TY - JOUR
T1 - Remarks on distance based topological indices for ℓ-apex trees
AU - Knor, Martin
AU - Imran, Muhammad
AU - Jamil, Muhammad Kamran
AU - Škrekovski, Riste
N1 - Funding Information:
Funding: The research was partially supported by Slovenian research agency ARRS, program no. P1-0383, Slovak research grants APVV-15-0220, APVV-17-0428, VEGA 1/0142/17 and VEGA 1/0238/19. The research was also supported by the UPAR Grants of United Arab Emirates University, Al Ain, UAE via Grant No. G00002590 and G00003271.
Publisher Copyright:
© 2020 by the authors.
PY - 2020/5/1
Y1 - 2020/5/1
N2 - A graph G is called an ℓ-apex tree if there exist a vertex subset A ⊂ V(G) with cardinality l such that G-A is a tree and there is no other subset of smaller cardinality with this property. In the paper, we investigate extremal values of several monotonic distance-based topological indices for this class of graphs, namely generalizedWiener index, and consequently for the Wiener index and the Harary index, and also for some newer indices as connective eccentricity index, generalized degree distance, and others. For the one extreme value we obtain that the extremal graph is a join of a tree and a clique. Regarding the other extreme value, which turns out to be a harder problem, we obtain results for ℓ = 1 and pose some open questions for higher ℓ. Symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including topological indices of graphs.
AB - A graph G is called an ℓ-apex tree if there exist a vertex subset A ⊂ V(G) with cardinality l such that G-A is a tree and there is no other subset of smaller cardinality with this property. In the paper, we investigate extremal values of several monotonic distance-based topological indices for this class of graphs, namely generalizedWiener index, and consequently for the Wiener index and the Harary index, and also for some newer indices as connective eccentricity index, generalized degree distance, and others. For the one extreme value we obtain that the extremal graph is a join of a tree and a clique. Regarding the other extreme value, which turns out to be a harder problem, we obtain results for ℓ = 1 and pose some open questions for higher ℓ. Symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including topological indices of graphs.
KW - Apex trees
KW - Distances in apex trees
KW - Extremal graphs
KW - Generalized wiener index
UR - http://www.scopus.com/inward/record.url?scp=85085638368&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85085638368&partnerID=8YFLogxK
U2 - 10.3390/SYM12050802
DO - 10.3390/SYM12050802
M3 - Article
AN - SCOPUS:85085638368
SN - 2073-8994
VL - 12
JO - Symmetry
JF - Symmetry
IS - 5
M1 - 802
ER -