The generation of epicyclic gear trains (EGTs) by application of graph theory has a major focus by researchers to avoid duplicity of mechanisms as well as topological structural and synthesis ...analysis. This paper presents the generation of EGTs with one degree of freedom up to eight links. The method has been formulated on the basis of conventional graphs, the graphs converted into adjacency matrices and then, obtained wiener number for structural synthesis and isomorphic property. The results obtained by this method have been fully satisfied with references
Denote by
the set of h-polygonal chains (where h is even) with n congruent regular h-polygons (
). For any
let
be the Wiener number of
In this paper, we show that
with the equalities on the left ...holding only if
and the equalities on the right holding only if
where
and
are extremal chains of type one and type two (their definitions are given in the main text), respectively. Thus we extend the known results of extremal benzenoid chains on Wiener number to a more general case.
The detection of isomorphism by graph theory in the epicyclic geared mechanisms (EGMs) and planer kinematic chains (PKCs) has a major issue with the duplicity of mechanism from the last few decades. ...In this paper, an innovative method based on Wiener number is presented to detect all distinct epicyclic geared mechanisms with multi-degree of freedoms and various numbers of links. The method presents a quantitative technique for the determination of isomorphic property by virtue of adjacency matrices and Wiener’s number. Finally, the proposed method is examined from various examples as four, five, and six-links to 1, 2-degree of freedom EGMs and six-links 1-DOF PKCs. All examples are fully satisfied with the references taken for the analysis.
The reciprocal complementary Wiener (RCW) number of a connected graph
G
is defined in mathematical chemistry as the sum of the weights
1
d
+
1
−
d
G
(
u
,
v
)
of all unordered pairs of distinct ...vertices, where
d
is the diameter and
d
G
(
u
,
v
)
is the distance between vertices
u
and
v
in
G
. Among others, we characterize the trees of fixed number of vertices and matching number with the smallest RCW number, and the trees that are not caterpillars on
n
≥
7
vertices with the smallest, the second-smallest and the third-smallest RCW numbers.
Let
G
be a connected graph and
η
(
G
)
=
S
z
(
G
)
−
W
(
G
)
, where
W
(
G
)
and
S
z
(
G
)
stand for the Wiener and Szeged numbers of
G
, respectively. A well-known result of Klavžar, Rajapakse and ...Gutman states that
η
(
G
)
≥
0
and by a result of Dobrynin and Gutman
η
(
G
)
=
0
if and only if each block of
G
is complete. In an earlier paper a classification of graphs with
η
(
G
)
≤
3
is presented. In this paper, we continue our earlier work to classify connected graphs which satisfy
η
(
G
)
=
4
or
5
.
Some physicists depicted the molecular structure
Sn
Cl
2
· 2(
H
2
O
) by a piece of an Archimedean tiling (4.8.8) that is a partial cube. Inspired by this fact, we determine Archimedean tilings whose ...connected subgraphs are all partial cubes. Actually there are only four Archimedean tilings, (4.4.4.4), (6.6.6), (4.8.8) and (4.6.12), which have this property. Furthermore, we obtain analytical expressions for Wiener numbers of some connected subgraphs of (4.8.8) and (4.6.12) tilings. In addition, we also discuss their asymptotic behaviors.
An algebraic approach to Wiener number Ghorbani, Modjtaba; Hakimi-Nezhaad, Mardjan; Barfaraz, Fatemeh Abbasi
Journal of applied mathematics & computing,
10/2017, Volume:
55, Issue:
1-2
Journal Article
Peer reviewed
An algebraic approach for generalizing the Wiener number by automorphism group of the graph is proposed by Graovać and Pisanski for the first time. In this paper, we continue this work to introduce a ...modification of some topological indices. These new topological indices are computed for an infinite class of fullerenes.
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