A paradigmatic approach to investigate restricted hyper totient graphs

Abstract

Nowadays, the problem of finding families of graphs for which one may ensure the existence of a vertex-labeling and/or an edge-labeling based on a certain class of integers, constitutes a challenge for researchers in both number and graph theory. In this paper, we focus on those vertexlabelings whose induced multiplicative edge-labeling assigns hyper totient numbers to the edges of the graph. In this way, we introduce and characterize the notions of hyper totient graph and restricted hyper totient graph. In particular, we prove that every finite graph is a hyper totient graph and we determine under which assumptions the following families of graphs constitute restricted hyper totient graphs: complete graphs, star graphs, complete bipartite graphs, wheel graphs, cycles, paths, fan graphs and friendship graphs.