Research Article
On the Eccentric Connectivity Polynomial of F-Sum of
Connected Graphs
Muhammad Imran ,
1
Shehnaz Akhter,
2
and Zahid Iqbal
2,3
1
Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O. Box 15551, Al Ain, UAE
2
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Sector H-12,
Islamabad, Pakistan
3
Department of Mathematics and Statistics, Institute of Southern Punjab, Multan, Pakistan
Correspondence should be addressed to Muhammad Imran; imrandhab@gmail.com
Received 17 February 2020; Revised 21 April 2020; Accepted 27 April 2020; Published 31 May 2020
Academic Editor: Honglei Xu
Copyright © 2020 Muhammad Imran et al. is is an open access article distributed under the Creative Commons Attribution
License,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
e eccentric connectivity polynomial (ECP) of a connected graph G �(V(G),E(G)) is described as
ξ
c
(G, y)�
a∈V(G)
deg
G
(a)y
ec
G
(a)
, where ec
G
(a) and deg
G
(a) represent the eccentricity and the degree of the vertex a, re-
spectively. e eccentric connectivity index (ECI) can also be acquired from ξ
c
(G, y) by taking its first derivatives at y � 1.e
ECIhasbeenwidelyusedforanalyzingboththeboilingpointandmeltingpointforchemicalcompoundsandmedicinaldrugsin
QSPR/QSAR studies. As the extension of ECI, the ECP also performs a pivotal role in pharmaceutical science and chemical
engineering. Graph products conveniently play an important role in many combinatorial applications, graph decompositions,
puremathematics,andappliedmathematics.Inthisarticle,weworkouttheECPof F-sumofgraphs.Moreover,wederivethe
explicit expressions of ECP for well-known graph products such as generalized hierarchical, cluster, and corona products of
graphs. We also apply these outcomes to deduce the ECP of some classes of chemical graphs.
1. Introduction
Let G be an n-vertex simple and connected graph with the
vertexset V(G) andtheedgeset E(G).Foragivengraph G,
the order and size are symbolized by |V(G)| and |E(G)|,
respectively. e degree of a ∈ V(G) is the number of ad-
jacentverticesto a in G,anditisrepresentedbydeg
G
(a).For
a
1
,a
2
∈ V(G),thedistancebetween a
1
and a
2
,denotedwith
d
G
(a
1
,a
2
), is defined as the length of the shortest path
among a
1
and a
2
in G, and the eccentricity ec
G
(a
1
) is the
largest distance among a
1
and any other vertex a
2
of G.We
use notions P
n
and C
n
for the n-vertex path and cycle,
respectively. e line graph denoted by L(G) of G is the
graphwhoseverticesaretheedgesoftheoriginalgraph;two
vertices e
1
and e
2
are connected if and only if they share a
common end vertex in G.ejoint G + G
′
of graphs G and
G
′
is the graph union G∪G
′
including all the edges joining
V(G) and V(G
′
).
Amoleculardescriptorisanumericmeasureofagraph
which characterizes its topology. In organic chemistry,
topological invariants have established many applications
in pharmaceutical drug design, QSAR/QSPR studies,
chemical documentation, and isomer discrimination.
Some effective topological classes such as degree based,
degree distance, eccentric connectivity indices, and so on
areestablishedasmolecularinvariants.Inrecentyears,the
study of eccentric invariants for chemical molecular
structure has become one of the flourishing lines of re-
search in theoretical chemistry.
e ECI of G is a newly discovered distance-based to-
pological invariant which was put forward by Sharma et al.
[1] and is defined as follows:
Hindawi
Complexity
Volume 2020, Article ID 5061682, 9 pages
https://doi.org/10.1155/2020/5061682