ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P2.03
https://doi.org/10.26493/2590-9770.1241.7f8
(Also available at http://adam-journal.eu)
Some remarks on Balaban and
sum-Balaban index*
Martin Knor
Slovak University of Technology in Bratislava, Faculty of Civil Engineering,
Department of Mathematics, Radlinského 11, 813 68, Bratislava, Slovakia
Jozef Komorník
Faculty of Management, Comenius University, Odbojárov 10, 831 04 Bratislava
Riste Škrekovski
FMF, University of Ljubljana, Jadranska ulica 19, 1000 Ljubljana, Slovenia, and
Faculty of Information Studies, Ljubljanska cesta 31a, 8000 Novo mesto, Slovenia, and
FAMNIT, University of Primorska, Glagoljaška, 6000 Koper, Slovenia
Aleksandra Tepeh†
Faculty of Information Studies, Ljubljanska cesta 31a, 8000 Novo mesto, Slovenia, and
Faculty of Electrical Engineering and Computer Science, University of Maribor,
Koroška cesta 46, 2000 Maribor, Slovenia
Received 3 February 2018, accepted 14 March 2019, published online 4 May 2020
Abstract
In the paper we study maximal values of Balaban and sum-Balaban index, and correct
some results appearing in the literature which are only partially correct. Henceforth, we
were able to solve a conjecture of M. Aouchiche, G. Caporossi and P. Hansen regarding
the comparison of Balaban and Randić index. In addition, we showed that for every k and
large enough n, the first k graphs of order n with the largest value of Balaban index are
trees. We conclude the paper with a result about the accumulation points of sum-Balaban
index.
Keywords: Topological index, Balaban index, sum-Balaban index, Randić index.
Math. Subj. Class.: 05C12, 05C90
*The authors acknowledge partial support by Slovak research grants VEGA 1/0142/17, VEGA 1/0238/19,
APVV-15-0220 and APVV-17-0428, Slovenian research agency ARRS, program no. P1–0383, and National
Scholarship Programme of the Slovak Republic SAIA.
†Corresponding author.
E-mail addresses: knor@math.sk (Martin Knor), jozef.komornik@fm.uniba.sk (Jozef Komorník),
skrekovski@gmail.com (Riste Škrekovski), aleksandra.tepeh@gmail.com (Aleksandra Tepeh)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P2.03
https://doi.org/10.26493/2590-9770.1241.7f8
(Dostopno tudi na http://adam-journal.eu)
Nekaj pripomb k Balabanovemu in
vsotno-Balabanovemu indeksu*
Martin Knor
Slovak University of Technology in Bratislava, Faculty of Civil Engineering,
Department of Mathematics, Radlinského 11, 813 68, Bratislava, Slovakia
Jozef Komorník
Faculty of Management, Comenius University, Odbojárov 10, 831 04 Bratislava
Riste Škrekovski
FMF, Univerza v Ljubljani, Jadranska ulica 19, 1000 Ljubljana, Slovenija, in
Fakulteta za informacijske študije, Ljubljanska cesta 31a, 8000 Novo mesto, Slovenija, in
FAMNIT, Univerza na Primorskem, Glagoljaška, 6000 Koper, Slovenija
Aleksandra Tepeh†
Fakulteta za informacijske študije, Ljubljanska cesta 31a, 8000 Novo mesto, Slovenija, in
Fakulteta za elektrotehniko in računalništvo, Univerza v Mariboru,
Koroška cesta 46, 2000 Maribor, Slovenija
Prejeto 3. februarja 2018, sprejeto 14. marca 2019, objavljeno na spletu 4. maja 2020
Povzetek
V članku raziskujemo maksimalne vrednosti Balabanovega in vsotno-Balabanovega in-
deksa in popravimo nekatere rezultate, ki se pojavljajo v literaturi in so samo delno pravilni.
Tako smo lahko dokazali domnevo M. Aouchicha, G. Caporossija in P. Hansena v zvezi z
Balabanovim in Randićevim indeksom. Poleg tega pokažemo, da za vsak k in dovolj velik
n velja: prvih k grafov reda n z velikimi vrednostmi Balabanovega indeksa so drevesa.
Članek sklenemo z rezultatom o stekališčih vsotno-Balabanovega indeksa.
Ključne besede: Topološki indeks, Balabanov indeks, vsotni-Balabanov indeks, Randićev indeks.
Math. Subj. Class.: 05C12, 05C90
*Avtorji priznavajo delno podporo s strani slovaških raziskovalnih dotacij VEGA 1/0142/17, VEGA
1/0238/19, APVV-15-0220 in APVV-17-0428, Javne agencije za raziskovalno dejavnost Republike Slovenije
ARRS, program št. P1–0383, in nacionalnega štipendijskega programa Slovaške republike SAIA.
†Kontaktni avtor.
E-poštni naslovi: knor@math.sk (Martin Knor), jozef.komornik@fm.uniba.sk (Jozef Komorník),
skrekovski@gmail.com (Riste Škrekovski), aleksandra.tepeh@gmail.com (Aleksandra Tepeh)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/