### Journals Information

**
Mathematics and Statistics Vol. 12(3), pp. 229 - 233 DOI: 10.13189/ms.2024.120302 Reprint (PDF) (331Kb) **

## On Questions Concerning Finite Prime Distance Graphs

**Ram Dayal ^{1}, A. Parthiban ^{2}^{,*}, P. Selvaraju ^{3}**

^{1}Department of Mathematics, Lovely Professional University, Phagwara 144 411, Punjab, India

^{2}Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India

^{3}Department of Mathematics, Rajalakshmi Institute of Technology, Chennai, India

**ABSTRACT**

Graph labeling is an allocation of labels (mostly integers) to the nodes/lines or both of a graph G_{α} subject to a few conditions. The field of graph theory, specifically graph labeling, plays a vital role in various fields. To name a few, graph labeling is utilized in coding, x−ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base management. It can also be applied to network security, network addressing, channel assignment process, and social networks. A graph G_{β} is a prime distance graph (PDG) if its nodes can be assigned with distinct integers such that for any two adjacent nodes, the positive difference of their labels is a prime number. A complete characterization of prime distance graphs is an open problem of high interest. This paper contributes partially towards the same. More specifically, Laison et al. raised the following questions. (1) Is there a family of graphs which are PDGs if and only if Goldbach’s Conjecture is true? (2) What other families of graphs are PDGs? In this paper, these questions are answered partially and also show certain families of graphs that admit prime distance labeling (PDL) if and only if the Twin Prime Conjecture holds, besides establishing PDL of some special graphs.

**KEYWORDS**

Prime Distance Labeling, Prime Distance Graph, Goldbach’s Conjecture, Twin Prime Conjecture

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Ram Dayal , A. Parthiban , P. Selvaraju , "On Questions Concerning Finite Prime Distance Graphs," Mathematics and Statistics, Vol. 12, No. 3, pp. 229 - 233, 2024. DOI: 10.13189/ms.2024.120302.

(b). APA Format:

Ram Dayal , A. Parthiban , P. Selvaraju (2024). On Questions Concerning Finite Prime Distance Graphs. Mathematics and Statistics, 12(3), 229 - 233. DOI: 10.13189/ms.2024.120302.