Distance-{0,1} Vertex Irregular Total Labeling of Triangular Book Graphs

Abstract

Irregularity strength labelings quantify, by a single integer, how far a graph is from admitting vertices with pairwise distinct weights, and total labelings sharpen this measure by letting both vertices and edges carry labels. Distance-{ vertex irregular total labeling is a variant of total labeling theory in which the weight of a vertex is formed from its own label, the labels of its neighboring vertices, and the labels of its incident edges, and every vertex is required to have a distinct weight. This study investigates the distance- vertex irregular total labeling of triangular book graphs and determines the total distance- vertex irregularity strength, i.e. the minimum value of the maximum label k for which such a labeling exists. A constructive analytical approach is employed by formulating vertex-weight expressions, deriving a lower bound, and constructing a labeling function that attains this bound. The results establish the explicit formula for n = 1 and  for , where denotes the triangular book graph with n pages. The graph with one page  is identified as a special case requiring a larger maximum label than the general lower bound, because with only one page vertex the three edge/vertex labels rather than by the pigeonhole argument that yields the general formula for . These findings extend the study of distance vertex irregular total labeling for book graphs and provide a foundation for future research on other graph classes and distance sets.