Narrow Results

Let $G=(V,E)$ be a non-empty, finite graph. If the vertices of G can be bijectively labeled by a set S of positive distinct real numbers with two vertices being adjacent if and only if the positive difference of the corresponding labels is in S, then G is called a proper monograph. The set S is called the signature of G and denoted as G(S). Not all proper monographs have the property that a set of idle vertices can be bijectively mapped to the maximum independent sets. As a result, in this paper, we present the proper monograph labelings of several classes of graphs that satisfy the property mentioned above. We present the proper monograph labelings of graphs such as cycles, $C_n\odot K_1$, cycles with paths attached to one or more vertices, and Cycles with an irreducible tree attached to one or more vertices.