# Asia Mathematika

An International Journal. ISSN: 2457-0834 (online)

Volume 3, Issue 2, August 2019, Pages: 34-52

## On Certain Distance Graphs and related Applications

V. Yegnanarayanan

School of Humanities and Sciences, SASTRA University, Thanjavur-613401, TN, India

### Abstract

Stimulated by the famous plane coloring problem Eggleton coined the term distance graph and studied widely the prime distance graphs. A prime distance graph (PDG) G(Z;D) is one whose vertex set V is the set of integers Z and the distance set D is a subset of the set of primes P . The edge set of G denoted E is the one whose elements (u; v) for any u; v ∊ V (G) are characterized by the property that d(u; v) ∊ D where d(u; v) = |u-v|. According to J.D.Laison, C. Starr and A. Walker a graph G is a PDG if there exists a 1-1 labeling f : V (G) → Z such that for any two adjacent vertices u and v the integer |f(u)-f(v)| is a prime. Further they called such a labelling of V(G) a prime distance labelling (PDL) of G. In this paper we prove certain existence and non-existence results concerning PDG and PDL and study the relationship between them. We also discuss certain applications besides raising some open problems.

### Keywords

Prime Distance Graph; Prime Distance Labeling; Chromatic number; Pythagorean number; Pythagorean triples and Pythagorean quadruples.

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