A note on the maximal inverse sum indeg index of trees | [Discrete Mathematics Letters • 2024]

Author(s):

1. Wei Gao: Department of Mathematics, Pennsylvania State University at Abington,Abington, PA,USA

Abstract:

The inverse sum indeg index (ISI index) of a graph G is defined as ISI(G) = Pvivj2E(G)(d(vi)d(vj))(d(vi) + d(vj)) 1, where d(vi) is the degree of a vertex vi. It is known that the star Sn uniquely minimizes the ISI index among trees of order n. However, characterizing trees of order n with the maximal ISI index (optimal trees, for convenience) appears to be difficult. Chen, Li, and Lin in [Appl. Math. Comput. 392 (2021) #125731] gave some structural properties and three conjectures regarding an optimal tree. In this paper, the trees within a set T Sn of trees of order n are investigated, where T Sn is defined in the main text and it is the set to which the optimal tree is conjectured to belong. Several structural properties associated with an optimal tree are presented. The findings of the present paper imply that if the second part of Conjecture 4.3 of the mentioned paper holds, then its remaining two conjectures are also valid.

Page(s): 36-43

DOI: 10.47443/dml.2024.025

Published: Journal: Discrete Mathematics Letters, Volume: 14, Issue: 0, Year: 2024

Keywords:

Graph , inverse sum indeg index , optimal tree

References:

[1] Ali A.,M. M. Matejic´ E. I. .2020 .Milovanovic´, I. Zˇ . Milovanovic´, Some new upper bounds for the inverse sum indeg index of graphs, Elect. J. Graph Theory Appl, 8 : 59-70.

[2] An M.,L. Xiong, M. .2018 .Some results on the inverse sum indeg index of a graph, Inform. , 134 : 42-46.

[3] Chen H.,Deng H. .2018 .The inverse sum indeg index of graphs with some given parameters. Discrete Math. Algorithms Appl, 10 : 1850006.

[4] Chen X.,Li X.,Lin W. .2021 .On connected graphs and trees with maximal inverse sum indeg index. Appl. Math. Comput, 392 : 125731.

[5] Falahati-Nezhad F.,Azari M.,T. M. .2017 .Dosˇlic´, Sharp bounds on the inverse sum indeg index. Discrete Appl. Math, 217 : 185-195.

[6] Gutman I.,M. M. Matejic´ E. I. .2020 .Lower bounds for inverse sum indeg index of graphs, Kragujevac J. , 44 : 551-562.

[7] Gutman I.,Rodr J. M.,Sigarreta J. M. .2019 .Linear and non-linear inequalities on the inverse sum indeg index. Discrete Appl. Math, 258 : 123-134.

[8] Jiang Y.,Chen X.,Lin W. .2021 .A note on chemical trees with maximal inverse sum indeg index. MATCH Commun. Math. Comput. Chem, 86 : 29-38.

[9] Lin W.,Fu P.,Wang Y. .2022 .On two conjectures concerning trees with maximal inverse sumindeg index. Comput. Appl, 41 : 252.

[10] M. M. Matejic´ E. I. .2017 .On relations between inverse sum indeg index and multiplicative sum Zagreb index. Appl. Math. Inform. Mech, 9 : 193-199.

[11] M. M. ,E. I. .2018 .Milovanovic´, Upper bounds for the inverse sum indeg index of graphs, Discrete Appl. , 251 : 258-267.

[12] Pattabiraman K. .2018 .Inverse sum indeg index of graphs. AKCE Int. J. Graphs Comb, 15 : 155-167.

[13] Sedlar J.,Vasilyev A. .2015 .On the inverse sum indeg index. Discrete Appl. Math, 184 : 202-212.

[14] D. ,Adriatic indices .2010 .Vukicˇevic´, M. Gasˇperov, Bond additive modeling 1. Croat. Chem. Acta, 83 : 243-260.

Citations

Downloads

Views