In this paper we prove that any distance-balanced graph G with D (G) ³|V (G)| -3 is regular. Also we define notion of distance- balanced closure of a graph and we find distance-balanced closures of trees T with D (T) ³ |V (T)| -3.

In this paper we investigate the geometric structures of M(n; 2) containing n points in R3 having two distinct DISTANCES. We will show that up to pseudo-equivalence there are 5 constructible models for M(4; 2) and 17 constructible models for M(5; 2).

Recently, E. Macajova and M. Skoviera proved that every bidirected Eulerian graph which admits a nowhere zero flow, admits a nowhere zero 4-flow. This result shows the validity of Bouchet's nowhere zero conjecture for Eulerian bidirected GRAPHS. In this paper we prove the same theorem in a different terminology and with a short and simple proof. More precisely, we prove that every Eulerian undirected graph which admits a zero-sum flow, admits a zero-sum 4-flow. As a conclusion we obtain a shorter proof for the previously mentioned result of Macajova and Skoviera.

The Quran is a wise book from The Wise (Ḥ akī m). Components of a wise writing are in connection with each other. Hence Distance (fā ṣ ilah: pl. fawā ṣ il), as the most important part of a verse, must be connected in a close meaning with its previous verse. Reaching this communication sometimes is easy and sometimes difficult which requires deep thought and contemplation; In some cases, it makes a challenge in the Quran exegesis. The present paper aims to find the aspects and causes of this challenge and then looks for solutions of the current crisis. Three aspects are “ mismatch between the beginning and the end of a verse” , “ redundancy and sequence” , and “ the duality and contradiction at the end of a verse” . To overcome the challenge, in addition to broadening the scope of thinking and contemplation, the content (sī ā ḳ ) of verses, semantic details and intricacies of a verse, and the structure of words should be regarded as well. This topic shows solidarity and harmony between the parts of the Quranic verses, which had not been studied before.

We study the blocks of interpoint DISTANCES, their distributions, correlations, independence and the homogeneity of their total variances. We discuss the exact and asymptotic distribution of the interpoint DISTANCES and their average under three models and provide connections between the correlation of interpoint DISTANCES with their vector correlation and test of sphericity. We discuss testing independence of the blocks based on the correlation of block interpoint DISTANCES. A homogeneity test of the total variances in each block and a simultaneous plot to visualize their relative ordering are presented.

Human always in interaction with their social environment, have consider some degree of privacy with different purposes, for themselves, the people around them and carry out their activities. Creating privacy depends on two elements; subjective meanings that ruling the creation of privacy, and the second sentence are person available facilities. Privacy is not seen, heard, smelled and availability of things that find objectivity with respect to spatial and temporal DISTANCES in the living environment. The main issue of this research is to inquire about the nature of privacy in residential areas, and its purpose is to understand the reasons for the formation of privacy in residential areas. The significance and necessity of this research can be expressed in a definition of the dwelling in such a way that the dwelling is the center of everyday human activities, which these activities are separated from each other by privacies. Therefore, it is necessary to examine the nature of these privacies, which defines the external credibility of the research in order to redefine this feature in the design of residential environments. The present research has been based on the observation and description of manmade environment, with reasons argues to the characteristics of privacy types, its origins and how it deals with the physical environment, which is adorned with objective evidence of Qashqai tribe living environment. The findings of this study show that the subjective meanings ruling on the creation of privacy in case study residential environments are affected by things such as "social situation" and "family social status", and facilities that are used in the environment to ensure privacy, are spatial and temporal characteristics in man-made environments. Spatial characteristics with aspects such as "spatial distance" and "forward / backward" afford privacy in the physical environment, and temporal characteristics get the aspects such as "temporal distance" and " temporal sequence of settings", that have rooted in respect smelling, hearing and visual DISTANCES in different cultures. The results of this research help to understand how human use of the environment affordances for interaction with their social environment.

We determine the forbidden induced subGRAPHS for the intersection of the classes of chordal bipartite GRAPHS and line GRAPHS of acyclic directed GRAPHS. This is a first step towards finding the forbidden induced subGRAPHS for the class of line GRAPHS of directed GRAPHS.

The edge-degree distance of a simple connected graphG is defined as the sum of the terms ( d (e|G)+d (f|G) ) d (e, f|G) over all unordered pairs fe, fg of edges of G, where d (e|G) and d (e, f|G) denote the degree of the edgee in G and the distance between the edges e and f in G, respectively. In this paper, we study the behavior of two versions of the edge-degree distance under two graph products called splice and link.