Further Study of Topological Integer Additive Set-Labeling of Graphs
U. M. Prajapati
Department of Mathematics, St. Xavier’s College, Ahmedabad, India.
K. R. Odedara
*
Department of Mathematics, R. R. Lalan College, Bhuj, India and Department of Mathematics, Gujarat University, Ahmedabad, India.
*Author to whom correspondence should be addressed.
Abstract
Topological integer additive set-labeling (TIASL), introduced by Sudev and Germina, integrates integer additive set-labeling with topological constraints on the ground set. While several foundational results on TIASLs have been established, deeper structural and operational aspects remain relatively less explored. In this paper, we establish a universality theorem by proving that every finite connected graph can be embedded into a connected TIASL graph via the addition of a single suitably labeled vertex. We then analyze the behavior of TIASL under classical graph operations including disjoint union, join, Cartesian product, corona, subdivision, and vertex identification, determining precise closure and non-closure conditions. Finally, we derive an explicit formula for the topological set-indexing number in terms of the order of the graph and its pendant vertex structure. These results extend the TIASL theory beyond existence and characterization, providing new structural and quantitative insights.
Keywords: Graphs, set-labeling, integer additive set-labeling, topological integer additive set-labeling, topological integer additive set-indexing number