On Modular Signatures of Some Autograph
G. V. Magalona, R. G. Panopio, and S. Arumugam (pp. 126-130)
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 uv∈E if and only if |α(u) − α(v)| = α(w) for some w∈V, then α is called an autograph labeling of G. The multiset S(G) = {α(v) : v∈V} 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 Kr ≠ K1 and complete bipartite graphs Kr,s ≠ K2,2 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.