On Antimode Graphs

The term mode graph was introduced by Boland, Kaufman, and Panrong to define a connected graph \( G \) such that, for every pair of vertices \( v, w \) in \( G \), the number of vertices with eccentricity \( e(v) \) is equal to the number of vertices with eccentricity \( e(w) \). As a natural extension to this work, the concept of an antimode graph was introduced to describe a graph for which, if \( e(v) \neq e(w) \), then the number of vertices with eccentricity \( e(v) \) is not equal to the number of vertices with eccentricity \( e(w) \). In this paper, we determine the existence of some classes of antimode graphs, namely equisequential and \((a, d)\)-antimode graphs.