On Antimode Graphs

Abstract

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 $e(v)\ne e(w)$, 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.