Communications in Mathematics and Applications Vol. 8, No. 3, pp. 229–240, 2017 ISSN 0975-8607 (online); 0976-5905 (print) Published by RGN Publications http://www.rgnpublications.com Research Article Some Remarks on the 2-Distance-Balanced Graphs Hassan Kharazi 1, * and Mehdi Alaeiyan 2 1 Department of Mathematics, Faculty of Science, Imam Hossein University, Tehran, Iran 2 School of Mathematics, Iran University of Science and Technology, Tehran, Iran *Corresponding author: hkharazi@ihu.ac.ir Abstract. The aim of this paper is to investigate the notion of 2-distance-balanced graphs as a generalized form of distance-balanced graphs. Furthermore, we introduce a subclass of such graphs so-called strongly 2-distance-balanced graphs and present some related results based on Cartesian and lexicographic products of two graphs. Keywords. 2-distance-balanced graph; Strongly 2-distance-balanced graph; Cartesian product; Lexicographic product MSC. 05C12; 05C40 Received: May 22, 2016 Accepted: June 10, 2017 Copyright © 2017 Hassan Kharazi and Mehdi Alaeiyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In 1999, Handa [4] considered initially distance-balanced partial cubes. Also, in 2005, distance- balanced graphs in the framework of various kinds of graph products has been studied by Jerebic et al. [7]. In 2006, Kutnar et al. [9] introduced a subclass of such graphs so-called strongly distance-balanced graphs by the concept of distance partition. Very recently, Kutnar et al. [11] also defined another class of such graphs as nicely distance-balanced graphs and classified them for diferent diameters. In this paper, by generalizing the concept of distance-balanced graphs we introduce a new class 2-distance-balanced graphs and study their properties. We also define strongly 2-distance-balanced graphs and discuss about their Cartesian and lexicographic products.