Manila Journal of Science 10 (2017), pp. 126-130

On Modular Signatures of Some Autographs

Abstract

Let G = (V, E) be a graph where the edge set E can be a multiset. If there exists a bijection α: V → S(G) where S(G) is a multiset of real numbers such that uvE if and only if |α(u) − α(v)| = α(w) for some wV, then α is called an autograph labeling of G. The multiset S(G) = {α(v) : vV} is called a signature of G. If the underlying set of S(G) is {0,1,2,…,n − 1} where n = |V|, then S(G) is called a modular signature of G. In this paper, we prove that complete graphs KrK1 and complete bipartite graphs Kr,sK2,2 have several modular signatures while K1 and K2,2 have unique modular signatures. We characterize paths, cycles, wheels, and fans that admit a modular signature. We also obtain several classes of graphs that do not have a modular signature.

 

Manila Journal Of Science 10 (2017), Pp 126-130 (492.2 KiB)