance of iterated line graphs, Graph Theory Notes of New York 37 (1999),
50–54.
[3] A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta
Appl. Math. 66 (2001), 211–249.
[4] A. A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener index of hexagonal systems, Acta
Appl. Math. 72 (2002), 247–294.
[5] R. C. Entringer, D. E. Jackson, D. A. Snyder, Distance in graphs, Czechoslovak Math. J. 26
(1976), 283–296.
[6] I. Gutman, S. Klavžar, B. Mohar (eds), Fifty years of the Wiener index, MATCH Commun
Math. Comput. Chem. 35 (1997), 1–259.
[7] I. Gutman, S. Klavžar, B. Mohar (eds), Fiftieth Anniversary of the Wiener index, Discrete Appl.
Math. 80 (1997), 1–113.
[8] I. Gutman, I. G. Zenkevich, Wiener index and vibrational energy, Z. Naturforsch. 57 (2002),
824–828.
[9] M. Knor, P. Potočnik, R. Škrekovski, Wiener index in iterated line graphs, submitted, (see also
IMFM preprint series 48 (2010), 1128, http://www.imfm.si/preprinti/index.
php?langlD=1).
[10] M. Knor, P. Potočnik, R. Škrekovski, On a conjecture about Wiener index in iterated line graphs
of trees, Discrete Math. 312 (2012), 1094–1105.
[11] J. Plesnı́k, On the sum of all distances in a graph or digraph, J. Graph Theory 8 (1984), 1–21.
[12] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947),
17–20.