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Submitted
July 22, 2017
Accepted
July 16, 2018
Published
March 1, 2019
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Abstract

Throughout this paper, we present a new strong property of graph so-
called nicely n-distance-balanced which is notably stronger than the concept of n-
distance-balanced recently given by the authors. We also initially introduce a new
graph invariant which modies Szeged index and is suitable to study n-distance-
balanced graphs. Looking for the graphs extremal with respect to the modied
Szeged index it turns out the n-distance-balanced graphs with odd integer n are
the only bipartite graphs which can maximize the modied Szeged index and this
also disproves a conjecture proposed by Khalifeh et al. [Khalifeh M.H.,Youse-
Azari H., Ashra A.R., Wagner S.G.: Some new results on distance-based graph
invariants, European J. Combin. 30 (2009) 1149-1163]. Furthermore, we gather
some facts concerning with the nicely n-distance-balanced graphs generated by some
well-known graph products. To enlighten the reader some examples are provided.
Moreover, a conjecture and a problem are presented within the results of this article.

Keywords

Nicely n-distance-balanced Szeged index lexicographic Cartesian and strong products.

Article Details

Author Biographies

Morteza Faghani, Payame Noor University

Department of Mathematics

Ehsan Pourhadi, School of Mathematics, Iran University of Science and Technology

Department of Mathematics
How to Cite
Faghani, M., & Pourhadi, E. (2019). Further Remarks on n-Distance-Balanced Graphs. Journal of the Indonesian Mathematical Society, 25(1), 44–61. https://doi.org/10.22342/jims.25.1.563.44-61
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