ON EDGE IRREGULAR LABELING WITH MINIMUM DISTINCT INTEGER LABELS FOR SOME STANDARD GRAPHS

Abstract

In this paper, we introduce a new variant of edge irregular labeling on graphs. For a simple graph G, let f be an edge labeling, where S is a finite set of nonnegative integers, and define the weight of each vertex as the sum of the labels of all incident edges. In the case where all vertex weights are pairwise distinct, then the labeling is called edge irregular. We define the minimum distinct edge-label number of G, denoted by md G, as the minimum cardinality of a label set S for which G admits such a labeling. This parameter may be viewed as a distinct irregularity type invariant, since it minimizes the number of distinct labels used rather than the largest label value. We establish exact values of md G for three standard graph families: paths, cycles, and stars. The obtained results demonstrate that this parameter exhibits different behavior across these graph classes and is strongly influenced by local degree structure and edge incidence patterns.